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+Toward Building General Foundation Models for Language, Vision, and
+Vision-Language Understanding Tasks
+Xinsong Zhang 1 Yan Zeng 1 Jipeng Zhang 2 Hang Li 1
+Abstract
+Foundation models or pre-trained models have
+substantially improved the performance of various
+language, vision, and vision-language understand-
+ing tasks. However, existing foundation models
+can only perform the best in one type of tasks,
+namely language, vision, or vision-language. It is
+still an open question whether it is possible to con-
+struct a foundation model performing the best for
+all the understanding tasks, which we call a gen-
+eral foundation model. In this paper, we propose
+a new general foundation model, X-FM (the X-
+Foundation Model). X-FM has one language en-
+coder, one vision encoder, and one fusion encoder,
+as well as a new training method. The training
+method includes two new techniques for learning
+X-FM from text, image, and image-text pair data.
+One is to stop gradients from the vision-language
+training when learning the language encoder. The
+other is to leverage the vision-language training
+to guide the learning of the vision encoder. Exten-
+sive experiments on benchmark datasets show that
+X-FM can significantly outperform existing gen-
+eral foundation models and perform better than or
+comparable to existing foundation models specif-
+ically for language, vision, or vision-language
+understanding.
+1. Introduction
+With the enormous power of foundation models, also known
+as pre-trained models, remarkable performance gains have
+recently been achieved in a variety of understanding tasks in
+natural language processing (NLP), computer vision (CV),
+and other fields (Devlin et al., 2019; Liu et al., 2019; Lewis
+et al., 2020; Raffel et al., 2020; Brown et al., 2020; Doso-
+vitskiy et al., 2021; He et al., 2022; Bao et al., 2021; Lu
+1ByteDance AI Lab 2The Hong Kong University of Science
+and Technology. Correspondence to: Xinsong Zhang <zhangxin-
+song.0320@bytedance.com>.
+Copyright 2023 by the author(s). The code and pre-trained models
+will be released upon publication.
+et al., 2019; Tan & Bansal, 2019a; Chen et al., 2020; Li
+et al., 2020; 2021a; Zeng et al., 2021; 2022) . Foundation
+models are usually equipped with Transformer (Vaswani
+et al., 2017) as the backbone, pre-trained with a tremendous
+amount of unlabeled data, and then fine-tuned with small
+amounts of labeled data in downstream tasks. The strong
+representation ability of the model, the massive amount of
+data, and the effective means of training make the founda-
+tion models powerful for successfully solving the tasks of
+vision, language, and vision-language (Li et al., 2021b;c;
+Singh et al., 2021; Wang et al., 2021b; 2022b; Diao et al.,
+2022; Wang et al., 2022a).
+The state-of-the-art foundation models usually work the
+best for one type of tasks, namely language, vision, and
+vision-language. For example, RoBERTa (Liu et al., 2019),
+BEiTv2 (Peng et al., 2022), and X-VLM (Zeng et al., 2021;
+2022) are language, vision, and vision-language founda-
+tion models respectively, and can achieve state-of-the-art
+performances for the specific type of tasks. It is still very
+challenging, however, to build a general foundation model
+that can perform the best in all types of tasks. Existing
+models, such as FLAVA (Singh et al., 2021), OFA (Wang
+et al., 2022b), DaVinci (Diao et al., 2022) and Uni-Perceiver-
+MoE (Zhu et al., 2022), are trying to achieve the goal. Their
+performances are still not satisfactory, however, when com-
+pared with the best performing foundation models for the
+individual types of tasks, as shown in Table 1. Previous
+work (Bingel & Søgaard, 2017; Wang et al., 2020) also
+shows that it is difficult to train a general foundation model
+in a multi-task learning setting that can effectively learn and
+utilize representations for all types of tasks. The reason is
+that language, vision, and vision-language are very different
+in nature, and a simple way of jointly training a model from
+language, vision, and vision-language data can easily create
+a suboptimal solution.
+To address the challenge, we propose a new general founda-
+tion model, X-FM (X-Foundation Model). X-FM consists of
+three modular encoders for language (text) encoding, vision
+(image) encoding, and fusion encoding, as shown in Fig 1.
+The language encoder, the vision encoder, and the entire
+model can be used in downstream tasks of language, vision,
+and vision-language understanding, respectively. All three
+arXiv:2301.05065v1  [cs.CV]  12 Jan 2023
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+Methods
+Text Tasks
+Vision Tasks
+Multi-modal Tasks (MSCOCO Retriveal & VQA)
+GLUE
+ImageNet
+Zero-Shot
+Fine-Tune
+MNLI
+RTE
+FT/LE
+TR
+IR
+TR
+IR
+VQA
+Foundation models specifically for language, vision, or vision-language understanding
+RoBERTa (Liu et al., 2019)
+87.6
+78.7
+–
+–
+–
+–
+–
+–
+BEiTv2 (Peng et al., 2022)
+–
+–
+85.5/80.1
+–
+–
+–
+–
+–
+X-VLM (Zeng et al., 2021)
+–
+–
+–
+70.8/92.1/96.5
+55.6/82.7/90.0
+80.4/95.5/98.2
+63.1/85.7/91.6
+78.1
+X2-VLM (Zeng et al., 2022)
+–
+–
+–
+–
+–
+80.5/95.5/97.8
+62.7/84.7/90.7
+79.2
+General foundation models
+UNIMO-2 (Li et al., 2021c)
+87.5
+–
+80.8/-
+–
+–
+–
+–
+76.3
+SimVLM (Wang et al., 2021c)
+83.4
+63.9
+-/80.6
+–
+–
+–
+–
+77.9
+FLAVA (Singh et al., 2021)
+80.3
+57.8
+-/75.5
+42.7/76.8/-
+38.4/67.5/-
+61.5/82.1/89.6
+50.1/74.4/83.2
+72.8
+OFA (Wang et al., 2022b)
+84.3
+70.8
+82.2/–
+–
+–
+–
+–
+78.0
+DaVinci (Diao et al., 2022)
+83.1
+64.2
+83.9/78.8
+–
+–
+–
+–
+76.3
+OmniVL (Wang et al., 2022a)
+–
+–
+–
+–
+–
+76.8/93.6/97.3
+58.5/82.6/89.5
+78.3
+Uni-Perceiver-MoE (Zhu et al., 2022)
+81.5
+75.8
+84.5/–
+64.6/–/–
+51.6/–/–
+70.5/–/–
+54.1/–/–
+–
+X-FMbase
+87.7
+83.2
+85.3/81.0
+73.8/93.9/97.2
+59.4/83.6/90.0
+81.8/96.0/98.3
+64.7/86.1/91.6
+79.1
+Table 1: Performance comparisons between foundation models. All results are from base-size models. MSCOCO is a
+cross-modal retrieval task, and IR and TR are image-retrieval and text-retrieval, respectively. MNLI results are average
+accuracies of MNLI-m and MNLI-mm. Accuracy is reported for RTE. For ImageNet1k classification, we report linear
+evaluation (LE) performance and fine-tuning (FT) performance, respectively. We report R@1/R@5/R@10 for all retrieval
+tasks at both zero-shot and fine-tune settings. We report the VQA test-dev result. bold denotes the best number across
+general foundation models. underline denotes the best across all models.
+encoders are stacked Transformer layers. The language en-
+coder and the vision encoder follow the implementations
+of BERT (Devlin et al., 2019) and ViT (Dosovitskiy et al.,
+2021), respectively. The fusion encoder has the same ar-
+chitecture as BERT except that there is a cross-attention
+sub-layer after the self-attention sub-layer in each Trans-
+former layer.
+In learning of X-FM, the language encoder, vision encoder,
+and fusion encoder are jointly trained with text data, im-
+age data, and image-text pair data as input. Given the text
+data, we train the language encoder by masked language
+modeling (MLM). Given the image data, we train the vi-
+sion encoder by masked image modeling (MIM). Given the
+image-text pair data, we train the fusion encoder by image
+text matching (ITM), image-conditioned masked language
+modeling (IMLM), bounding box prediction (BBP), train
+the vision encoder and the language encoder by image-text
+contrastive learning (ITC), and train the vision encoder by
+MIM. (See Fig 1.)
+The essential thinking of our learning method is that lan-
+guage is more abstract than vision, and there is an asymmet-
+ric relationship between language and vision. Therefore, we
+separate the learning of the three encoders. The language
+encoder is trained mainly from text data and is isolated from
+the training of the fusion encoder. The vision encoder is
+simultaneously trained from image data and image-text pair
+data, guided by the vision-language training. The fusion
+encoder is trained from image-text pair data.
+Our learning method includes two new techniques. One
+technique is to stop gradients from the vision-language train-
+ing when learning the language encoder. The gradient flow
+is stopped from the fusion encoder to the language encoder
+in training, while the activation flow from the language en-
+coder to the fusion encoder is as usual. As a result, the
+language encoder is not affected by training of the fusion
+encoder with image-text pair data. Moreover, the training of
+the fusion encoder concentrates on learning the alignments
+between language and vision features.
+The other technique is to leverage the vision-language train-
+ing to guide the learning of the vision encoder with masked
+image modeling (MIM). In MIM, the masked image is com-
+pared with the original image by the differences between the
+predicted representations and target representations at the
+masked and [CLS] positions. The vision encoder creates
+both the predicated and target representations, while there
+is gradient flow from the predicted representations but no
+gradient flow from the target representations. The vision
+encoder can create the target representations because it is
+also trained in the vision-language training.
+We conduct experiments on a variety of twenty-two tasks of
+language, vision, and vision-language understanding. X-FM
+can outperform other general foundation models by a large
+margin and can even achieve better or comparable perfor-
+mance than SOTA foundation models specifically designed
+for language, vision, or vision-language understanding tasks,
+as shown in Table 1.
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+2. Related Work
+Following the success of language model pre-training, vi-
+sion pre-training and vision-language pre-training with
+Transformer as the backbone (Vaswani et al., 2017) have
+also made significant progress recently, pushing the state-of-
+the-art of various understanding tasks of language, vision,
+and vision-language.
+In language understanding, BERT (Devlin et al., 2019) is
+the first model adopting masked language modeling (MLM)
+for pre-training, which achieves remarkable performance
+on a wide range of tasks. Several other models are then
+developed to improve training robustness (Liu et al., 2019),
+sample efficiency (Sun et al., 2019; Joshi et al., 2020; Clark
+et al., 2020), and prediction accuracy of BERT (Lan et al.,
+2020; Zhang et al., 2020; He et al., 2021).
+In vision understanding, ViT (Dosovitskiy et al., 2021; Tou-
+vron et al., 2021) is proposed, utilizing Transformer as the
+backbone. Inspired by MLM, subsequent work proposes
+using masked image modeling (MIM) with the objective of
+recovering masked images. The learning targets vary from
+pixels (He et al., 2022) to image tokens (Bao et al., 2021;
+Peng et al., 2022).
+In vision-language understanding, there are generally two
+approaches. One is “dual encoders,” in which image and text
+are encoded separately, followed by a shallow interaction
+layer. The other is “fusion encoder(s)” in which attention
+or self-attention is used to fuse information from the two
+modalities after encoding. The former approach includes
+CLIP (Radford et al., 2021) and ALIGN (Jia et al., 2021)
+and performs well in vision tasks and cross-modal retrieval
+tasks. However, it cannot perform so well in multi-modal
+fusion tasks such as visual question answering (VQA (Goyal
+et al., 2017)) and visual reasoning (NLVR2 (Suhr et al.,
+2019b)). The latter approach varies depending on the way
+of using image features. Early work feeds pre-extracted
+object features along with texts into Transformer models
+and trains the models to make multi-modal modeling and
+multi-modal alignments with suitable objectives (Lu et al.,
+2019; Tan & Bansal, 2019b; Li et al., 2020; Chen et al.,
+2020; Cho et al., 2021; Zhang et al., 2021). Later work
+uses patch embeddings directly with new architectures such
+as vision Transformer (Li et al., 2021a; 2022) or multiway
+Transformer (Wang et al., 2021a; Bao et al., 2022) and uses
+new objectives such as bounding box prediction (Zeng et al.,
+2021; 2022).
+Recently, the fact that Transformer can model multi-modal
+data within a single architecture has inspired research to
+develop general foundation models that can solve lan-
+guage, vision, and vision-language tasks at the same time.
+UNIMO (Li et al., 2021b;c) jointly learns from image
+and text data vision representations, language representa-
+tions, and vision-language alignments in a shared space.
+FLAVA (Singh et al., 2021), a general foundation model, per-
+forms pre-training with masked uni-modal and multi-modal
+modeling objectives. OFA (Wang et al., 2022c) formulates
+vision-language tasks as sequence-to-sequence (seq2seq)
+problems and pre-trains a seq2seq model in multi-task learn-
+ing. SimVLM (Wang et al., 2021c) pre-trains a seq2seq
+model with a single objective of language generation (prefix
+language modeling). DaVinci (Diao et al., 2022) combines
+prefix language modeling and prefix image modeling to
+learn a general foundation model for a wide range of tasks.
+Uni-Perceiver (Zhu et al., 2021; 2022) builds a unified per-
+ception architecture that processes various modalities and
+tasks with a single Transformer network and shared parame-
+ters.
+Previous studies on general foundation models have shown
+that different capabilities can be established with only one
+model. Still, few studies demonstrate that the best perfor-
+mance can be achieved in all tasks with one model. In this
+paper, we propose a new general foundation model and show
+that it can perform the best for all the understanding tasks
+of language, vision, and vision-language. We compare our
+model extensively with recent general foundation models
+on multiple dimensions, as shown in Appendix A.
+Several super-large foundation models (over 1B parame-
+ters) are proposed recently, most of which are trained on
+super-large in-house datasets (over 400M image-text pairs).
+The authors do not report results at the base (about 280M
+parameters) and large (about 800M parameters) scale on
+public datasets, which we consider in this paper. CoCa (Yu
+et al., 2022) pre-trains an image-text sequence-to-sequence
+model with contrastive loss and captioning loss. BEiT-
+3 (Wang et al., 2022d) uses a multi-way Transformer and a
+unified objective of masked “language” modeling for learn-
+ing from image (Imglish1), text, and image-text pair data.
+Florence (Yuan et al., 2021) first scales the web-scale image-
+text pairs to 900M representations and then adapts to vari-
+ous computer vision tasks. Flamingo (Alayrac et al., 2022)
+makes use of a large language model in vision-language
+pre-training to solve the “in-context learning” problem for
+vision-language tasks. PaLI (Chen et al., 2022) jointly scales
+up the vision encoder and language encoder to cover a vari-
+ety of language, vision, vision-language, and multilingual
+tasks.
+3. Method
+3.1. Model Architecture and Training Process
+We propose a new general foundation model X-FM, having
+a language encoder, a vision encoder, and a fusion encoder,
+shown as Fig 1. The language encoder is a stack of Trans-
+1They view the image as a foreign language.
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+Text Encoder
+Cross-Attention
+Fusion Encoder
+Feed Forward
+Self-Attention
+Vision Encoder
+Feed Forward
+Self-Attention
+two
+brown
+and
+white
+dogs.
+Mask
+Feed Forward
+Self-Attention
+stop grad
+MIM
+MLM
+IMLM, ITM, BBP
+ITC
+x M
+x N
+x L
+Mask
+target
+Figure 1: The architecture and pre-training process of X-FM, a Transformer-based general foundation model. Given
+a text, we learn the language encoder by MLM. Given an image, we learn the vision encoder by MIM. Given an image-text
+pair, we learn the fusion encoder by BBP, ITM, IMLM and ITC, and further learn the vision encoder by MIM. The gradients
+of BBP, ITM, and IMLM are stopped from the fusion encoder to the language encoder. The vision encoder is trained by
+MIM with both the image-text pair data and the image data. M, N and P denote numbers of encoder layers.
+former layers like that of BERT (Devlin et al., 2019), while
+the vision encoder is a stack of Transformer layers like that
+of ViT (Dosovitskiy et al., 2021). The language encoder uses
+a post-layer-norm, while the vision encoder uses a pre-layer-
+norm. The fusion encoder is similar to that of ALBEF (Li
+et al., 2021a) and X-VLM (Zeng et al., 2021), in which
+each layer has an attention sub-layer after a self-attention
+sub-layer. In the self-attention sub-layers, the queries are
+from language and the keys & values are from vision.
+We propose a new method for learning X-FM, also shown in
+Fig 1. Text, image, and image-text pair data are used as input
+to train X-FM. The language encoder is trained by masked
+language modeling (MLM) and image text contrastive learn-
+ing (ITC). The vision encoder is trained by masked image
+modeling (MIM) and ITC. The fusion encoder is trained
+by image text matching (ITM), image-conditioned masked
+language modeling (IMLM), and bounding box prediction
+(BBP). There are two new techniques developed for the
+training.
+Stop Gradient. We stop gradients from the vision-language
+training when learning the language encoder. Specifically,
+when the fusion encoder is trained with image-text pair
+data by ITM, IMLM, and BBP, there are forward flows
+(activations) from the language encoder to the fusion en-
+coder, but there are no backward flows (gradients) from the
+fusion encoder to the language encoder. In this way, the
+language encoder is only trained with text data by MLM
+and with image-text pair data by ITC. The former helps the
+language encoder to learn text representations, and the latter
+helps the language encoder and the vision encoder to make
+alignments between their respective text representations and
+image representations. Meanwhile, the training of the fusion
+encoder is performed separately with the focus of learning
+from image-text pair data.
+Masked Image Modeling. The training of vision encoder
+by MIM is carried out as follows. The image data is first
+masked and then predicted by the vision encoder. The dif-
+ferences between predicted representations and ‘target’ rep-
+resentations at masked positions and [CLS] position are
+then measured with MSE (mean squared error) loss. The
+target representations are obtained from the same image
+data (without masking) by the vision encoder. There are
+no gradients from the target representations in the learning
+of the vision encoder. The vision encoder can create target
+representations because it is also trained with image-text
+pair data. In this way, the vision encoder is trained by both
+the cross-modal objectives (ITC, ITM, BBP, IMLM) with
+image-text pair data and the uni-modal objective (MIM)
+with image data. The representations obtained from the
+vision-language training are highly semantic, which is nec-
+essary for MIM as demonstrated in previous work (Bao
+et al., 2021; Peng et al., 2022; Wei et al., 2022a;b).
+There are three advantages by exploiting the new MIM
+technique. First, it becomes possible to leverage image data
+for learning of the vision encoder, which is relatively easy
+to obtain. Second, it is convenient to conduct MIM with the
+signals from the vision-language training. Note that most
+previous work for MIM makes use of an external image
+tokenizer such as VQ-VAE (Bao et al., 2021; Singh et al.,
+2021), CLIP (Wei et al., 2022b), and VQ-KL (Peng et al.,
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+2022). Third, the learning of the vision encoder and that
+of the fusion encoder are mutually enhanced. Once the
+vision encoder is trained, it is also utilized to train the fusion
+encoder.
+3.2. Pre-training Objectives
+We explain six objectives in learning of X-FM. Here, T
+represents the distribution of text data, I represents the
+distribution of image data, and D represents the distribution
+of image-text pair data.
+Masked Language Modeling (MLM) We perform MLM
+on text data to learn the language encoder of X-FM. Specifi-
+cally we recover the masked tokens in a text by minimizing
+the cross entropy loss below.
+Lmlm = ET ∼T H(⃗y( ¯T), ˆ⃗p( ¯T))
+(1)
+where T denotes a text, ¯T denotes the masked text of T, ˆ⃗p
+denotes the predicted probability vectors of masked tokens
+of ¯T, ⃗y denotes the one-hot vectors representing the original
+tokens of ¯T, and H denotes cross-entropy.
+Image-Text Contrastive Learning (ITC). We use an
+image-text contrastive loss as in CLIP (Radford et al., 2021)
+to learn the alignments between images and texts in ITC.
+Given a batch of images and texts, we calculate the cosine
+similarities between all image-text pairs. For each image,
+there is one text matched and the rest is unmatched. For each
+text, there is one image matched and the rest is unmatched.
+The contrastive loss is defined as follows.
+Litc = 1
+2E(I,T )∼D
+�
+H(⃗yi2t(I), ⃗pi2t(I))
++ H(⃗yt2i(T), ⃗pt2i(T))
+�
+(2)
+where (I, T) denotes an image-text pair, ⃗pi2t(I) denotes the
+in-batch image-to-text similarities, ⃗pt2i(T) denotes the in-
+batch text-to-image similarities, ⃗yi2t(I) denotes the one-hot
+vectors representing the image-to-text matching relations,
+⃗yt2i(T) denotes the one-hot vectors representing the text-to-
+image matching relations, and H denotes cross-entropy.
+Image-Text Matching (ITM). We also learn the align-
+ments between images and texts in ITM, using a loss in-
+dicating whether an image-text pair is matched. For each
+image in a batch there is a matched (positive) text, and we
+sample an unmatched (negative) text in the batch. For each
+text there is a matched (positive) image, and we sample
+an unmatched image in the batch. The loss is defined as
+follows.
+Litm = E(I,T )∼D
+�
+H(pmatch(I, T))
++H(pmatch(˜I, T))
+(3)
++H(pmatch(I, ˜T))
+�
+where (I, T) denotes a positive image-text pair, (˜I, T) and
+(I, ˜T) denote negative image-text pairs, pmatch(I, T) de-
+notes a predicted matching probability of (I, T), and H
+denotes logistic loss.
+Image-conditioned
+Masked
+Language
+Modeling
+(IMLM) We conduct IMLM on image-text pair data to
+learn the fusion encoder.
+Specifically, we recover the
+masked tokens of the text given for an image-text pair by
+minimizing the cross entropy loss below.
+Limlm = E(I,T )∼DH(⃗y( ¯T), ˆ⃗p(I, ¯T))
+(4)
+where (I, T) denotes an image-text pair, ¯T denotes the
+masked text of T, ˆ⃗p(I, ¯T) denotes the predicted probability
+vectors of the masked tokens of ¯T based on I, ⃗y denotes the
+one-hot vectors representing the original tokens of ¯T, and
+H denotes cross-entropy.
+Bounding Box Prediction (BBP) We adopt the BBP in X-
+VLM (Zeng et al., 2021; 2022), which locates the visual
+concept in the image by a bounding box given the text. With
+BBP we learn the alignments between the images and texts
+in multi-granularity. In BBP, two losses are simultaneously
+minimized to measure the differences between the predicted
+bounding box and the ground-truth bounding box. One
+is generalized intersection over union GIoU (Rezatofighi
+et al., 2019) and the other is ℓ1 distance.
+Lbbp = E(I,T )∼D{GIoU(⃗b,ˆ⃗b) + ∥⃗b − ˆ⃗b∥1}
+(5)
+where⃗b = (cx, cy, w, h) denotes the ground truth bounding
+box, ˆ⃗b = ( ˆcx, ˆcy, ˆw, ˆh) denotes the predicted bounding box.
+A bounding box is represented by two coordinates, width,
+and height.
+Masked Image Modeling (MIM) We perform MIM on im-
+age data and image-text pair data to learn the vision encoder.
+Specifically, we recover the masked image patches in an
+image by minimizing the loss below.
+Lmim = E(I,T )∼D||⃗v(¯I) − ˆ⃗v(¯I)||2 + EI∼I||⃗v(¯I) − ˆ⃗v(¯I)||2
+(6)
+where (I, T) and I denote an image-text pair and a single
+image respectively, ¯I denotes the masked image I, ˆ⃗v(¯I) de-
+notes the predicted representations at the masked positions
+and [CLS] of ¯I, and ⃗v(¯I) denotes the target representa-
+tions at the masked positions and [CLS] of ¯I. ||˙||2 is the
+MSE loss. We employ block masking following previous
+work (Bao et al., 2021; Peng et al., 2022). Note that (I, T)
+and I are independently sampled from D and I, and the
+sample sizes are not necessarily equal.
+Finally, the pre-training objective of X-FM is defined as the
+sum of the losses described above.
+L = Lmlm + Litc + Litm + Limlm + Lbbp + Lmim (7)
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+Base-Size Models
+Large-Size Models
+RoBERTa
+BEiTv2
+X2-VLM
+UNIMO-2
+FLAVA
+SimVLM
+OFA
+DaVinci
+Uni-Per.
+OmniVL
+X-FM
+RoBERTa
+BEiTv2
+X2-VLM
+SimVLM
+OFA
+Uni-Per.
+X-FM
+Task
+Eval.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+MNLI
+FT
+87.6
+–
+–
+87.5
+80.3
+83.4
+84.3
+82.3
+81.5
+–
+87.7
+90.2
+–
+–
+–
+84.3
+85.7
+90.4
+CoLA
+FT
+63.6
+–
+–
+62.1
+50.7
+46.7
+52.3
+52.1
+52.2
+–
+65.3
+68.0
+–
+–
+–
+52.3
+57.4
+69.9
+MRPC
+FT
+90.2
+–
+–
+–
+84.2
+79.8
+88.7
+83.1
+–
+–
+91.7
+90.9
+–
+–
+–
+88.7
+–
+92.4
+QQP
+FT
+91.9
+–
+–
+–
+88.7
+90.4
+91.3
+88.2
+–
+–
+91.8
+92.2
+–
+–
+–
+91.3
+–
+92.2
+SST-2
+FT
+94.8
+–
+–
+94.7
+90.9
+90.9
+92.7
+90.5
+90.9
+–
+95.0
+96.4
+–
+–
+–
+92.7
+93.4
+96.7
+QNLI
+FT
+92.8
+–
+–
+–
+87.3
+88.6
+91.1
+87.2
+88.2
+–
+92.9
+94.7
+–
+–
+–
+91.1
+91.9
+94.8
+RTE
+FT
+78.70
+–
+–
+–
+57.8
+63.9
+70.8
+60.7
+75.8
+–
+83.8
+86.6
+–
+–
+–
+70.8
+78.4
+87.4
+STS-B
+FT
+91.2
+–
+–
+91.2
+85.7
+87.2
+–
+86.3
+–
+–
+90.8
+92.4
+–
+–
+–
+–
+–
+92.1
+Language Avg.
+86.4
+–
+–
+–
+78.2
+78.9
+–
+78.8
+–
+–
+87.4
+88.9
+–
+–
+–
+–
+–
+89.5
+ImageNet
+FT
+–
+85.5
+–
+80.8
+–
+–
+82.2
+83.9
+84.5
+–
+85.3
+–
+87.3
+–
+–
+–
+86.4
+86.3
+ImageNet
+LE
+–
+80.1
+–
+–
+75.5
+80.6
+71.4†
+75.9
+–
+–
+81.0
+–
+66.8†
+–
+82.3
+74.7†
+–
+81.0
+Food101
+LE
+–
+88.2†
+–
+–
+88.5
+–
+75.2†
+89.3
+–
+87.4
+88.7
+–
+52.2†
+–
+–
+81.6†
+–
+88.9
+CIFAR10
+LE
+–
+95.3†
+–
+–
+92.9
+–
+86.1†
+93.0
+–
+96.2
+97.2
+–
+63.5†
+–
+–
+91.9†
+–
+97.2
+CIFAR100
+LE
+–
+81.5†
+–
+–
+77.7
+–
+66.7†
+79.0
+–
+83.2
+86.7
+–
+39.7†
+–
+–
+75.6†
+–
+85.1
+Pets
+LE
+–
+93.1†
+–
+–
+84.8
+–
+81.0†
+85.5
+–
+87.1
+90.8
+–
+38.9†
+–
+–
+86.8†
+–
+90.0
+DTD
+LE
+–
+78.4†
+–
+–
+77.3
+–
+70.3†
+77.1
+–
+76.2
+78.4
+–
+44.4†
+–
+–
+74.4†
+–
+79.0
+Flowers102
+LE
+–
+95.7†
+–
+–
+96.4
+–
+86.3†
+96.1
+–
+89.8
+97.1
+–
+66.6†
+–
+–
+92.6†
+–
+95.8
+Vision Avg.
+–
+88.7
+–
+–
+86.3
+–
+79.2
+86.7
+–
+86.7
+89.8
+–
+50.9
+–
+–
+83.8
+–
+89.3
+VQAv2
+FT
+–
+–
+79.2
+76.3
+72.5
+77.9
+78.0
+73.9
+–
+78.3
+79.1
+–
+–
+80.5
+79.3
+80.3
+–
+79.5
+NLVR2
+FT
+–
+–
+86.1
+–
+–
+81.8
+–
+77.9
+–
+–
+86.7
+–
+–
+87.6
+84.8
+–
+–
+87.8
+Flickr30K TR R@1
+ZS
+–
+–
+85.1†
+88.5
+67.7
+–
+–
+–
+82.1
+–
+90.1
+–
+–
+86.8†
+–
+–
+83.6
+89.7
+Flickr30K IR R@1
+ZS
+–
+–
+77.3†
+72.7
+65.2
+–
+–
+–
+72.4
+–
+79.1
+–
+–
+80.5†
+–
+–
+75.9
+79.1
+Flickr30K TR R@1
+FT
+–
+–
+97.4
+92.0
+–
+–
+–
+–
+93.6
+94.9
+97.4
+–
+–
+99.1
+–
+–
+94.1
+97.9
+Flickr30K IR R@1
+FT
+–
+–
+90.0
+80.1
+–
+–
+–
+–
+79.8
+83.4
+88.6
+–
+–
+91.1
+–
+–
+83.7
+89.4
+COCO TR R@1
+ZS
+–
+–
+68.4†
+–
+42.7
+–
+–
+–
+64.6
+–
+73.8
+–
+–
+69.7†
+–
+–
+67.9
+74.4
+COCO IR R@1
+ZS
+–
+–
+55.2†
+–
+38.4
+–
+–
+–
+51.6
+–
+59.4
+–
+–
+58.3†
+–
+–
+55.3
+59.4
+COCO TR R@1
+FT
+–
+–
+80.5
+–
+–
+–
+–
+–
+70.5
+76.8
+81.8
+–
+–
+82.3
+–
+–
+74.7
+82.1
+COCO IR R@1
+FT
+–
+–
+62.7
+–
+–
+–
+–
+–
+52.6
+58.5
+64.7
+–
+–
+65.2
+–
+–
+57.1
+65.4
+Vision-Language Avg.
+–
+–
+78.2
+–
+–
+–
+–
+–
+–
+–
+80.1
+–
+–
+80.1
+–
+–
+–
+80.5
+Table 2: Experimental results on vision, language and vision-language tasks. MNLI results are average of MNLI-m
+and MNLI-mm. MRPC results are average accuracies and F1 scores. Matthews correlation coefficient (MCC) is reported for
+CoLA, and Pearson correlation coefficient (PCC) is reported for STS-B. We report accuracies for all the vision and multi-
+modal tasks. FT is short for fine-tuning, LE for linear evaluation, ZS for zero-shot, TR for text retrieval, and IR for image
+retrieval. Results for RoBERTa are from its corresponding paper (Liu et al., 2019), and they use the mid-training (Phang
+et al., 2018) on MNLI for RTE, MRPC, and STS-B while other models (e.g., BERT, SimVLM, DaVinci, X-FM) do not
+use this trick. Language Avg. is the average score of all the language tasks, while Vision Avg. is the average score of six
+line evaluation tasks except ImageNet. Vision-Language Avg. is the average score of all vision-language tasks. † are our
+reproduced results with the officially released models. Uni-Per. stands for Uni-Perceiver-MoE (Zhu et al., 2022).
+4. Experiments
+4.1. Pre-training Datasets
+We conduct our experiments on several widely used pub-
+lic datasets, which consist of two in-domain datasets,
+COCO (Lin et al., 2014) and Visual Genome (VG) (Kr-
+ishna et al., 2017), and two out-of-domain datasets, SBU
+Captions (Ordonez et al., 2011) and Conceptual Captions
+(CC) (Sharma et al., 2018). Following X-VLM (Zeng et al.,
+2021; 2022), we also include annotations of objects and
+regions from RefCOCO (Yu et al., 2016), Objects365 (Shao
+et al., 2019) and OpenImages (Kuznetsova et al., 2018).
+Since we assume also using uni-modal data, we include
+RoBERTa corpus (Liu et al., 2019), C4 datasets (Raffel
+et al., 2020) and Imagenet21K (Ridnik et al., 2021). All
+pre-training datasets are listed in Table 3.
+4.2. Implementation Details
+Pre-training
+Our model is of base size and large size, and
+the parameters are listed in Table 5. The vision encoder is
+initialized with BEiTv2 (Peng et al., 2022). The language
+encoder is initialized with RoBERTa (Liu et al., 2019). The
+fusion encoder is trained from scratch. X-FM is pre-trained
+at image resolution of 224 × 224 with patch size of 16 × 16.
+Dataset
+# Images
+# Texts
+# Objects
+# Regions
+COCO
+0.11M
+0.55M
+0.45M
+-
+VG
+0.10M
+-
+2.0M
+3.7M
+SBU
+0.86M
+0.86M
+-
+-
+CC-3M
+2.9M
+2.9M
+-
+-
+Objects365
+0.58M
+-
+2.0M
+-
+OpenImages
+1.7M
+-
+4.2M
+-
+C4
+-
+800GB
+-
+-
+RoBERTa Corpus
+-
+160GB
+-
+-
+ImageNet-21k
+14M
+-
+-
+-
+Table 3: Statistics of the pre-training datasets.
+We pre-train X-FMbase for 200K steps with a batch size of
+3072 image-text pairs, 3072 images, and 8192 sentences on
+32 A100 and pre-train X-FMlarge with the same batch for
+160K steps on 64 A100, which takes about six days. The
+learning rate for both models is warmed-up to 1e−4 in the
+first 2500 steps and decayed following a linear schedule. We
+set the maximum number of text tokens to 30 for image-text
+pairs, while that of pure text corpus is set to 128. We apply
+mixed precision for pre-training.
+Fine-tuning
+We choose widely used downstream tasks
+whose details are shown in Appendix B. We report overall
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+Model
+# Params
+MSCOCO (5K test set)
+Flickr30K (1K test set)
+MSCOCO (5K test set)
+Flickr30K (1K test set)
+TR-Fine-Tune
+IR-Fine-Tune
+TR-Fine-Tune
+IR-Fine-Tune
+TR-Zero-Shot
+IR-Zero-Shot
+TR-Zero-Shot
+IR-Zero-Shot
+R@1/R@5/R@10
+R@1/R@5/R@10
+R@1/R@5/R@10
+R@1/R@5/R@10
+R@1/R@5/R@10
+R@1/R@5/R@10
+R@1/R@5/R@10
+R@1/R@5/R@10
+ALBEF
+210M
+73.1/91.4/96.0
+56.8/81.5/89.2
+94.3/99.4/99.8
+82.8/96.7/98.4
+–
+–
+90.5/98.8/99.7
+76.8/93.7/96.7
+VLMobase
+175M
+74.8/93.1/96.9
+57.2/82.6/89.8
+92.3/99.4/99.9
+79.3/95.7/97.8
+–
+–
+–
+–
+VL-BEiT
+175M
+79.5/–/–
+61.5/–/–
+95.8/–/–
+83.9/–/–
+–
+–
+–
+–
+OmniVL
+288M
+76.8/93.6/97.3
+58.5/82.6/89.5
+94.9/9.6/99.9
+83.4/97.0/98.6
+–
+–
+–
+–
+X-VLM
+216M
+80.4/95.5/98.2
+63.1/85.7/91.6
+96.8/99.8/100
+86.1/97.4/98.7
+70.8/92.1/96.5
+55.6/82.7/90.0
+85.3/97.8/99.6
+71.9/93.3/96.4
+X2-VLMbase
+255M
+80.5/95.5/97.8
+62.7/84.7/90.7
+97.4/99.9/100
+90.0/98.6/99.3
+68.4†/92.5†/96.8†
+55.2†/82.2†/89.3†
+85.1†/99.2†/100.0†
+77.3†/95.3†/97.6†
+X-FMbase
+284M
+81.8/96.0/98.3
+64.7/86.1/91.6
+97.4/100/100
+88.6/97.9/98.9
+73.8/93.9/97.2
+59.4/83.6/90.0
+90.1/99.2/99.9
+79.1/95.2/97.3
+VLMolarge
+562M
+78.2/94.4/97.4
+60.6/84.4/91.0
+95.3/99.9/100
+84.5/97.3/98.6
+–
+–
+–
+–
+X2-VLMlarge
+593M
+82.3/96.2/98.3
+65.2/86.4/91.9
+99.1/100/100
+91.1/98.6/99.4
+69.7†/93.0†/97.2†
+58.3†/83.8†/90.5†
+86.8†/98.9†/99.9†
+80.5†/96.4†/98.3†
+X-FMlarge
+807M
+82.1/96.2/98.2
+65.4/86.6/91.9
+97.9/100/100
+89.4/98.2/99.1
+74.4/94.1/97.3
+59.4/84.4/90.7
+89.7/99.1/100
+79.1/95.4/97.9
+Super-Large Models or Super-Large Datasets
+CLIP
+490M
+–
+–
+88.7/98.0/99.2
+76.7/93.6/96.4
+58.4/81.5/88.1
+37.8/62.4/72.2
+88.0/98.7/99.4
+68.7/90.6/95.2
+ALIGN
+490M
+77.0/93.5/96.9
+59.9/83.3/89.8
+95.3/99.8/100
+84.9/97.4/98.6
+58.6/83.0/89.7
+45.6/69.8/78.6
+88.6/98.7/99.7
+75.7/93.8/96.8
+Florence
+893M
+81.8/95.2/–
+63.2/85.7/–
+97.2/99.9/–
+87.9/98.1/–
+64.7/85.9/–
+47.2/71.4/–
+90.9/99.1/–
+76.7/93.6/–
+CoCa
+2.1B
+–
+–
+–
+–
+66.3/86.2/91.8
+51.2/74.2/82.0
+92.5/99.5/99.9
+80.4/95.7/97.7
+BEiT-3
+1.9B
+84.8/96.5/98.3
+67.2/87.7/92.8
+98.0/100/100
+90.3/98.7/99.5
+–
+–
+94.9/99.9/100.0
+81.5/95.6/97.8
+X2-VLMlarge
+593M
+84.4/96.5/98.5
+67.7/87.5/92.5
+98.8/100/100
+91.8/98.6/99.5
+–
+–
+–
+–
+Table 4: Results of text-retrieval (TR) and image-retrieval (IR) on COCO and Flickr30K. † denotes our reproduced results
+with the officially released models. Giant models with over 1B parameters (e.g., BEiT-3) and models are pre-trained with
+over 400M data (e.g., CLIP and X2-VLMlarge) are in grey since they are not directly comparable with other models.
+Model
+Param
+Hidden
+Layers
+Vision
+Text
+Fusion
+X-FMbase
+284
+768
+12
+12
+12
+X-FMlarge
+807
+1024
+24
+24
+12
+Table 5: Size variants of X-FM. All modules consist of
+transformer layers. Param indicates the parameter number
+of transformer layers.
+performance on eight language tasks from GLUE (Wang
+et al., 2019), eight vision tasks following OmniVL (Wang
+et al., 2022a), four multi-modal tasks, which are text-
+image retrieval on MSCOCO and Flickr, visual question
+answering (VQA (Goyal et al., 2017)) and visual reason-
+ing (NLVR2 (Suhr et al., 2019b)). For image-text retrieval
+task, we report both zero-shot results and fine-tuned results.
+For the ImageNet classification task, we report both linear
+evaluation results and fine-tuning results. The other vision
+tasks are evaluated in the linear evaluation setting. All the
+other tasks are evaluated in the fine-tuning setting. Because
+the image resolution differs between pre-training and fine-
+tuning, the position parameters are adapted using linear
+interpolation. For all downstream tasks, we apply random
+resize crops and horizontal flips augmentation for the im-
+ages during training. More details of network architectures
+and hyper-parameters setups are given in Appendix C.
+4.3. Comparison with SOTA Foundation Models
+We extensively compare the performance of X-FM with
+state-of-the-art foundation models on vision, language, and
+multi-modal tasks. We first compare our model with general
+foundation models, including UNIMO-v2 (Li et al., 2021c),
+FLAVA (Singh et al., 2021), SimVLM (Wang et al., 2021c),
+OFA (Wang et al., 2022b), DaVinci (Diao et al., 2022), Om-
+niVL (Wang et al., 2022a), and Uni-Perceiver-MoE (Zhu
+et al., 2022). We also include comparisons with SOTA
+foundation models specifically designed for language, vi-
+sion, or vision-language tasks, RoBERTa (Liu et al., 2019),
+BEiTv2 (Peng et al., 2022), and X2-VLM (Zeng et al., 2022).
+There are several observations in Table 2. First, X-FMbase
+(column 11) outperforms all the previous general foundation
+models (column 4-10) across almost all tasks by a large mar-
+gin, becoming a new and stronger general foundation model.
+Compared to the previous general foundation models, X-
+FMbase improves at least 3.2% and the most even to 9.7%
+on the average of all the reported numbers. Second, we com-
+pare X-FM with state-of-the-art foundation models specif-
+ically designed for language, vision, and vision-language
+tasks, RoBERTa, BEiTv2 and X2-VLM. We observe that
+X-FM is also better than or comparable with the foundation
+models at both base and large scale (column 1,2,3 vs 11 and
+12,13,14 vs 18).
+4.4. Comparison with SOTA Vision-Language Models
+In addition to general foundation models, we also compare
+X-FM with state-of-the-art vision-language models. The re-
+sults are shown in Table 4 and Table 7. X-FM demonstrates
+its superiority on MSCOCO retrieval and NLVR2, while
+achieving competitive performance on Flickr retrieval and
+VQA. Note that X-FMbase outperforms CLIP, ALIGN and
+Florence on image-text retrieval tasks with fewer parame-
+ters and much less training data. Compared to the recently
+released SOTA vision-language model, X2-VLM, X-FM is
+much better on image-text retrieval tasks at the zero-shot
+setting.
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+X-FMbase
+RoBERTa†
+S-MLM
+S-ITM
+wostop
+BEiTv2†
+woMIM
+wBEiTv2 Tokenizer
+X2-VLM†
+Multi-task
+ALL
+Task
+Eval.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+MNLI
+FT
+87.7
+87.4
+87.3
+87.7
+–
+–
+–
+–
+87.4
+87.6
+CoLA
+FT
+63.2
+61.6
+63.6
+64.2
+–
+–
+–
+–
+62.2
+65.2
+MRPC
+FT
+90.7
+92.2
+91.1
+90.7
+–
+–
+–
+–
+92.0
+92.5
+QQP
+FT
+91.5
+91.6
+91.6
+91.6
+–
+–
+–
+–
+91.6
+91.6
+SST-2
+FT
+95.0
+95.1
+94.2
+94.6
+–
+–
+–
+–
+94.4
+95.3
+QNLI
+FT
+93.1
+93.0
+93.2
+92.5
+–
+–
+–
+–
+92.8
+92.9
+RTE
+FT
+80.9
+79.1
+81.6
+81.2
+–
+–
+–
+–
+79.8
+81.9
+STS-B
+FT
+90.9
+90.7
+90.7
+90.4
+–
+–
+–
+–
+90.1
+90.8
+Language Avg.
+86.6
+86.4
+86.7
+86.6
+–
+–
+–
+–
+86.3
+87.2
+ImageNet
+FT
+–
+–
+–
+–
+85.5
+84.8
+85.0
+–
+85.0
+85.3
+ImageNet
+LE
+–
+–
+–
+–
+80.5
+79.1
+79.4
+–
+79.3
+81.1
+Food101
+LE
+–
+–
+–
+–
+88.2
+86.9
+87.2
+–
+86.9
+88.7
+CIFAR10
+LE
+–
+–
+–
+–
+95.3
+96.6
+96.5
+–
+96.6
+97.5
+CIFAR100
+LE
+–
+–
+–
+–
+81.5
+83.3
+83.9
+–
+84.1
+86.9
+Pets
+LE
+–
+–
+–
+–
+93.1
+88.1
+88.5
+–
+88.2
+90.7
+DTD
+LE
+–
+–
+–
+–
+78.4
+77.7
+76.9
+–
+78.0
+78.7
+Flowers102
+LE
+–
+–
+–
+–
+95.7
+94.1
+94.5
+–
+94.2
+97.1
+Vision Avg.
+–
+–
+–
+–
+87.3
+86.3
+86.5
+–
+86.5
+88.2
+VQAv2
+FT
+–
+78.8
+78.5
+78.7
+–
+78.3
+78.2
+78.0
+78.2
+78.6
+NLVR2
+FT
+–
+86.3
+86.0
+86.4
+–
+85.9
+85.5
+86.2
+86.1
+86.7
+Flickr30K TR R@1
+ZS
+–
+88.3
+87.2
+87.1
+–
+87.1
+87.2
+87.7
+85.0
+89.3
+Flickr30K IR R@1
+ZS
+–
+76.6
+74.9
+75.8
+–
+76.1
+75.3
+75.1
+75.6
+77.4
+Flickr30K TR R@1
+FT
+–
+97.5
+97.0
+97.2
+–
+96.4
+96.7
+97.0
+97.0
+97.7
+Flickr30K IR R@1
+FT
+–
+87.4
+86.9
+87.3
+–
+86.2
+86.6
+86.2
+86.4
+87.4
+COCO TR R@1
+ZS
+–
+72.0
+72.1
+70.5
+–
+73.0
+72.1
+73.2
+69.9
+72.8
+COCO IR R@1
+ZS
+–
+58.4
+57.1
+57.7
+–
+58.2
+57.7
+57.7
+56.5
+59.0
+COCO TR R@1
+FT
+–
+81.2
+80.2
+80.9
+–
+80.6
+80.1
+80.3
+80.0
+81.2
+COCO IR R@1
+FT
+–
+64.2
+63.4
+63.6
+–
+63.7
+63.0
+63.1
+63.0
+64.0
+Vision-Language Avg.
+–
+79.1
+78.3
+78.5
+–
+78.6
+78.2
+78.5
+77.8
+79.4
+Table 6: Ablation studies on vision, language, and vision-language tasks. We use the same settings as Table 2. “ALL” for
+X-FMbase is trained with the same data under the same settings for pre-training and fine-tuning compared to all the variants.
+Language Avg. is the average of all language tasks, while Vision Avg. is the average of all vision tasks. Vision-Language
+Avg. is the average of all vision-language tasks. Note that performance of “ALL” is slightly different from X-FMbase in
+Table 2, because we use less training steps (160k) for ablation to save the computational resources.
+4.5. Ablation Study
+To verify the contributions of different modules in our frame-
+work, we ablate them and evaluate the performance of X-
+FM on all downstream tasks. The results are shown in
+Table 6. We first explain several abbreviations in the table.
+S-MLM means that we only separate the language represen-
+tations in the learning IMLM task, while S-ITM means that
+language representations for computing ITM and BBP are
+separated. wostop indicates without stopping the gradients
+of all language representations. woMIM means that we do
+not learn by MIM, while wBEiTv2 tokenizer means that
+we learn by MIM with the image tokenizer used in BEiTv2.
+Multi-task is a variation that uses straightforward multi-task
+learning to optimize the three encoders in X-FM. To make
+a fair comparison, we also train RoBERTa, BEiTv2 and
+X2-VLM with the same data noted as RoBERTa†, BEiTv2†
+and X2-VLM†. Note that we also increase the fusion layers
+in X2-VLM† to make the parameter sizes comparable to
+our models. RoBERTa†, BEiTv2† and X2-VLM† all have
+slightly better results on average than the official ones. From
+the results, we have the following observations.
+First, both designs (stop gradient and masked image mod-
+eling) bring improvements, and the combination can make
+further improvements on all three downstream tasks (col-
+umn 10 vs. others). Second, without separated language
+representations, models always perform worse on language
+understanding tasks (column 10 vs. 2,3,4). Besides, the sep-
+arate language representations in the IMLM task on image-
+text data are helpful for multi-modal tasks (column 2 vs. 4).
+As we point out in section 1, the fusion encoder can learn
+better cross-modal feature alignments by IMLM task from
+image-text pairs instead of utilizing text tokens. Although S-
+ITM shows slight side effects (column 4 vs. 3), stopping the
+gradients of language representation in the fusion encoder
+is necessary to simultaneously achieve strong language un-
+derstanding and vision-language understanding capability.
+Third, the MIM task is useful for vision-language and vi-
+sion learning (column 10 vs. 6). Meanwhile, the targets
+in our MIM task are better than the BEiTv2 tokenizer (col-
+umn 10 vs. 7). Four, X-FM is much better than a naive
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+Method
+# Params
+VQA
+NLVR2
+test-dev
+test-std
+dev
+test-P
+ALBEF
+210M
+74.5
+74.7
+80.2
+80.5
+VLMobase
+175M
+76.6
+76.9
+82.8
+83.3
+METER
+341M
+77.7
+77.6
+82.3
+83.1
+VL-BEiT
+175M
+77.5
+77.8
+81.9
+82.7
+BLIPbase
+240M
+78.2
+78.2
+82.5
+83.1
+X-VLM
+216M
+78.1
+78.1
+84.2
+84.2
+OFAbase
+182M
+78.0
+78.1
+-
+-
+OmniVL
+288M
+78.3
+78.4
+-
+-
+X2-VLMbase
+255M
+79.2
+79.3
+85.9
+86.1
+X-FMbase
+284M
+79.1
+79.2
+86.3
+86.5
+VLMolarge
+562M
+79.9
+80.0
+85.6
+86.9
+OFAlarge
+472M
+80.3
+80.5
+-
+-
+X2-VLMlarge
+593M
+80.5
+80.5
+87.2
+87.6
+X-FMlarge
+807M
+79.5
+79.6
+86.2
+87.8
+Super-Large Models or Super-Large Datasets
+SimVLMbase
+273M
+77.9
+78.1
+81.7
+X2-VLMbase
+255M
+80.4
+80.2
+86.2
+87.0
+SimVLMlarge
+783M
+79.3
+79.6
+84.1
+84.8
+X2-VLMlarge
+593M
+81.9
+81.8
+88.7
+89.4
+Florence
+893M
+80.2
+80.3
+–
+–
+CoCa
+2.1B
+82.3
+82.3
+86.1
+87.0
+BEiT-3
+1.9B
+84.2
+84.0
+91.5
+92.6
+Table 7: Results on VQA and visual reasoning. Giant mod-
+els with over 1B parameters (e.g., CoCa and BEiT-3) or
+models are pre-trained with over 400M data (e.g., SimVLM
+and X2-VLMlarge) are in grey because they are not directly
+comparable with other models.
+multi-task learning strategy for a foundation model (column
+10 vs. 8). Compared with the straightforward multi-task
+strategy, X-FMbase improves an average of 0.9%, 1.7%
+and 1.6% on language, vision, and vision-language tasks,
+respectively. Five, X-FM is also slightly better than foun-
+dation models specifically designed for language, vision,
+and vision-language tasks with the same training corpus
+(column 10 vs. 1,5,8).
+5. Conclusion and Limitation
+5.1. Conclusion
+In this work, we address the problem of how to build a
+general foundation model that can perform the best for all
+the understanding tasks of language, vision, and vision-
+language. We propose a general foundation model with two
+new and effective training techniques, X-FM, to learn rich
+language, vision and vision-language representations at the
+same time. Experimental results explicitly imply that X-FM
+outperforms other general foundation models by a large
+margin. Moreover, X-FM can even be better or comparable
+compared with the SOTA foundation models specifically
+designed for language, vision, or vision-language under-
+standing tasks.
+5.2. Limitation
+Like most existing work on foundation models, the entire
+project consumed over 5 A100 GPU years on a computing
+cluster with high electricity costs, although we only tested
+base and large models. There is still potential for efficiency
+improvement through sparse attention (Zaheer et al., 2020)
+or the lottery ticket hypothesis (Frankle & Carbin, 2018).
+We will explore the techniques to improve the training effi-
+ciency and reduce the carbon footprint so that we can adhere
+to the proposals on “green” deep learning (Schwartz et al.,
+2020; Xu et al., 2021).
+Due to considerations of fair comparisons and computa-
+tional resources, we did not try super-large models which
+use at least 1.9B or more parameters like BEITv3 (Wang
+et al., 2022d), CoCa (Yu et al., 2022) and PaLI (Chen et al.,
+2022). However, scalability is also an important factor for
+foundation models. We leave the investigations to future
+work.
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+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+A. Comparison of Recent Foundation Models
+Table 8 shows an extensive comparison of recent foundation
+models and X-FM on multiple axes. Previous work either (i)
+perform best on uni-modal tasks (Liu et al., 2019; Peng et al.,
+2022) or vision-language tasks (Zeng et al., 2021; 2022); (2)
+target a specific uni-modal domain along with part of vision-
+and-language tasks (Wang et al., 2021a; Radford et al., 2021;
+Jia et al., 2021; Wang et al., 2021c; Yu et al., 2022; Wang
+et al., 2022b; Diao et al., 2022); or (3) target all domains
+but cannot perform best on all the tasks (Li et al., 2021c;
+Singh et al., 2021; Zhu et al., 2022). Our model, X-FM, is
+a general foundation model that can perform the best for
+all the understanding tasks of language, vision, and vision
+language.
+B. Details of Downstream Tasks
+Language Understanding.
+We conduct experiments on GLUE benchmark including
+MNLI (Williams et al., 2018), CoLA (Warstadt et al., 2019),
+MRPC (Dolan & Brockett, 2005), QQP (Iyer et al., 2017),
+SST-2 (Socher et al., 2013), QNLI (Rajpurkar et al., 2016),
+RTE (Dagan et al., 2005; Haim et al., 2006; Giampiccolo
+et al., 2007; Bentivogli et al., 2009), and STS-B (Agirre
+et al., 2007). We follow the practice of BERT (Devlin et al.,
+2019; Liu et al., 2019) and feed the input into the language
+encoder, and the hidden state of the [CLS] is fed into a
+new multi-class linear classifier or regression head.
+Vision Understanding.
+We conduct vision experiments on both fine-tuning and lin-
+ear evaluation (linear eval). The linear evaluation follows
+a common practice (Caron et al., 2021; He et al., 2020;
+Singh et al., 2021) in self-supervised learning to evaluate
+the representation quality, where the pre-trained backbone
+model is frozen, and an MLP head is appended on top of it.
+We choose 7 popular datasets following OmnVL (Wang
+et al., 2022a):
+ImageNet (Russakovsky et al., 2015),
+Food101 (Bossard et al., 2014), CIFAR10 (Krizhevsky et al.,
+2009), CIFAR100 (Krizhevsky et al., 2009), DTD (Cimpoi
+et al., 2014), Pets (Parkhi et al., 2012) and Flowers102 (Nils-
+back & Zisserman, 2008).
+Vision-Language Understanding.
+Image-Text Retrieval We evaluate X-FM on both
+MSCOCO and Flickr30K datasets. We adopt the widely
+used Karpathy split (Karpathy & Li, 2015) for both datasets.
+Following the previous work (Li et al., 2021a; Zeng et al.,
+2021; 2022), we first encode images and texts separately
+and calculate s(I, T) to obtain the top-k candidates, and
+then use the fusion encoder to re-rank the candidates.
+Visual Question Answering The task requires the model
+to predict an answer given an image and a question. We
+evaluate X-FM on the VQA v2.0 dataset (Goyal et al., 2017).
+Following the previous work (Zeng et al., 2021), we use
+a Transformer decoder to generate answers based on the
+outputs of the fusion module. The decoder network shares
+the same network architecture with the fusion encoder. Note
+that we use an image resolution of 768*768 for the final
+result of X-FMbase, and use an image resolution of 480*480
+for X-FMlarge and X-FMbase in ablation studies for efficient
+fine-tuning.
+Visual Reasoning We evaluate X-FM on a widely used
+benchmark NLVR2 (Suhr et al., 2019a). The task allows the
+model to determine whether a text describes the relations
+between two images. Following previous work (Wang et al.,
+2021a; Bao et al., 2022), we formulate the triplet input into
+two image-text pairs, each containing the text description
+and an image. We then concatenate the final output [CLS]
+features of the fusion module of the two pairs to predict the
+label.
+C. Details of hyper parameters
+Pre-training
+X-FMbase is implemented with a 12-layer
+language encoder, a 12-layer vision encoder, and a 12-layer
+fusion encoder, 768 dimensions for hidden states, 3072 for
+intermediate size, and 128 for maximum input length. X-
+FMlarge is implemented with a 24-layer language encoder, a
+24-layer vision encoder, and a 12-layer fusion encoder, 1024
+dimensions for hidden states, 4096 for intermediate size, and
+128 for maximum input length. We initialize the language
+encoder with RoBERTa and the vision encoder with BEiTv2.
+The weight decay is set to 0.01 with β1 = 0.9, β2 = 0.98.
+The learning rate is 1e-4 with a warm-up period for the first
+2500 steps and then linearly decayed to 0. In each batch,
+there are 3072 image-text pairs, 3072 images, and 8192 text-
+only sentences. We use center-crop to resize each image
+to the size of 224×224. The default settings are shown in
+Table 9.
+Fine-tuning
+The learning rate is ∈ {1e-5, 2e-5, 5e-5} and
+our model is optimized by AdamW. Because the image
+resolution differs between pre-training and fine-tuning, the
+position parameters are adapted using linear interpolation.
+For all downstream tasks, we apply random resize crops
+and horizontal flips augmentation during training. The de-
+fault settings for text classification, image classification and
+vision-language understanding are shown in Tables 10, 11,
+12 and 13, respectively. Note that the resolution for VQA is
+different as described in Section B.
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+Methods
+Multimodal data
+Pretraining Objectives
+Fusion Arch.
+Target Modalities
+public
+dataset(s)
+size
+Contr.
+ITM
+BBP
+(M/P)LM
+Unimodal
+ST
+CT
+MT
+V
+CV&L
+MV&L
+L
+RoBERTa (Liu et al., 2019)
+–
+–
+–
+–
+–
+–
+–
+MLM
+–
+–
+–
+–
+–
+–
+�
+BEiTv2 (Peng et al., 2022)
+–
+–
+–
+–
+–
+–
+–
+MIM
+–
+–
+–
+�
+–
+–
+–
+X-VLM (Zeng et al., 2021; 2022)
+�
+Combination
+5M
+�
+�
+�
+MLM
+–
+–
+�
+–
+–
+�
+�
+–
+VLMo (Wang et al., 2021a)
+�
+Combination
+5M
+�
+�
+–
+MLM
+MLM+MIM
+–
+–
+�
+–
+�
+�
+–
+CLIP (Radford et al., 2021)
+�
+WebImageText
+400M
+�
+–
+–
+–
+–
+–
+–
+–
+�
+�
+–
+–
+ALIGN (Jia et al., 2021)
+�
+JFT
+1.8B
+�
+–
+–
+–
+–
+–
+–
+–
+�
+�
+–
+–
+SimVLM (Wang et al., 2021c)
+�
+JFT
+1.8B
+–
+–
+–
+PrefixLM
+PrefixLM
+�
+–
+–
+∗
+–
+�
+�
+CoCa (Yu et al., 2022)
+�
+JFT
+4.8B
+�
+–
+–
+LM
+–
+�
+–
+–
+�
+�
+�
+–
+UNIMO-2 (Li et al., 2021c)
+�
+Combination
+5M
+–
+�
+–
+MLM
+VCL
+�
+–
+–
+�
+�
+�
+�
+OFA (Wang et al., 2022b)
+�
+Combination
+15M
+–
+–
+–
+LM
+LM
+�
+–
+–
+∗
+–
+�
+�
+DaVinci (Diao et al., 2022)
+�
+Combination
+46M
+–
+–
+–
+PrefixLM + PrefixIM
+PrefixLM
+�
+–
+–
+�
+–
+�
+�
+FLAVA (Singh et al., 2021)
+�
+Combination
+70M
+�
+�
+–
+MLM
+MLM+MIM
+�
+–
+–
+�
+�
+�
+�
+Uni-Perceiver-MoE (Zhu et al., 2022)
+�
+Combination
+116M
+–
+�
+–
+LM+MLM
+LM+MLM+Classify.
+�
+–
+–
+�
+�
+�
+�
+X-FM
+�
+Combination
+5M
+�
+�
+�
+MLM+MIM
+MLM+MIM
+–
+�
+–
+�
+�
+�
+�
+Super-Large Models
+Flamingo (Alayrac et al., 2022)
+�
+Combination
+2.2B
+–
+–
+–
+LM
+–
+�
+–
+–
+–
+�
+�
+–
+BEiT-v3 (Wang et al., 2022d)
+�
+Combination
+21M
+–
+–
+–
+MLM
+MLM+MIM
+–
+–
+�
+∗
+�
+�
+–
+PaLI (Chen et al., 2022)
+�
+WebImageText
+41B
+–
+–
+–
+LM
+–
+�
+–
+–
+�
+�
+�
+�
+Table 8: Comparison of recent foundation models in different modalities. Contr. indicates contrastive learning. ITM is
+short for image-text matching. BBP represents boundary box prediction. (M/P)LM means image-conditioned (masked/prefix)
+language modeling. V, CV&L, MV&L and L stand for vision tasks, cross-modal retrieval tasks, multi-modal fusion tasks
+and language tasks respectively. ST, CT and MT are abbreviations for single Transformer, cross-attention Transformer and
+multiway Transformer. VCL stands for visual contrastive learning. ∗ means the modality is partially targeted (SimVLM
+and OFA include ImageNet.). Giant models with over 1B parameters (e.g. BEiT-3) are in grey since they are not directly
+comparable with other models.
+config
+value
+optimizer
+AdamW
+learning rate
+1e-4
+weight decay
+0.01
+optimizer momentum
+β1, β2=0.9, 0.999
+language batch size
+8192
+vision batch size
+3072
+vision-language batch size
+3072
+learning rate schedule
+linear decay
+warmup steps
+2500
+training steps
+200k
+augmentation
+RandomResizedCrop
+image res
+224*224
+patch size
+16
+text length for MLM
+128
+text length for IMLM
+30
+Table 9: Pre-training setting.
+config
+value
+optimizer
+AdamW
+learning rate
+{1e-5, 2e-5, 5e-5}
+weight decay
+0.0
+optimizer momentum
+β1, β2=0.9, 0.999
+batch size
+{16, 32, 64}
+learning rate schedule
+linear decay
+warmup ratio
+0.0
+training epochs
+{5, 10, 20}
+Table 10: Text classification: GLUE setting.
+config
+value
+optimizer
+AdamW
+learning rate
+[2e-5, 4e-5]
+weight decay
+0.01
+optimizer momentum
+β1, β2=0.9, 0.999
+batch size
+[256, 2048]
+learning rate schedule
+linear decay
+warmup rate
+0.1
+training epochs
+100
+augmentation
+RandomResizedCrop
+image res
+224*224
+patch size
+16
+Table 11: Image classification: Linear probing setting.
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+config
+value
+optimizer
+AdamW
+learning rate
+4e-5
+minimal learning rate
+1e-7
+weight decay
+0.01
+optimizer momentum
+β1, β2=0.9, 0.999
+batch size
+1024
+learning rate schedule
+linear decay
+warmup rate
+0.1
+training epochs
+100
+augmentation
+RandomResizedCrop
+image res
+224*224
+patch size
+16
+label smoothing
+0.1
+mixup prob.
+1.0
+cutmix prob.
+1.0
+Table 12: ImageNet classification: Fine-tuning setting.
+config
+value
+optimizer
+AdamW
+learning rate
+{1e-5, 2e-5, 5e-5}
+weight decay
+0.01
+optimizer momentum
+β1, β2=0.9, 0.999
+batch size
+{64, 192, 512}
+learning rate schedule
+linear decay
+warmup rate
+0.1
+training epochs
+{10, 15, 20}
+augmentation
+RandomResizedCrop
+image res
+384*384
+patch size
+16
+Table 13: Vision-Language understanding: fine-tuning set-
+ting.
+
diff --git a/2dE4T4oBgHgl3EQfagyV/content/tmp_files/load_file.txt b/2dE4T4oBgHgl3EQfagyV/content/tmp_files/load_file.txt
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@@ -0,0 +1,2701 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf,len=2700
+page_content='Toward Building General Foundation Models for Language, Vision, and  Vision-Language Understanding Tasks  Xinsong Zhang 1 Yan Zeng 1 Jipeng Zhang 2 Hang Li 1  Abstract  Foundation models or pre-trained models have  substantially improved the performance of various  language, vision, and vision-language understand-  ing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' However, existing foundation models  can only perform the best in one type of tasks,  namely language, vision, or vision-language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' It is  still an open question whether it is possible to con-  struct a foundation model performing the best for  all the understanding tasks, which we call a gen-  eral foundation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' In this paper, we propose  a new general foundation model, X-FM (the X-  Foundation Model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' X-FM has one language en-  coder, one vision encoder, and one fusion encoder,  as well as a new training method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The training  method includes two new techniques for learning  X-FM from text, image, and image-text pair data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  One is to stop gradients from the vision-language  training when learning the language encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The  other is to leverage the vision-language training  to guide the learning of the vision encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Exten-  sive experiments on benchmark datasets show that  X-FM can significantly outperform existing gen-  eral foundation models and perform better than or  comparable to existing foundation models specif-  ically for language, vision, or vision-language  understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Introduction  With the enormous power of foundation models, also known  as pre-trained models, remarkable performance gains have  recently been achieved in a variety of understanding tasks in  natural language processing (NLP), computer vision (CV),  and other fields (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Lewis  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Doso-  vitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Lu  1ByteDance AI Lab 2The Hong Kong University of Science  and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Correspondence to: Xinsong Zhang <zhangxin-  song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0320@bytedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='com>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Copyright 2023 by the author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The code and pre-trained models  will be released upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Tan & Bansal, 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 2022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Foundation  models are usually equipped with Transformer (Vaswani  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2017) as the backbone, pre-trained with a tremendous  amount of unlabeled data, and then fine-tuned with small  amounts of labeled data in downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The strong  representation ability of the model, the massive amount of  data, and the effective means of training make the founda-  tion models powerful for successfully solving the tasks of  vision, language, and vision-language (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Diao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  The state-of-the-art foundation models usually work the  best for one type of tasks, namely language, vision, and  vision-language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' For example, RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019),  BEiTv2 (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022), and X-VLM (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  2022) are language, vision, and vision-language founda-  tion models respectively, and can achieve state-of-the-art  performances for the specific type of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' It is still very  challenging, however, to build a general foundation model  that can perform the best in all types of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Existing  models, such as FLAVA (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021), OFA (Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022b), DaVinci (Diao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022) and Uni-Perceiver-  MoE (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022), are trying to achieve the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Their  performances are still not satisfactory, however, when com-  pared with the best performing foundation models for the  individual types of tasks, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Previous  work (Bingel & Søgaard, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020) also  shows that it is difficult to train a general foundation model  in a multi-task learning setting that can effectively learn and  utilize representations for all types of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The reason is  that language, vision, and vision-language are very different  in nature, and a simple way of jointly training a model from  language, vision, and vision-language data can easily create  a suboptimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  To address the challenge, we propose a new general founda-  tion model, X-FM (X-Foundation Model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' X-FM consists of  three modular encoders for language (text) encoding, vision  (image) encoding, and fusion encoding, as shown in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  The language encoder, the vision encoder, and the entire  model can be used in downstream tasks of language, vision,  and vision-language understanding, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' All three  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='05065v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='CV]  12 Jan 2023  Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks  Methods  Text Tasks  Vision Tasks  Multi-modal Tasks (MSCOCO Retriveal & VQA)  GLUE  ImageNet  Zero-Shot  Fine-Tune  MNLI  RTE  FT/LE  TR  IR  TR  IR  VQA  Foundation models specifically for language, vision, or vision-language understanding  RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019)  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  –  –  –  –  –  –  BEiTv2 (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022)  –  –  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5/80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  –  –  –  –  –  X-VLM (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021)  –  –  –  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8/92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1/96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6/82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7/90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4/95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5/98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1/85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7/91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  X2-VLM (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022)  –  –  –  –  –  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5/95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5/97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7/84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7/90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  General foundation models  UNIMO-2 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021c)  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  –  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8/-  –  –  –  –  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  SimVLM (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021c)  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9  /80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  –  –  –  –  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9  FLAVA (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021)  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8  /75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7/76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8/-  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4/67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5/-  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5/82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1/89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4/83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8  OFA (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022b)  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2/–  –  –  –  –  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  DaVinci (Diao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022)  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8  –  –  –  –  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  OmniVL (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022a)  –  –  –  –  –  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8/93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6/97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5/82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6/89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  Uni-Perceiver-MoE (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022)  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5/–  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6/–/–  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6/–/–  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5/–/–  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1/–/–  –  X-FMbase  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3/81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8/93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9/97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4/83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6/90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8/96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0/98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7/86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1/91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  Table 1: Performance comparisons between foundation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' All results are from base-size models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' MSCOCO is a  cross-modal retrieval task, and IR and TR are image-retrieval and text-retrieval, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' MNLI results are average  accuracies of MNLI-m and MNLI-mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Accuracy is reported for RTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' For ImageNet1k classification, we report linear  evaluation (LE) performance and fine-tuning (FT) performance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We report R@1/R@5/R@10 for all retrieval  tasks at both zero-shot and fine-tune settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We report the VQA test-dev result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' bold denotes the best number across  general foundation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' underline denotes the best across all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  encoders are stacked Transformer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The language en-  coder and the vision encoder follow the implementations  of BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019) and ViT (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2021), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The fusion encoder has the same ar-  chitecture as BERT except that there is a cross-attention  sub-layer after the self-attention sub-layer in each Trans-  former layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  In learning of X-FM, the language encoder, vision encoder,  and fusion encoder are jointly trained with text data, im-  age data, and image-text pair data as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Given the text  data, we train the language encoder by masked language  modeling (MLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Given the image data, we train the vi-  sion encoder by masked image modeling (MIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Given the  image-text pair data, we train the fusion encoder by image  text matching (ITM), image-conditioned masked language  modeling (IMLM), bounding box prediction (BBP), train  the vision encoder and the language encoder by image-text  contrastive learning (ITC), and train the vision encoder by  MIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' (See Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=')  The essential thinking of our learning method is that lan-  guage is more abstract than vision, and there is an asymmet-  ric relationship between language and vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Therefore, we  separate the learning of the three encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The language  encoder is trained mainly from text data and is isolated from  the training of the fusion encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The vision encoder is  simultaneously trained from image data and image-text pair  data, guided by the vision-language training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The fusion  encoder is trained from image-text pair data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Our learning method includes two new techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' One  technique is to stop gradients from the vision-language train-  ing when learning the language encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The gradient flow  is stopped from the fusion encoder to the language encoder  in training, while the activation flow from the language en-  coder to the fusion encoder is as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' As a result, the  language encoder is not affected by training of the fusion  encoder with image-text pair data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Moreover, the training of  the fusion encoder concentrates on learning the alignments  between language and vision features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  The other technique is to leverage the vision-language train-  ing to guide the learning of the vision encoder with masked  image modeling (MIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' In MIM, the masked image is com-  pared with the original image by the differences between the  predicted representations and target representations at the  masked and [CLS] positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The vision encoder creates  both the predicated and target representations, while there  is gradient flow from the predicted representations but no  gradient flow from the target representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The vision  encoder can create the target representations because it is  also trained in the vision-language training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  We conduct experiments on a variety of twenty-two tasks of  language, vision, and vision-language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' X-FM  can outperform other general foundation models by a large  margin and can even achieve better or comparable perfor-  mance than SOTA foundation models specifically designed  for language, vision, or vision-language understanding tasks,  as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Related Work  Following the success of language model pre-training, vi-  sion pre-training and vision-language pre-training with  Transformer as the backbone (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2017) have  also made significant progress recently, pushing the state-of-  the-art of various understanding tasks of language, vision,  and vision-language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  In language understanding, BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019) is  the first model adopting masked language modeling (MLM)  for pre-training, which achieves remarkable performance  on a wide range of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Several other models are then  developed to improve training robustness (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019),  sample efficiency (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Clark  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020), and prediction accuracy of BERT (Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  In vision understanding, ViT (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Tou-  vron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021) is proposed, utilizing Transformer as the  backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Inspired by MLM, subsequent work proposes  using masked image modeling (MIM) with the objective of  recovering masked images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The learning targets vary from  pixels (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022) to image tokens (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  In vision-language understanding, there are generally two  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' One is “dual encoders,” in which image and text  are encoded separately, followed by a shallow interaction  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The other is “fusion encoder(s)” in which attention  or self-attention is used to fuse information from the two  modalities after encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The former approach includes  CLIP (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021) and ALIGN (Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021)  and performs well in vision tasks and cross-modal retrieval  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' However, it cannot perform so well in multi-modal  fusion tasks such as visual question answering (VQA (Goyal  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2017)) and visual reasoning (NLVR2 (Suhr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2019b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The latter approach varies depending on the way  of using image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Early work feeds pre-extracted  object features along with texts into Transformer models  and trains the models to make multi-modal modeling and  multi-modal alignments with suitable objectives (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Tan & Bansal, 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Later work  uses patch embeddings directly with new architectures such  as vision Transformer (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 2022) or multiway  Transformer (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022) and uses  new objectives such as bounding box prediction (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Recently, the fact that Transformer can model multi-modal  data within a single architecture has inspired research to  develop general foundation models that can solve lan-  guage, vision, and vision-language tasks at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  UNIMO (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='c) jointly learns from image  and text data vision representations, language representa-  tions, and vision-language alignments in a shared space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  FLAVA (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021), a general foundation model, per-  forms pre-training with masked uni-modal and multi-modal  modeling objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' OFA (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022c) formulates  vision-language tasks as sequence-to-sequence (seq2seq)  problems and pre-trains a seq2seq model in multi-task learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' SimVLM (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021c) pre-trains a seq2seq  model with a single objective of language generation (prefix  language modeling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' DaVinci (Diao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022) combines  prefix language modeling and prefix image modeling to  learn a general foundation model for a wide range of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Uni-Perceiver (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 2022) builds a unified per-  ception architecture that processes various modalities and  tasks with a single Transformer network and shared parame-  ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Previous studies on general foundation models have shown  that different capabilities can be established with only one  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Still, few studies demonstrate that the best perfor-  mance can be achieved in all tasks with one model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' In this  paper, we propose a new general foundation model and show  that it can perform the best for all the understanding tasks  of language, vision, and vision-language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We compare our  model extensively with recent general foundation models  on multiple dimensions, as shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Several super-large foundation models (over 1B parame-  ters) are proposed recently, most of which are trained on  super-large in-house datasets (over 400M image-text pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  The authors do not report results at the base (about 280M  parameters) and large (about 800M parameters) scale on  public datasets, which we consider in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' CoCa (Yu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022) pre-trains an image-text sequence-to-sequence  model with contrastive loss and captioning loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' BEiT-  3 (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022d) uses a multi-way Transformer and a  unified objective of masked “language” modeling for learn-  ing from image (Imglish1), text, and image-text pair data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Florence (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021) first scales the web-scale image-  text pairs to 900M representations and then adapts to vari-  ous computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Flamingo (Alayrac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022)  makes use of a large language model in vision-language  pre-training to solve the “in-context learning” problem for  vision-language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' PaLI (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022) jointly scales  up the vision encoder and language encoder to cover a vari-  ety of language, vision, vision-language, and multilingual  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Method  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Model Architecture and Training Process  We propose a new general foundation model X-FM, having  a language encoder, a vision encoder, and a fusion encoder,  shown as Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The language encoder is a stack of Trans-  1They view the image as a foreign language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks  Text Encoder  Cross-Attention  Fusion Encoder  Feed Forward  Self-Attention  Vision Encoder  Feed Forward  Self-Attention  two  brown  and  white  dogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Mask  Feed Forward  Self-Attention  stop grad  MIM  MLM  IMLM, ITM, BBP  ITC  x M  x N  x L  Mask  target  Figure 1: The architecture and pre-training process of X-FM, a Transformer-based general foundation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Given  a text, we learn the language encoder by MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Given an image, we learn the vision encoder by MIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Given an image-text  pair, we learn the fusion encoder by BBP, ITM, IMLM and ITC, and further learn the vision encoder by MIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The gradients  of BBP, ITM, and IMLM are stopped from the fusion encoder to the language encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The vision encoder is trained by  MIM with both the image-text pair data and the image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' M, N and P denote numbers of encoder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  former layers like that of BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019), while  the vision encoder is a stack of Transformer layers like that  of ViT (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The language encoder uses  a post-layer-norm, while the vision encoder uses a pre-layer-  norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The fusion encoder is similar to that of ALBEF (Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021a) and X-VLM (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021), in which  each layer has an attention sub-layer after a self-attention  sub-layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' In the self-attention sub-layers, the queries are  from language and the keys & values are from vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  We propose a new method for learning X-FM, also shown in  Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Text, image, and image-text pair data are used as input  to train X-FM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The language encoder is trained by masked  language modeling (MLM) and image text contrastive learn-  ing (ITC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The vision encoder is trained by masked image  modeling (MIM) and ITC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The fusion encoder is trained  by image text matching (ITM), image-conditioned masked  language modeling (IMLM), and bounding box prediction  (BBP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' There are two new techniques developed for the  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Stop Gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We stop gradients from the vision-language  training when learning the language encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Specifically,  when the fusion encoder is trained with image-text pair  data by ITM, IMLM, and BBP, there are forward flows  (activations) from the language encoder to the fusion en-  coder, but there are no backward flows (gradients) from the  fusion encoder to the language encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' In this way, the  language encoder is only trained with text data by MLM  and with image-text pair data by ITC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The former helps the  language encoder to learn text representations, and the latter  helps the language encoder and the vision encoder to make  alignments between their respective text representations and  image representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Meanwhile, the training of the fusion  encoder is performed separately with the focus of learning  from image-text pair data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Masked Image Modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The training of vision encoder  by MIM is carried out as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The image data is first  masked and then predicted by the vision encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The dif-  ferences between predicted representations and ‘target’ rep-  resentations at masked positions and [CLS] position are  then measured with MSE (mean squared error) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The  target representations are obtained from the same image  data (without masking) by the vision encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' There are  no gradients from the target representations in the learning  of the vision encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The vision encoder can create target  representations because it is also trained with image-text  pair data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' In this way, the vision encoder is trained by both  the cross-modal objectives (ITC, ITM, BBP, IMLM) with  image-text pair data and the uni-modal objective (MIM)  with image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The representations obtained from the  vision-language training are highly semantic, which is nec-  essary for MIM as demonstrated in previous work (Bao  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  There are three advantages by exploiting the new MIM  technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' First, it becomes possible to leverage image data  for learning of the vision encoder, which is relatively easy  to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Second, it is convenient to conduct MIM with the  signals from the vision-language training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Note that most  previous work for MIM makes use of an external image  tokenizer such as VQ-VAE (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2021), CLIP (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022b), and VQ-KL (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Third, the learning of the vision encoder and that  of the fusion encoder are mutually enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Once the  vision encoder is trained, it is also utilized to train the fusion  encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Pre-training Objectives  We explain six objectives in learning of X-FM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Here, T  represents the distribution of text data, I represents the  distribution of image data, and D represents the distribution  of image-text pair data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Masked Language Modeling (MLM) We perform MLM  on text data to learn the language encoder of X-FM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Specifi-  cally we recover the masked tokens in a text by minimizing  the cross entropy loss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Lmlm = ET ∼T H(⃗y( ¯T), ˆ⃗p( ¯T))  (1)  where T denotes a text, ¯T denotes the masked text of T, ˆ⃗p  denotes the predicted probability vectors of masked tokens  of ¯T, ⃗y denotes the one-hot vectors representing the original  tokens of ¯T, and H denotes cross-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Image-Text Contrastive Learning (ITC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We use an  image-text contrastive loss as in CLIP (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021)  to learn the alignments between images and texts in ITC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Given a batch of images and texts, we calculate the cosine  similarities between all image-text pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' For each image,  there is one text matched and the rest is unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' For each  text, there is one image matched and the rest is unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  The contrastive loss is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Litc = 1  2E(I,T )∼D  �  H(⃗yi2t(I), ⃗pi2t(I))  + H(⃗yt2i(T), ⃗pt2i(T))  �  (2)  where (I, T) denotes an image-text pair, ⃗pi2t(I) denotes the  in-batch image-to-text similarities, ⃗pt2i(T) denotes the in-  batch text-to-image similarities, ⃗yi2t(I) denotes the one-hot  vectors representing the image-to-text matching relations,  ⃗yt2i(T) denotes the one-hot vectors representing the text-to-  image matching relations, and H denotes cross-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Image-Text Matching (ITM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We also learn the align-  ments between images and texts in ITM, using a loss in-  dicating whether an image-text pair is matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' For each  image in a batch there is a matched (positive) text, and we  sample an unmatched (negative) text in the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' For each  text there is a matched (positive) image, and we sample  an unmatched image in the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The loss is defined as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Litm = E(I,T )∼D  �  H(pmatch(I, T))  +H(pmatch(˜I, T))  (3)  +H(pmatch(I, ˜T))  �  where (I, T) denotes a positive image-text pair, (˜I, T) and  (I, ˜T) denote negative image-text pairs, pmatch(I, T) de-  notes a predicted matching probability of (I, T), and H  denotes logistic loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Image-conditioned  Masked  Language  Modeling  (IMLM) We conduct IMLM on image-text pair data to  learn the fusion encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Specifically, we recover the  masked tokens of the text given for an image-text pair by  minimizing the cross entropy loss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Limlm = E(I,T )∼DH(⃗y( ¯T), ˆ⃗p(I, ¯T))  (4)  where (I, T) denotes an image-text pair, ¯T denotes the  masked text of T, ˆ⃗p(I, ¯T) denotes the predicted probability  vectors of the masked tokens of ¯T based on I, ⃗y denotes the  one-hot vectors representing the original tokens of ¯T, and  H denotes cross-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Bounding Box Prediction (BBP) We adopt the BBP in X-  VLM (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 2022), which locates the visual  concept in the image by a bounding box given the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' With  BBP we learn the alignments between the images and texts  in multi-granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' In BBP, two losses are simultaneously  minimized to measure the differences between the predicted  bounding box and the ground-truth bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' One  is generalized intersection over union GIoU (Rezatofighi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019) and the other is ℓ1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Lbbp = E(I,T )∼D{GIoU(⃗b,ˆ⃗b) + ∥⃗b − ˆ⃗b∥1}  (5)  where⃗b = (cx, cy, w, h) denotes the ground truth bounding  box, ˆ⃗b = ( ˆcx, ˆcy, ˆw, ˆh) denotes the predicted bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  A bounding box is represented by two coordinates, width,  and height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Masked Image Modeling (MIM) We perform MIM on im-  age data and image-text pair data to learn the vision encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Specifically, we recover the masked image patches in an  image by minimizing the loss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Lmim = E(I,T )∼D||⃗v(¯I) − ˆ⃗v(¯I)||2 + EI∼I||⃗v(¯I) − ˆ⃗v(¯I)||2  (6)  where (I, T) and I denote an image-text pair and a single  image respectively, ¯I denotes the masked image I, ˆ⃗v(¯I) de-  notes the predicted representations at the masked positions  and [CLS] of ¯I, and ⃗v(¯I) denotes the target representa-  tions at the masked positions and [CLS] of ¯I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' ||˙||2 is the  MSE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We employ block masking following previous  work (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Note that (I, T)  and I are independently sampled from D and I, and the  sample sizes are not necessarily equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Finally, the pre-training objective of X-FM is defined as the  sum of the losses described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  L = Lmlm + Litc + Litm + Limlm + Lbbp + Lmim (7)  Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks  Base-Size Models  Large-Size Models  RoBERTa  BEiTv2  X2-VLM  UNIMO-2  FLAVA  SimVLM  OFA  DaVinci  Uni-Per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  OmniVL  X-FM  RoBERTa  BEiTv2  X2-VLM  SimVLM  OFA  Uni-Per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  X-FM  Task  Eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  MNLI  FT  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  –  –  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  –  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  –  –  –  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4  CoLA  FT  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  –  –  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  –  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='4  Vision-Language Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  –  –  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  –  –  –  –  –  –  –  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  –  –  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  –  –  –  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  Table 2: Experimental results on vision, language and vision-language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' MNLI results are average of MNLI-m  and MNLI-mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' MRPC results are average accuracies and F1 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Matthews correlation coefficient (MCC) is reported for  CoLA, and Pearson correlation coefficient (PCC) is reported for STS-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We report accuracies for all the vision and multi-  modal tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' FT is short for fine-tuning, LE for linear evaluation, ZS for zero-shot, TR for text retrieval, and IR for image  retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Results for RoBERTa are from its corresponding paper (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019), and they use the mid-training (Phang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2018) on MNLI for RTE, MRPC, and STS-B while other models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', BERT, SimVLM, DaVinci, X-FM) do not  use this trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Language Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' is the average score of all the language tasks, while Vision Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' is the average score of six  line evaluation tasks except ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Vision-Language Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' is the average score of all vision-language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' † are our  reproduced results with the officially released models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Uni-Per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' stands for Uni-Perceiver-MoE (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Pre-training Datasets  We conduct our experiments on several widely used pub-  lic datasets, which consist of two in-domain datasets,  COCO (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2014) and Visual Genome (VG) (Kr-  ishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2017), and two out-of-domain datasets, SBU  Captions (Ordonez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2011) and Conceptual Captions  (CC) (Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Following X-VLM (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 2022), we also include annotations of objects and  regions from RefCOCO (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2016), Objects365 (Shao  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019) and OpenImages (Kuznetsova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Since we assume also using uni-modal data, we include  RoBERTa corpus (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019), C4 datasets (Raffel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020) and Imagenet21K (Ridnik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' All  pre-training datasets are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Implementation Details  Pre-training  Our model is of base size and large size, and  the parameters are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The vision encoder is  initialized with BEiTv2 (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The language  encoder is initialized with RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The  fusion encoder is trained from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' X-FM is pre-trained  at image resolution of 224 × 224 with patch size of 16 × 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Dataset  # Images  # Texts  # Objects  # Regions  COCO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='11M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='55M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='45M VG  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='10M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='7M  SBU  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='86M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='58M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0M OpenImages  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2M C4 800GB RoBERTa Corpus 160GB ImageNet-21k  14M Table 3: Statistics of the pre-training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  We pre-train X-FMbase for 200K steps with a batch size of  3072 image-text pairs, 3072 images, and 8192 sentences on  32 A100 and pre-train X-FMlarge with the same batch for  160K steps on 64 A100, which takes about six days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The  learning rate for both models is warmed-up to 1e−4 in the  first 2500 steps and decayed following a linear schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We  set the maximum number of text tokens to 30 for image-text  pairs, while that of pure text corpus is set to 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We apply  mixed precision for pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Fine-tuning  We choose widely used downstream tasks  whose details are shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We report overall  Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks  Model  # Params  MSCOCO (5K test set)  Flickr30K (1K test set)  MSCOCO (5K test set)  Flickr30K (1K test set)  TR-Fine-Tune  IR-Fine-Tune  TR-Fine-Tune  IR-Fine-Tune  TR-Zero-Shot  IR-Zero-Shot  TR-Zero-Shot  IR-Zero-Shot  R@1/R@5/R@10  R@1/R@5/R@10  R@1/R@5/R@10  R@1/R@5/R@10  R@1/R@5/R@10  R@1/R@5/R@10  R@1/R@5/R@10  R@1/R@5/R@10  ALBEF  210M  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1/91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='0  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='2  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3/99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='8  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8/96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='4  –  –  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5/98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='7  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='7  VLMobase  175M  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8/93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='9  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='8  –  –  –  –  VL-BEiT  175M  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='8/–/–  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9/–/–  –  –  –  –  OmniVL  288M  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='5  –  –  –  –  Table 4: Results of text-retrieval (TR) and image-retrieval (IR) on COCO and Flickr30K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' † denotes our reproduced results  with the officially released models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Giant models with over 1B parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', BEiT-3) and models are pre-trained with  over 400M data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', CLIP and X2-VLMlarge) are in grey since they are not directly comparable with other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Model  Param  Hidden  Layers  Vision  Text  Fusion  X-FMbase  284  768  12  12  12  X-FMlarge  807  1024  24  24  12  Table 5: Size variants of X-FM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' All modules consist of  transformer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Param indicates the parameter number  of transformer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  performance on eight language tasks from GLUE (Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019), eight vision tasks following OmniVL (Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022a), four multi-modal tasks, which are text-  image retrieval on MSCOCO and Flickr, visual question  answering (VQA (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2017)) and visual reason-  ing (NLVR2 (Suhr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' For image-text retrieval  task, we report both zero-shot results and fine-tuned results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  For the ImageNet classification task, we report both linear  evaluation results and fine-tuning results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The other vision  tasks are evaluated in the linear evaluation setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' All the  other tasks are evaluated in the fine-tuning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Because  the image resolution differs between pre-training and fine-  tuning, the position parameters are adapted using linear  interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' For all downstream tasks, we apply random  resize crops and horizontal flips augmentation for the im-  ages during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' More details of network architectures  and hyper-parameters setups are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Comparison with SOTA Foundation Models  We extensively compare the performance of X-FM with  state-of-the-art foundation models on vision, language, and  multi-modal tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We first compare our model with general  foundation models, including UNIMO-v2 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021c),  FLAVA (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021), SimVLM (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021c),  OFA (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022b), DaVinci (Diao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022), Om-  niVL (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022a), and Uni-Perceiver-MoE (Zhu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We also include comparisons with SOTA  foundation models specifically designed for language, vi-  sion, or vision-language tasks, RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019),  BEiTv2 (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022), and X2-VLM (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  There are several observations in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' First, X-FMbase  (column 11) outperforms all the previous general foundation  models (column 4-10) across almost all tasks by a large mar-  gin, becoming a new and stronger general foundation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Compared to the previous general foundation models, X-  FMbase improves at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2% and the most even to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7%  on the average of all the reported numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Second, we com-  pare X-FM with state-of-the-art foundation models specif-  ically designed for language, vision, and vision-language  tasks, RoBERTa, BEiTv2 and X2-VLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We observe that  X-FM is also better than or comparable with the foundation  models at both base and large scale (column 1,2,3 vs 11 and  12,13,14 vs 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Comparison with SOTA Vision-Language Models  In addition to general foundation models, we also compare  X-FM with state-of-the-art vision-language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The re-  sults are shown in Table 4 and Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' X-FM demonstrates  its superiority on MSCOCO retrieval and NLVR2, while  achieving competitive performance on Flickr retrieval and  VQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Note that X-FMbase outperforms CLIP, ALIGN and  Florence on image-text retrieval tasks with fewer parame-  ters and much less training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Compared to the recently  released SOTA vision-language model, X2-VLM, X-FM is  much better on image-text retrieval tasks at the zero-shot  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks  X-FMbase  RoBERTa†  S-MLM  S-ITM  wostop  BEiTv2†  woMIM  wBEiTv2 Tokenizer  X2-VLM†  Multi-task  ALL  Task  Eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  1  2  3  4  5  6  7  8  9  10  MNLI  FT  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  –  –  –  –  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  CoLA  FT  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  –  –  –  –  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  MRPC  FT  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  –  –  –  –  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  QQP  FT  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  –  –  –  –  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  SST-2  FT  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  –  –  –  –  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  QNLI  FT  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  –  –  –  –  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9  RTE  FT  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  –  –  –  –  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9  STS-B  FT  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4  –  –  –  –  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8  Language Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='4  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='6  –  –  –  –  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  ImageNet  FT  –  –  –  –  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  –  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='3  ImageNet  LE  –  –  –  –  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='4  –  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='3  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  Food101  LE  –  –  –  –  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='4  Table 6: Ablation studies on vision, language, and vision-language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We use the same settings as Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' “ALL” for  X-FMbase is trained with the same data under the same settings for pre-training and fine-tuning compared to all the variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Language Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' is the average of all language tasks, while Vision Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' is the average of all vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Vision-Language  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' is the average of all vision-language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Note that performance of “ALL” is slightly different from X-FMbase in  Table 2, because we use less training steps (160k) for ablation to save the computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Ablation Study  To verify the contributions of different modules in our frame-  work, we ablate them and evaluate the performance of X-  FM on all downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The results are shown in  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We first explain several abbreviations in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  S-MLM means that we only separate the language represen-  tations in the learning IMLM task, while S-ITM means that  language representations for computing ITM and BBP are  separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' wostop indicates without stopping the gradients  of all language representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' woMIM means that we do  not learn by MIM, while wBEiTv2 tokenizer means that  we learn by MIM with the image tokenizer used in BEiTv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Multi-task is a variation that uses straightforward multi-task  learning to optimize the three encoders in X-FM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' To make  a fair comparison, we also train RoBERTa, BEiTv2 and  X2-VLM with the same data noted as RoBERTa†, BEiTv2†  and X2-VLM†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Note that we also increase the fusion layers  in X2-VLM† to make the parameter sizes comparable to  our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' RoBERTa†, BEiTv2† and X2-VLM† all have  slightly better results on average than the official ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' From  the results, we have the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  First, both designs (stop gradient and masked image mod-  eling) bring improvements, and the combination can make  further improvements on all three downstream tasks (col-  umn 10 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Second, without separated language  representations, models always perform worse on language  understanding tasks (column 10 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 2,3,4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Besides, the sep-  arate language representations in the IMLM task on image-  text data are helpful for multi-modal tasks (column 2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  As we point out in section 1, the fusion encoder can learn  better cross-modal feature alignments by IMLM task from  image-text pairs instead of utilizing text tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Although S-  ITM shows slight side effects (column 4 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 3), stopping the  gradients of language representation in the fusion encoder  is necessary to simultaneously achieve strong language un-  derstanding and vision-language understanding capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Third, the MIM task is useful for vision-language and vi-  sion learning (column 10 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Meanwhile, the targets  in our MIM task are better than the BEiTv2 tokenizer (col-  umn 10 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Four, X-FM is much better than a naive  Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks  Method  # Params  VQA  NLVR2  test-dev  test-std  dev  test-P  ALBEF  210M  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  VLMobase  175M  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='9  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='3  METER  341M  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='1  X-VLM  216M  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='4 X2-VLMbase  255M  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='1  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='5 X2-VLMlarge  593M  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content='2  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='5  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6  Table 7: Results on VQA and visual reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Giant mod-  els with over 1B parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', CoCa and BEiT-3) or  models are pre-trained with over 400M data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', SimVLM  and X2-VLMlarge) are in grey because they are not directly  comparable with other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  multi-task learning strategy for a foundation model (column  10 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Compared with the straightforward multi-task  strategy, X-FMbase improves an average of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='7%  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='6% on language, vision, and vision-language tasks,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Five, X-FM is also slightly better than foun-  dation models specifically designed for language, vision,  and vision-language tasks with the same training corpus  (column 10 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 1,5,8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Conclusion and Limitation  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Conclusion  In this work, we address the problem of how to build a  general foundation model that can perform the best for all  the understanding tasks of language, vision, and vision-  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We propose a general foundation model with two  new and effective training techniques, X-FM, to learn rich  language, vision and vision-language representations at the  same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Experimental results explicitly imply that X-FM  outperforms other general foundation models by a large  margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Moreover, X-FM can even be better or comparable  compared with the SOTA foundation models specifically  designed for language, vision, or vision-language under-  standing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Limitation  Like most existing work on foundation models, the entire  project consumed over 5 A100 GPU years on a computing  cluster with high electricity costs, although we only tested  base and large models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' There is still potential for efficiency  improvement through sparse attention (Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020)  or the lottery ticket hypothesis (Frankle & Carbin, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  We will explore the techniques to improve the training effi-  ciency and reduce the carbon footprint so that we can adhere  to the proposals on “green” deep learning (Schwartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Due to considerations of fair comparisons and computa-  tional resources, we did not try super-large models which  use at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9B or more parameters like BEITv3 (Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022d), CoCa (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022) and PaLI (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' However, scalability is also an important factor for  foundation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We leave the investigations to future  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+page_content=' arXiv preprint  arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='11869, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  and Dai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Uni-perceiver-moe: Learning sparse gen-  eralist models with conditional moes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' arXiv preprint  arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='04674, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', Wang,  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', and Dai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Uni-perceiver: Pre-training unified archi-  tecture for generic perception for zero-shot and few-shot  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' arXiv preprint arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='01522, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Comparison of Recent Foundation Models  Table 8 shows an extensive comparison of recent foundation  models and X-FM on multiple axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Previous work either (i)  perform best on uni-modal tasks (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2022) or vision-language tasks (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' (2)  target a specific uni-modal domain along with part of vision-  and-language tasks (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Diao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' or (3) target all domains  but cannot perform best on all the tasks (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Our model, X-FM, is  a general foundation model that can perform the best for  all the understanding tasks of language, vision, and vision  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Details of Downstream Tasks  Language Understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  We conduct experiments on GLUE benchmark including  MNLI (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2018), CoLA (Warstadt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019),  MRPC (Dolan & Brockett, 2005), QQP (Iyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2017),  SST-2 (Socher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2013), QNLI (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2016),  RTE (Dagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Haim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Giampiccolo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Bentivogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2009), and STS-B (Agirre  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We follow the practice of BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019) and feed the input into the language  encoder, and the hidden state of the [CLS] is fed into a  new multi-class linear classifier or regression head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Vision Understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  We conduct vision experiments on both fine-tuning and lin-  ear evaluation (linear eval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The linear evaluation follows  a common practice (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021) in self-supervised learning to evaluate  the representation quality, where the pre-trained backbone  model is frozen, and an MLP head is appended on top of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  We choose 7 popular datasets following OmnVL (Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022a):  ImageNet (Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2015),  Food101 (Bossard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2014), CIFAR10 (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2009), CIFAR100 (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2009), DTD (Cimpoi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2014), Pets (Parkhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2012) and Flowers102 (Nils-  back & Zisserman, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Vision-Language Understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Image-Text Retrieval We evaluate X-FM on both  MSCOCO and Flickr30K datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We adopt the widely  used Karpathy split (Karpathy & Li, 2015) for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Following the previous work (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 2022), we first encode images and texts separately  and calculate s(I, T) to obtain the top-k candidates, and  then use the fusion encoder to re-rank the candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Visual Question Answering The task requires the model  to predict an answer given an image and a question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We  evaluate X-FM on the VQA v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0 dataset (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Following the previous work (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021), we use  a Transformer decoder to generate answers based on the  outputs of the fusion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The decoder network shares  the same network architecture with the fusion encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Note  that we use an image resolution of 768*768 for the final  result of X-FMbase, and use an image resolution of 480*480  for X-FMlarge and X-FMbase in ablation studies for efficient  fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Visual Reasoning We evaluate X-FM on a widely used  benchmark NLVR2 (Suhr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The task allows the  model to determine whether a text describes the relations  between two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Following previous work (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=',  2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022), we formulate the triplet input into  two image-text pairs, each containing the text description  and an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We then concatenate the final output [CLS]  features of the fusion module of the two pairs to predict the  label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Details of hyper parameters  Pre-training  X-FMbase is implemented with a 12-layer  language encoder, a 12-layer vision encoder, and a 12-layer  fusion encoder, 768 dimensions for hidden states, 3072 for  intermediate size, and 128 for maximum input length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' X-  FMlarge is implemented with a 24-layer language encoder, a  24-layer vision encoder, and a 12-layer fusion encoder, 1024  dimensions for hidden states, 4096 for intermediate size, and  128 for maximum input length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We initialize the language  encoder with RoBERTa and the vision encoder with BEiTv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  The weight decay is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='01 with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  The learning rate is 1e-4 with a warm-up period for the first  2500 steps and then linearly decayed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' In each batch,  there are 3072 image-text pairs, 3072 images, and 8192 text-  only sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' We use center-crop to resize each image  to the size of 224×224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The default settings are shown in  Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Fine-tuning  The learning rate is ∈ {1e-5, 2e-5, 5e-5} and  our model is optimized by AdamW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Because the image  resolution differs between pre-training and fine-tuning, the  position parameters are adapted using linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  For all downstream tasks, we apply random resize crops  and horizontal flips augmentation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' The de-  fault settings for text classification, image classification and  vision-language understanding are shown in Tables 10, 11,  12 and 13, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Note that the resolution for VQA is  different as described in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks  Methods  Multimodal data  Pretraining Objectives  Fusion Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Target Modalities  public  dataset(s)  size  Contr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  ITM  BBP  (M/P)LM  Unimodal  ST  CT  MT  V  CV&L  MV&L  L  RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2019)  –  –  –  –  –  –  –  MLM  –  –  –  –  –  –  �  BEiTv2 (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022)  –  –  –  –  –  –  –  MIM  –  –  –  �  –  –  –  X-VLM (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' 2022)  �  Combination  5M  �  �  �  MLM  –  –  �  –  –  �  �  –  VLMo (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021a)  �  Combination  5M  �  �  –  MLM  MLM+MIM  –  –  �  –  �  �  –  CLIP (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021)  �  WebImageText  400M  �  –  –  –  –  –  –  –  �  �  –  –  ALIGN (Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021)  �  JFT  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8B  �  –  –  –  –  –  –  –  �  �  –  –  SimVLM (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021c)  �  JFT  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8B  –  –  –  PrefixLM  PrefixLM  �  –  –  ∗  –  �  �  CoCa (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022)  �  JFT  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='8B  �  –  –  LM  –  �  –  –  �  �  �  –  UNIMO-2 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021c)  �  Combination  5M  –  �  –  MLM  VCL  �  –  –  �  �  �  �  OFA (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022b)  �  Combination  15M  –  –  –  LM  LM  �  –  –  ∗  –  �  �  DaVinci (Diao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022)  �  Combination  46M  –  –  –  PrefixLM + PrefixIM  PrefixLM  �  –  –  �  –  �  �  FLAVA (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2021)  �  Combination  70M  �  �  –  MLM  MLM+MIM  �  –  –  �  �  �  �  Uni-Perceiver-MoE (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022)  �  Combination  116M  –  �  –  LM+MLM  LM+MLM+Classify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  �  –  –  �  �  �  �  X-FM  �  Combination  5M  �  �  �  MLM+MIM  MLM+MIM  –  �  –  �  �  �  �  Super-Large Models  Flamingo (Alayrac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022)  �  Combination  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='2B  –  –  –  LM  –  �  –  –  –  �  �  –  BEiT-v3 (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022d)  �  Combination  21M  –  –  –  MLM  MLM+MIM  –  –  �  ∗  �  �  –  PaLI (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=', 2022)  �  WebImageText  41B  –  –  –  LM  –  �  –  –  �  �  �  �  Table 8: Comparison of recent foundation models in different modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Contr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' indicates contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' ITM is  short for image-text matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' BBP represents boundary box prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' (M/P)LM means image-conditioned (masked/prefix)  language modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' V, CV&L, MV&L and L stand for vision tasks, cross-modal retrieval tasks, multi-modal fusion tasks  and language tasks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' ST, CT and MT are abbreviations for single Transformer, cross-attention Transformer and  multiway Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' VCL stands for visual contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' ∗ means the modality is partially targeted (SimVLM  and OFA include ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' Giant models with over 1B parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content=' BEiT-3) are in grey since they are not directly  comparable with other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  config  value  optimizer  AdamW  learning rate  1e-4  weight decay  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='01  optimizer momentum  β1, β2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='999  language batch size  8192  vision batch size  3072  vision-language batch size  3072  learning rate schedule  linear decay  warmup steps  2500  training steps  200k  augmentation  RandomResizedCrop  image res  224*224  patch size  16  text length for MLM  128  text length for IMLM  30  Table 9: Pre-training setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  config  value  optimizer  AdamW  learning rate  {1e-5, 2e-5, 5e-5}  weight decay  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  optimizer momentum  β1, β2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='999  batch size  {16, 32, 64}  learning rate schedule  linear decay  warmup ratio  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  training epochs  {5, 10, 20}  Table 10: Text classification: GLUE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  config  value  optimizer  AdamW  learning rate  [2e-5, 4e-5]  weight decay  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='01  optimizer momentum  β1, β2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='999  batch size  [256, 2048]  learning rate schedule  linear decay  warmup rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  training epochs  100  augmentation  RandomResizedCrop  image res  224*224  patch size  16  Table 11: Image classification: Linear probing setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks  config  value  optimizer  AdamW  learning rate  4e-5  minimal learning rate  1e-7  weight decay  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='01  optimizer momentum  β1, β2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='999  batch size  1024  learning rate schedule  linear decay  warmup rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  training epochs  100  augmentation  RandomResizedCrop  image res  224*224  patch size  16  label smoothing  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  mixup prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  cutmix prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='0  Table 12: ImageNet classification: Fine-tuning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='  config  value  optimizer  AdamW  learning rate  {1e-5, 2e-5, 5e-5}  weight decay  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='01  optimizer momentum  β1, β2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='999  batch size  {64, 192, 512}  learning rate schedule  linear decay  warmup rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
+page_content='1  training epochs  {10, 15, 20}  augmentation  RandomResizedCrop  image res  384*384  patch size  16  Table 13: Vision-Language understanding: fine-tuning set-  ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfagyV/content/2301.05065v1.pdf'}
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+arXiv:2301.00990v1  [math.NA]  3 Jan 2023
+The energy method for high-order invariants in shallow water wave equations
+Qifeng Zhanga, Tong Yana, Guang-hua Gaob
+aDepartment of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China
+bDepartment of Mathematics, Nanjing University of Posts and Telecommunications, Nanjing, 210096, China
+Abstract
+Third order dispersive evolution equations are widely adopted to model one-dimensional long waves and have
+extensive applications in fluid mechanics, plasma physics and nonlinear optics. Among them are the KdV equation,
+the Camassa–Holm equation and the Degasperis–Procesi equation. They share many common features such as
+complete integrability, Lax pairs and bi-Hamiltonian structure. In this paper we revisit high-order invariants
+for these three types of shallow water wave equations by the energy method in combination of a skew-adjoint
+operator (1 − ∂xx)−1. Several applications to seek high-order invariants of the Benjamin-Bona-Mahony equation,
+the regularized long wave equation and the Rosenau equation are also presented.
+Keywords: Energy method; High-order invariant; Shallow water wave equation
+1. Introduction
+A family of third order dispersive evolution equations of the form
+ut − α2uxxt + γuxxx + c0ux = (c1u2 + c2u2
+x + c3uuxx)x,
+x ∈ R, t > 0
+(1.1)
+frequently appeared in the simulation of the shallow water waves, see e.g., [1], where α, γ and ci (i = 0, 1, 2, 3) are
+real constants; u denotes a horizontal velocity field with the independent spatial variable x and temporal variable t.
+A typical such equation (1.1) with α2 = c0 = c2 = c3 = 0, c1 = 2, γ = −2 is the KdV equation
+ut − 4uux − 2uxxx = 0,
+x ∈ R, t > 0,
+(1.2)
+which describes the unidirectional propagation of waves at the free surface of shallow water under the influence
+of gravity. The first four invariants of (1.2) are respectively as (see e.g., [2], although there is a minor typo in the
+coefficient of the fourth invariant, it does not affect the reading of this classic review)
+M1 =
+�
+R
+udx,
+M2 =
+�
+R
+u2dx,
+M3 =
+�
+R
+�
+u2
+x − 2
+3u3�
+dx,
+M4 =
+�
+R
+�
+u2
+xx − 10
+3 uu2
+x + 5
+9u4�
+dx.
+Taking α2 = c3 = 1, γ = c0 = 0, c1 = − 3
+2, c2 =
+1
+2, we have another example called the Camassa–Holm
+equation [3]
+ut − uxxt + 3uux = 2uxuxx + uuxxx,
+x ∈ R, t > 0,
+(1.3)
+which models the unidirectional propagation of shallow water waves over a flat bottom. The first three invariants
+are listed as follows
+E1 =
+�
+R
+(u − uxx)dx,
+E2 = 1
+2
+�
+R
+(u2 + u2
+x)dx,
+E3 = 1
+2
+�
+R
+u(u2 + u2
+x)dx.
+The third example by assigning α2 = c2 = c3 = 1, γ = c0 = 0, c1 = −2 is called the Degasperis–Procesi
+equation
+ut − uxxt + 4uux = 3uxuxx + uuxxx,
+x ∈ R, t > 0,
+(1.4)
+∗E-mail address: zhangqifeng0504@gmail.com (Q. Zhang), tyan0320@mails.zstu.edu.cn (Tong Yan), gaogh@njupt.edu.cn (G. Gao)
+Preprint submitted to Elsevier
+January 4, 2023
+
+which can be regarded as a model for nonlinear shallow water dynamics [4]. The frequently discussed invariants
+are
+H1 =
+�
+R
+(u − uxx)dx,
+H2 =
+�
+R
+(u − uxx)vdx,
+H3 =
+�
+R
+u3dx,
+where 4v − vxx = u.
+Up to now, there have been thousands of papers focusing on the theoretical and numerical studies on these three
+equations. It is worth mentioning that the invariant-preserving property is a key index of the success for numerical
+methods. However, high-order invariants are usually difficult to preserve numerically. Liu et al. also pointed out “it
+appears a rather difficult task to preserve all three conservation laws” in [5]. In this work, higher-order invariants of
+these equations will be re-derived in view of the energy method, which may be possible to provide some thoughts
+for invariant-preserving numerical methods. Actually, the energy method originated from conservation laws in
+physics was first proposed in 1928 by Courant, Friedrichs and Lewy [6]. From then on, it has been widely applied
+to the mathematical and numerical analysis of nonlinear evolution equations. We trust the readers with [7] instead
+of a long list of references to relevant works.
+The rest of the paper is arranged as follows.
+In Section 2, combining the energy method and a skew-
+adjoint operator, we show the high-order invariants for the KdV equation, the Camassa–Holm equation and the
+Degasperis–Procesi equation, respectively. Then we list several applications for seeking some high-order invariants
+of other types of the shallow water wave equations in Section 3.
+2. Main results
+In what follows, we directly show that Mi (i = 1, 2, 3, 4), Ei (i = 1, 2, 3) and Hi (i = 1, 2, 3) are invariants of
+(1.2), (1.3) and (1.4) subjected to the periodic boundary conditions based on the energy method, respectively.
+2.1. Invariants of the KdV equation
+Proof: (I) Multiplying by 1, u and (u2 + uxx), respectively, with (1.2), we have Mi (i = 1, 2, 3). In what follows, we
+show the fourth invariant M4 of the KdV equation by the energy method.
+Multiplying both sides of (1.2) by 2uxxxx + 10
+3 u2
+x + 20
+3 uuxx + 20
+9 u3 and integrating the result, we have
+0 =
+�
+R
+�
+2uxxxx + 10
+3 u2
+x + 20
+3 uuxx + 20
+9 u3�
+· utdx
+−
+�
+R
+�
+2uxxxx + 10
+3 u2
+x + 20
+3 uuxx + 20
+9 u3�
+· (4uux + 2uxxx)dx
+=
+�
+R
+�
+2uxxuxxt − 10
+3 (utu2
+x + 2uuxuxt) + 20
+9 u3ut
+�
+dx
+−
+�
+R
+�
+2uxxxx + 10
+3 u2
+x + 20
+3 uuxx + 20
+9 u3�
+· (4uux + 2uxxx)dx
+= d
+dt M4 − 8
+�
+R
+uuxuxxxxdx − 40
+3
+�
+R
+uu3
+xdx − 80
+3
+�
+R
+u2uxuxxdx − 80
+9
+�
+R
+u4uxdx
+− 4
+�
+R
+uxxxuxxxxdx − 20
+3
+�
+R
+uxxxu2
+xdx − 40
+3
+�
+R
+uuxxuxxxdx − 40
+9
+�
+R
+u3uxxxdx.
+(2.1)
+It remains to check that the sum of all the integral terms in the above equation is zero. Calculating each term in
+(2.1) using the integration by parts, we have
+− 8
+�
+R
+uuxuxxxxdx = −20
+�
+R
+uxu2
+xxdx,
+(2.2)
+− 80
+3
+�
+R
+u2uxuxxdx = 80
+3
+�
+R
+uu3
+xdx,
+(2.3)
+− 80
+9
+�
+R
+u4uxdx = 0,
+(2.4)
+− 4
+�
+R
+uxxxuxxxxdx = 0,
+(2.5)
+2
+
+− 20
+3
+�
+R
+uxxxu2
+xdx = 40
+3
+�
+R
+uxu2
+xxdx,
+(2.6)
+− 40
+3
+�
+R
+uuxxuxxxdx = 20
+3
+�
+R
+uxu2
+xxdx,
+(2.7)
+− 40
+9
+�
+R
+u3uxxxdx = −40
+3
+�
+R
+uu3
+xdx.
+(2.8)
+Substituting (2.2)–(2.8) into (2.1), we have d
+dt M4 = 0, which completes the proof.
+Remark 1. Suppose the general form of the KdV equation is
+ut − auux − buxxx = 0,
+and the corresponding high-order invariant
+M(t) =
+�
+R
+(u2
+xx − Auu2
+x + Bu4)dx.
+Using the same method above, we could derive
+� 5a = 3Ab,
+12Bb = Aa,
+which can be rewritten as
+a
+b = 3A
+5 = 12B
+A .
+Therefore, it follows
+A2 = 20B.
+For instance, when a = −6, b = −1, we have A = 10, B = 5, which deduces to the KdV equation as
+ut + 6uux + uxxx = 0,
+with a fourth-order invariant
+M(t) =
+�
+R
+(u2
+xx − 10uu2
+x + 5u4)dx.
+2.2. Invariants of the Camassa–Holm equation
+Proof: Multiplying by 1 and u on both sides of (1.3), respectively, and then integrating the results, which implies
+E1 and E2 through the integration by parts. Below, we prove E3 by the energy method. Firstly, noticing that (1.3)
+can be written with a skew-adjoint operator (1 − ∂xx)−1 as
+ut + uux + ∂x(1 − ∂xx)−1�
+u2 + 1
+2u2
+x
+�
+= 0.
+Let g = (1 − ∂xx)−1�
+u2 + 1
+2u2
+x
+�
+. Then we see from the above equation that (1.3) is equivalent to
+
+ut + uux + gx = 0,
+(2.9)
+g − gxx = u2 + 1
+2u2
+x.
+(2.10)
+Multiplying (2.9) by 3u2 + u2
+x − 2(uux)x and integrating the result on both sides, we have
+0 =
+�
+R
+(ut + uux + gx) · (3u2 + u2
+x − 2(uux)x)dx
+=
+�
+R
+ut · (3u2 + u2
+x − 2(uux)x)dx +
+�
+R
+(uux + gx) · (3u2 + u2
+x − 2(uux)x)dx
+≜ A + B.
+(2.11)
+3
+
+Calculating each term derives that
+A =
+�
+R
+ut · (3u2 + u2
+x − 2(uux)x)dx
+=
+�
+R
+ut · (3u2 + u2
+x)dx +
+�
+R
+2uux · uxtdx
+=
+�
+R
+ut · 3u2dx +
+�
+R
+ut · u2
+xdx +
+�
+R
+u · (u2
+x)tdx
+=
+�
+R
+(u3)tdx +
+�
+R
+(u · u2
+x)tdx
+= d
+dt
+�
+R
+(u3 + uu2
+x)dx
+(2.12)
+and
+B =
+�
+R
+(uux + gx) · (3u2 + u2
+x − 2(uux)x)dx
+=
+�
+R
+u · u3
+xdx +
+�
+R
+gx · (3u2 + u2
+x)dx −
+�
+R
+gx · 2(uux)xdx
+=
+�
+R
+u · u3
+xdx +
+�
+R
+gx · (3u2 + u2
+x)dx + 2
+�
+R
+gxx · uuxdx
+=
+�
+R
+u · u3
+xdx +
+�
+R
+gx · (3u2 + u2
+x)dx + 2
+�
+R
+(g − u2 − 1
+2u2
+x) · uuxdx
+=
+�
+R
+gx · (3u2 + u2
+x)dx + 2
+�
+R
+g · uuxdx
+=
+�
+R
+gx · (3u2 + u2
+x)dx −
+�
+R
+gx · u2dx
+=
+�
+R
+gx · (2u2 + u2
+x)dx
+= 2
+�
+R
+gx · (g − gxx)dx = 0.
+(2.13)
+Substituting (2.12) and (2.13) into (2.11), we have
+d
+dt
+�
+R
+(u3 + uu2
+x)dx = 0,
+which implies E3.
+2.3. Invariants of the Degasperis–Procesi equation
+Proof: Integrating on both sides of (1.4), it easily obtains H1. Then we show invariants H2 and H3 of (1.4),
+respectively. Firstly let g = (1 − ∂xx)−1� 3
+2u2�
+, then (1.4) is equivalent to
+
+ut + uux + gx = 0,
+(2.14)
+g − gxx = 3
+2u2.
+(2.15)
+Multiplying by 2u − 6v on both sides of (2.14) and then integrating the result, we have
+0 =
+�
+R
+(ut + uux + gx) · (2u − 6v)dx
+=
+�
+R
+ut · (2u − 6v)dx +
+�
+R
+uux · (2u − 6v)dx +
+�
+R
+gx · (2u − 6v)dx
+≜ C + D.
+(2.16)
+4
+
+The each term in the above identity is estimated as
+C =
+�
+R
+ut · (2u − 6v)dx = 2
+�
+R
+ut · udx − 6
+�
+R
+ut · vdx = 2
+�
+R
+ut · udx − 6
+�
+R
+(4vt − vxxt) · vdx
+= 2
+�
+R
+ut · udx − 24
+�
+R
+vt · vdx − 6
+�
+R
+vxt · vxdx = d
+dt
+�
+R
+(u2 − 12v2 − 3v2
+x)dx
+= d
+dt
+�
+R
+�
+u2 − 3(4v − vxx) · v
+�
+dx = d
+dt
+�
+R
+(u2 − 3uv)dx = d
+dt
+�
+R
+u · (u − 3v)dx
+= d
+dt
+�
+R
+u · (v − vxx)dx = d
+dt
+�
+R
+(u − uxx) · vdx
+(2.17)
+and
+D =
+�
+R
+uux · (2u − 6v)dx +
+�
+R
+gx · (2u − 6v)dx
+= −6
+�
+R
+uux · vdx +
+�
+R
+gx · (2u − 6v)dx
+= 3
+�
+R
+u2 · vxdx +
+�
+R
+gx · (2u − 6v)dx
+= 2
+�
+R
+(g − gxx) · vxdx +
+�
+R
+gx · (2u − 6v)dx
+= 2
+�
+R
+g · vxdx − 2
+�
+R
+gxx · vxdx +
+�
+R
+gx · (2v − 2vxx)dx
+= 2
+�
+R
+g · vxdx + 2
+�
+R
+gx · vdx − 2
+�
+R
+gxx · vxdx − 2
+�
+R
+gx · vxxdx
+= 2
+�
+R
+(gv)xdx − 2
+�
+R
+(gx · vx)xdx = 0.
+(2.18)
+Substituting (2.17) and (2.18) into (2.16), we have
+d
+dt
+�
+R
+(u − uxx) · vdx = 0,
+which implies H2.
+Finally, we show H3. Multiplying (2.14) on both sides by u2 and integrating the result, it yields by noting (2.15)
+0 =
+�
+R
+(ut + uux + gx) · u2dx
+=
+�
+R
+ut · u2dx +
+�
+R
+u3 · uxdx +
+�
+R
+gx · u2dx
+=
+�
+R
+�1
+3u3�
+tdx + 2
+3
+�
+gx · (g − gxx)dx
+= 1
+3
+d
+dt
+�
+R
+u3dx,
+which implies the invariant H3.
+3. Applications to other periodic nonlinear dispersive waves
+3.1. Benjamin-Bona-Mahony equation
+Consider the Benjamin-Bona-Mahony equation [8] of the form
+ut − uxxt + ux + εuux = 0,
+x ∈ R.
+(3.1)
+5
+
+It can be written as
+ut + ∂x(1 − ∂xx)−1�
+u + ε
+2u2�
+= 0,
+x ∈ R.
+Let g = (1 − ∂xx)−1�
+u + ε
+2u2�
+, then the equation (3.1) turns out to be
+
+ut + gx = 0,
+(3.2)
+g − gxx = u + ε
+2u2.
+(3.3)
+Multiplying both sides of (3.2) by u2 and integrating the result, and then using (3.3), we have
+0 =
+�
+R
+(ut + gx) · u2dx =
+�
+R
+ut · u2dx +
+�
+R
+gx · u2dx
+=
+�
+R
+ut · u2dx + 2
+ε
+�
+R
+gx · (g − gxx − u)dx =
+�
+R
+ut · u2dx − 2
+ε
+�
+R
+gx · udx
+=
+�
+R
+ut · u2dx + 2
+ε
+�
+R
+ut · udx = d
+dt
+�
+R
+�1
+3u3 + 1
+εu2�
+dx,
+which indicates
+�
+R
+1
+3
+�
+u3 + 1
+εu2�
+dx
+is a three-order invariant for (3.1).
+3.2. Regularized long wave equation
+Consider the regularized long wave equation [9] of the form
+ut − µuxxt + ux + upux = 0,
+(3.4)
+where µ > 0 is a positive constant. When p = 2, it is called modified regularized long wave equation; when p ⩾ 3,
+it is called generalized regularized long wave equation. Similar to the foregoing argument, (3.4) can be written as
+an equivalent form of
+
+ut + gx = 0,
+(3.5)
+g − µgxx = u +
+1
+p + 1up+1.
+(3.6)
+Multiplying both sides of (3.5) by up+1, integrating the result, and then using (3.6), we have
+0 =
+�
+R
+(ut + gx) · up+1dx =
+�
+R
+ut · up+1dx +
+�
+R
+gx · up+1dx
+=
+�
+R
+ut · up+1dx + (p + 1)
+�
+R
+gx · (g − µgxx − u)dx
+=
+�
+R
+ut · up+1dx − (p + 1)
+�
+R
+gx · udx
+=
+�
+R
+ut · up+1dx + (p + 1)
+�
+R
+ut · udx
+= d
+dt
+�
+R
+�
+1
+p + 2up+2 + p + 1
+2
+u2�
+dx,
+which indicates
+�
+R
+�
+1
+p + 2up+2 + p + 1
+2
+u2�
+dx
+is a high-order invariant for (3.4). This corrects an invariant I3 in Example 4 appeared in [10] (pp. 492).
+6
+
+3.3. Rosenau equation
+Consider the Rosenau equation [11]
+ut + uxxxxt + ux + uux = 0,
+(3.7)
+which is equivalent to
+
+ut + gx = 0,
+(3.8)
+g + gxxxx = u + 1
+2u2.
+(3.9)
+Multiplying both sides of (3.8) by u2 and noticing (3.9), similar to the argument in the above, we have a third-order
+invariant for (3.7) of the form
+�
+R
+�1
+3u3 + u2�
+dx.
+Acknowledgement
+We appreciate Prof. Zhi-zhong Sun for many useful discussions. This work is dedicated to Prof. Zhi-zhong Sun
+on the occasion of his 60th birthday. The work is supported by Natural Science Foundation of Zhejiang Province
+(Grant No. LZ23A010007).
+References
+References
+[1] J. Escher, Y. Liu, Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation. J. Funct. Anal., 241 (2006)
+457–485.
+[2] T. Tao, Low-regularity global solutions to nonlinear dispersive equations. Surveys in analysis and operator theory (Canberra, 2001),
+19–48, Proc. Centre Math. Appl. Austral. Nat. Univ., 40, Austral. Nat. Univ., Canberra, (2002).
+[3] R. Camassa, D. D. Holm, An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71 (1993) 1661–1664.
+[4] A. Degasperis, M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.). Symmetry and Perturbation Theory, World
+Scientific, Singapore, (1999) 23–37.
+[5] H. Liu, Y. Xing, An invariant preserving discontinuous Galerkin method for the Camassa-Holm equation. SIAM J. Sci. Comput., 38
+(2016) A1919–A1934.
+[6] R. Courant, K.O. Friedrichs, H. Lewy, ¨Uber die partiellen Differenzenglei-chungen der mathematischen physik, Math. Ann., 100 (1928)
+32–74.
+[7] Z. Sun. Finite Difference Methods for Nonlinear Evolution Equations, Science Press, Beijing, (2018).
+[8] L.A. Medeiros, G.P. Menzala, Existence and uniqueness for periodic solutions of the Benjamin-Bona-Mahony equation. SIAM J. Math.
+Anal., 8(5) (1977) 792–799.
+[9] C.E. Seyler, D.L. Fenstermacher, A symmetric regularized-long-wave equation. The Physics of Fluids, 27(4) (1984) 4–7.
+[10] A. Ghiloufi, K. Omrani, New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear
+dispersive waves. Numer. Methods Partial Differential Equation, 34 (2018) 451–500.
+[11] M.A. Park, On the Rosenau equation. Math. Appl. Comput., 9 (1990) 145–152.
+7
+
diff --git a/49AzT4oBgHgl3EQfEPqD/content/tmp_files/load_file.txt b/49AzT4oBgHgl3EQfEPqD/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
+++ b/49AzT4oBgHgl3EQfEPqD/content/tmp_files/load_file.txt
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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='00990v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='NA]  3 Jan 2023  The energy method for high-order invariants in shallow water wave equations  Qifeng Zhanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Tong Yana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Guang-hua Gaob  aDepartment of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Zhejiang Sci-Tech University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' 310018,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' China  bDepartment of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Nanjing University of Posts and Telecommunications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' 210096,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' China  Abstract  Third order dispersive evolution equations are widely adopted to model one-dimensional long waves and have  extensive applications in fluid mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' plasma physics and nonlinear optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Among them are the KdV equation,  the Camassa–Holm equation and the Degasperis–Procesi equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' They share many common features such as  complete integrability, Lax pairs and bi-Hamiltonian structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' In this paper we revisit high-order invariants  for these three types of shallow water wave equations by the energy method in combination of a skew-adjoint  operator (1 − ∂xx)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Several applications to seek high-order invariants of the Benjamin-Bona-Mahony equation,  the regularized long wave equation and the Rosenau equation are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Keywords: Energy method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' High-order invariant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Shallow water wave equation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Introduction  A family of third order dispersive evolution equations of the form  ut − α2uxxt + γuxxx + c0ux = (c1u2 + c2u2  x + c3uuxx)x,  x ∈ R, t > 0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1)  frequently appeared in the simulation of the shallow water waves, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=', [1], where α, γ and ci (i = 0, 1, 2, 3) are  real constants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' u denotes a horizontal velocity field with the independent spatial variable x and temporal variable t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  A typical such equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1) with α2 = c0 = c2 = c3 = 0, c1 = 2, γ = −2 is the KdV equation  ut − 4uux − 2uxxx = 0,  x ∈ R, t > 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2)  which describes the unidirectional propagation of waves at the free surface of shallow water under the influence  of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' The first four invariants of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2) are respectively as (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=', [2], although there is a minor typo in the  coefficient of the fourth invariant, it does not affect the reading of this classic review)  M1 =  �  R  udx,  M2 =  �  R  u2dx,  M3 =  �  R  �  u2  x − 2  3u3�  dx,  M4 =  �  R  �  u2  xx − 10  3 uu2  x + 5  9u4�  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Taking α2 = c3 = 1, γ = c0 = 0, c1 = − 3  2, c2 =  1  2, we have another example called the Camassa–Holm  equation [3]  ut − uxxt + 3uux = 2uxuxx + uuxxx,  x ∈ R, t > 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3)  which models the unidirectional propagation of shallow water waves over a flat bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' The first three invariants  are listed as follows  E1 =  �  R  (u − uxx)dx,  E2 = 1  2  �  R  (u2 + u2  x)dx,  E3 = 1  2  �  R  u(u2 + u2  x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  The third example by assigning α2 = c2 = c3 = 1, γ = c0 = 0, c1 = −2 is called the Degasperis–Procesi  equation  ut − uxxt + 4uux = 3uxuxx + uuxxx,  x ∈ R, t > 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4)  ∗E-mail address: zhangqifeng0504@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='com (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Zhang), tyan0320@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='zstu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='cn (Tong Yan), gaogh@njupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='cn (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Gao)  Preprint submitted to Elsevier  January 4, 2023  which can be regarded as a model for nonlinear shallow water dynamics [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' The frequently discussed invariants  are  H1 =  �  R  (u − uxx)dx,  H2 =  �  R  (u − uxx)vdx,  H3 =  �  R  u3dx,  where 4v − vxx = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Up to now, there have been thousands of papers focusing on the theoretical and numerical studies on these three  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' It is worth mentioning that the invariant-preserving property is a key index of the success for numerical  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' However, high-order invariants are usually difficult to preserve numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' also pointed out “it  appears a rather difficult task to preserve all three conservation laws” in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' In this work, higher-order invariants of  these equations will be re-derived in view of the energy method, which may be possible to provide some thoughts  for invariant-preserving numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Actually, the energy method originated from conservation laws in  physics was first proposed in 1928 by Courant, Friedrichs and Lewy [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' From then on, it has been widely applied  to the mathematical and numerical analysis of nonlinear evolution equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' We trust the readers with [7] instead  of a long list of references to relevant works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  The rest of the paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  In Section 2, combining the energy method and a skew-  adjoint operator, we show the high-order invariants for the KdV equation, the Camassa–Holm equation and the  Degasperis–Procesi equation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Then we list several applications for seeking some high-order invariants  of other types of the shallow water wave equations in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Main results  In what follows, we directly show that Mi (i = 1, 2, 3, 4), Ei (i = 1, 2, 3) and Hi (i = 1, 2, 3) are invariants of  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4) subjected to the periodic boundary conditions based on the energy method, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Invariants of the KdV equation  Proof: (I) Multiplying by 1, u and (u2 + uxx), respectively, with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2), we have Mi (i = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' In what follows, we  show the fourth invariant M4 of the KdV equation by the energy method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Multiplying both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2) by 2uxxxx + 10  3 u2  x + 20  3 uuxx + 20  9 u3 and integrating the result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' we have  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='0 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2uxxxx + 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='x + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 uuxx + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9 u3�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='utdx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2uxxxx + 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='x + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 uuxx + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9 u3�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='(4uux + 2uxxx)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2uxxuxxt − 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 (utu2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='x + 2uuxuxt) + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9 u3ut  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2uxxxx + 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='x + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 uuxx + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9 u3�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='(4uux + 2uxxx)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='= d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='dt M4 − 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='uuxuxxxxdx − 40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='uu3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='xdx − 80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='u2uxuxxdx − 80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='u4uxdx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='− 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='uxxxuxxxxdx − 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='uxxxu2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='xdx − 40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='uuxxuxxxdx − 40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='u3uxxxdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1)  It remains to check that the sum of all the integral terms in the above equation is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Calculating each term in  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1) using the integration by parts, we have  − 8  �  R  uuxuxxxxdx = −20  �  R  uxu2  xxdx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2)  − 80  3  �  R  u2uxuxxdx = 80  3  �  R  uu3  xdx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3)  − 80  9  �  R  u4uxdx = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4)  − 4  �  R  uxxxuxxxxdx = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='5)  2  − 20  3  �  R  uxxxu2  xdx = 40  3  �  R  uxu2  xxdx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='6)  − 40  3  �  R  uuxxuxxxdx = 20  3  �  R  uxu2  xxdx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='7)  − 40  9  �  R  u3uxxxdx = −40  3  �  R  uu3  xdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='8)  Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='8) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1), we have d  dt M4 = 0, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Suppose the general form of the KdV equation is  ut − auux − buxxx = 0,  and the corresponding high-order invariant  M(t) =  �  R  (u2  xx − Auu2  x + Bu4)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Using the same method above, we could derive  � 5a = 3Ab,  12Bb = Aa,  which can be rewritten as  a  b = 3A  5 = 12B  A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Therefore, it follows  A2 = 20B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  For instance, when a = −6, b = −1, we have A = 10, B = 5, which deduces to the KdV equation as  ut + 6uux + uxxx = 0,  with a fourth-order invariant  M(t) =  �  R  (u2  xx − 10uu2  x + 5u4)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Invariants of the Camassa–Holm equation  Proof: Multiplying by 1 and u on both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3), respectively, and then integrating the results, which implies  E1 and E2 through the integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Below, we prove E3 by the energy method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Firstly, noticing that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3)  can be written with a skew-adjoint operator (1 − ∂xx)−1 as  ut + uux + ∂x(1 − ∂xx)−1�  u2 + 1  2u2  x  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Let g = (1 − ∂xx)−1�  u2 + 1  2u2  x  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Then we see from the above equation that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3) is equivalent to  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  ut + uux + gx = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9)  g − gxx = u2 + 1  2u2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='10)  Multiplying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9) by 3u2 + u2  x − 2(uux)x and integrating the result on both sides, we have  0 =  �  R  (ut + uux + gx) · (3u2 + u2  x − 2(uux)x)dx  =  �  R  ut · (3u2 + u2  x − 2(uux)x)dx +  �  R  (uux + gx) · (3u2 + u2  x − 2(uux)x)dx  ≜ A + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='11)  3  Calculating each term derives that  A =  �  R  ut · (3u2 + u2  x − 2(uux)x)dx  =  �  R  ut · (3u2 + u2  x)dx +  �  R  2uux · uxtdx  =  �  R  ut · 3u2dx +  �  R  ut · u2  xdx +  �  R  u · (u2  x)tdx  =  �  R  (u3)tdx +  �  R  (u · u2  x)tdx  = d  dt  �  R  (u3 + uu2  x)dx  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='12)  and  B =  �  R  (uux + gx) · (3u2 + u2  x − 2(uux)x)dx  =  �  R  u · u3  xdx +  �  R  gx · (3u2 + u2  x)dx −  �  R  gx · 2(uux)xdx  =  �  R  u · u3  xdx +  �  R  gx · (3u2 + u2  x)dx + 2  �  R  gxx · uuxdx  =  �  R  u · u3  xdx +  �  R  gx · (3u2 + u2  x)dx + 2  �  R  (g − u2 − 1  2u2  x) · uuxdx  =  �  R  gx · (3u2 + u2  x)dx + 2  �  R  g · uuxdx  =  �  R  gx · (3u2 + u2  x)dx −  �  R  gx · u2dx  =  �  R  gx · (2u2 + u2  x)dx  = 2  �  R  gx · (g − gxx)dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='13)  Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='12) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='13) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='11), we have  d  dt  �  R  (u3 + uu2  x)dx = 0,  which implies E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Invariants of the Degasperis–Procesi equation  Proof: Integrating on both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4), it easily obtains H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Then we show invariants H2 and H3 of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Firstly let g = (1 − ∂xx)−1� 3  2u2�  , then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4) is equivalent to  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  ut + uux + gx = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='14)  g − gxx = 3  2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='15)  Multiplying by 2u − 6v on both sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='14) and then integrating the result, we have  0 =  �  R  (ut + uux + gx) · (2u − 6v)dx  =  �  R  ut · (2u − 6v)dx +  �  R  uux · (2u − 6v)dx +  �  R  gx · (2u − 6v)dx  ≜ C + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='16)  4  The each term in the above identity is estimated as  C =  �  R  ut · (2u − 6v)dx = 2  �  R  ut · udx − 6  �  R  ut · vdx = 2  �  R  ut · udx − 6  �  R  (4vt − vxxt) · vdx  = 2  �  R  ut · udx − 24  �  R  vt · vdx − 6  �  R  vxt · vxdx = d  dt  �  R  (u2 − 12v2 − 3v2  x)dx  = d  dt  �  R  �  u2 − 3(4v − vxx) · v  �  dx = d  dt  �  R  (u2 − 3uv)dx = d  dt  �  R  u · (u − 3v)dx  = d  dt  �  R  u · (v − vxx)dx = d  dt  �  R  (u − uxx) · vdx  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='17)  and  D =  �  R  uux · (2u − 6v)dx +  �  R  gx · (2u − 6v)dx  = −6  �  R  uux · vdx +  �  R  gx · (2u − 6v)dx  = 3  �  R  u2 · vxdx +  �  R  gx · (2u − 6v)dx  = 2  �  R  (g − gxx) · vxdx +  �  R  gx · (2u − 6v)dx  = 2  �  R  g · vxdx − 2  �  R  gxx · vxdx +  �  R  gx · (2v − 2vxx)dx  = 2  �  R  g · vxdx + 2  �  R  gx · vdx − 2  �  R  gxx · vxdx − 2  �  R  gx · vxxdx  = 2  �  R  (gv)xdx − 2  �  R  (gx · vx)xdx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='18)  Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='18) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='16), we have  d  dt  �  R  (u − uxx) · vdx = 0,  which implies H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Finally, we show H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Multiplying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='14) on both sides by u2 and integrating the result, it yields by noting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='15)  0 =  �  R  (ut + uux + gx) · u2dx  =  �  R  ut · u2dx +  �  R  u3 · uxdx +  �  R  gx · u2dx  =  �  R  �1  3u3�  tdx + 2  3  �  gx · (g − gxx)dx  = 1  3  d  dt  �  R  u3dx,  which implies the invariant H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Applications to other periodic nonlinear dispersive waves  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Benjamin-Bona-Mahony equation  Consider the Benjamin-Bona-Mahony equation [8] of the form  ut − uxxt + ux + εuux = 0,  x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1)  5  It can be written as  ut + ∂x(1 − ∂xx)−1�  u + ε  2u2�  = 0,  x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Let g = (1 − ∂xx)−1�  u + ε  2u2�  , then the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1) turns out to be  \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3  ut + gx = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2)  g − gxx = u + ε  2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3)  Multiplying both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2) by u2 and integrating the result, and then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3), we have  0 =  �  R  (ut + gx) · u2dx =  �  R  ut · u2dx +  �  R  gx · u2dx  =  �  R  ut · u2dx + 2  ε  �  R  gx · (g − gxx − u)dx =  �  R  ut · u2dx − 2  ε  �  R  gx · udx  =  �  R  ut · u2dx + 2  ε  �  R  ut · udx = d  dt  �  R  �1  3u3 + 1  εu2�  dx,  which indicates  �  R  1  3  �  u3 + 1  εu2�  dx  is a three-order invariant for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Regularized long wave equation  Consider the regularized long wave equation [9] of the form  ut − µuxxt + ux + upux = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4)  where µ > 0 is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' When p = 2, it is called modified regularized long wave equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' when p ⩾ 3,  it is called generalized regularized long wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Similar to the foregoing argument, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4) can be written as  an equivalent form of  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  ut + gx = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='5)  g − µgxx = u +  1  p + 1up+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='6)  Multiplying both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='5) by up+1, integrating the result, and then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='6), we have  0 =  �  R  (ut + gx) · up+1dx =  �  R  ut · up+1dx +  �  R  gx · up+1dx  =  �  R  ut · up+1dx + (p + 1)  �  R  gx · (g − µgxx − u)dx  =  �  R  ut · up+1dx − (p + 1)  �  R  gx · udx  =  �  R  ut · up+1dx + (p + 1)  �  R  ut · udx  = d  dt  �  R  �  1  p + 2up+2 + p + 1  2  u2�  dx,  which indicates  �  R  �  1  p + 2up+2 + p + 1  2  u2�  dx  is a high-order invariant for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' This corrects an invariant I3 in Example 4 appeared in [10] (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' 492).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Rosenau equation  Consider the Rosenau equation [11]  ut + uxxxxt + ux + uux = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='7)  which is equivalent to  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  ut + gx = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='8)  g + gxxxx = u + 1  2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9)  Multiplying both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='8) by u2 and noticing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9), similar to the argument in the above, we have a third-order  invariant for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='7) of the form  �  R  �1  3u3 + u2�  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Acknowledgement  We appreciate Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Zhi-zhong Sun for many useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' This work is dedicated to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Zhi-zhong Sun  on the occasion of his 60th birthday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' The work is supported by Natural Science Foundation of Zhejiang Province  (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' LZ23A010007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  References  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Escher, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
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+page_content='  [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Tao, Low-regularity global solutions to nonlinear dispersive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Surveys in analysis and operator theory (Canberra, 2001),  19–48, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Centre Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Austral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=', 40, Austral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=', Canberra, (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Camassa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Holm, An integrable shallow water equation with peaked solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=', 71 (1993) 1661–1664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Degasperis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Procesi, Asymptotic integrability, in: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Degasperis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Gaeta (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Symmetry and Perturbation Theory, World  Scientific, Singapore, (1999) 23–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Xing, An invariant preserving discontinuous Galerkin method for the Camassa-Holm equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=', 38  (2016) A1919–A1934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
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+Quantum sensing of electric field distributions of liquid electrolytes with NV-centers
+in nanodiamonds
+M. Hollendonner,1, 2 S. Sharma,2 D. B. R. Dasari,3 A. Finkler,4 S. V. Kusminskiy,2, 5 and R. Nagy1, ∗
+1Friedrich-Alexander-University Erlangen-Nuremberg, 91058 Erlangen, Germany
+2Max Planck Institute for the Science of Light, 91058 Erlangen, Germany
+33rd Institute of Physics, IQST, and Research Center SCoPE,
+University of Stuttgart, 70569 Stuttgart, Germany
+4Department of Chemical and Biological Physics,
+Weizmann Institute of Science, Rehovot 7610001, Israel
+5Institute for Theoretical Solid State Physics, RWTH Aachen University, 52074 Aachen, Germany
+(Dated: January 12, 2023)
+To use batteries as large-scale energy storage systems it is necessary to measure and understand
+their degradation in-situ and in-operando. As a battery’s degradation is often the result of molecular
+processes inside the electrolyte, a sensing platform which allows to measure the ions with a high
+spatial resolution is needed. Primary candidates for such a platform are NV-centers in diamonds.
+We propose to use a single NV-center to deduce the electric field distribution generated by the
+ions inside the electrolyte through microwave pulse sequences. We show that the electric field can
+be reconstructed with great accuracy by using a protocol which includes different variations of the
+Free Induction Decay to obtain the mean electric field components and a modified Hahn-echo pulse
+sequence to measure the electric field’s standard deviation σE. From a semi-analytical ansatz we find
+that for a lithium ion battery there is a direct relationship between σE and the ionic concentration.
+Our results show that it is therefore possible to use NV-centers as sensors to measure both the
+electric field distribution and the local ionic concentration inside electrolytes.
+I.
+INTRODUCTION
+Rechargeable batteries play an important role for our
+society and are a key ingredient for the transition to-
+wards renewable energy sources [1–3]. As the production
+of batteries is accompanied with a considerable use of re-
+sources, recyclable [4] batteries with a long lifetime are
+needed. The latter is limited by degradation mechanisms,
+such as the formation of solid-electrolyte interfaces [5] or
+lithium-plating [6] which can reduce the battery’s capac-
+ity with increasing cell age [7]. As these processes happen
+on a molecular level within nanometer scales [5], a sensor
+which is capable of monitoring the ionic concentration
+in-situ and in-operando with high spatial and temporal
+resolutions is needed. Even though MRI allows to recon-
+struct transport properties [8, 9] of a battery, tools which
+allow to perform measurements inside the electrolyte are
+still absent [5].
+It has been demonstrated that nitrogen-vacancy (NV)
+centers in diamond (see Fig. 1(b)) are high-resolution
+quantum sensors, which can detect oscillating or fluctu-
+ating [10–13] magnetic fields with nano- [14, 15] and even
+subpico-Tesla [16] sensitivities. Besides this, NV-centers
+have great ability for the detection of electric fields. They
+can not only detect DC [17, 18] or AC [19] electric fields
+with remarkable precision, but are additionally capable
+of detecting single fundamental charges [20] even within
+the diamond lattice [21]. This electric field sensitivity was
+used by Ref. [22] to show that, based on theoretical con-
+∗ roland.nagy@fau.de
+siderations, bulk NV-centers can work as electrochemical
+sensors if they are in contact with an electrolyte solution.
+Here we show that nanodiamonds equipped with sin-
+gle NV-centers can act as in-situ electric field sensors
+inside liquid electrolytes (Fig. 1(a)). By exploiting how
+transverse and axial electric fields act on the NV-center’s
+ground state spin states, we find variations of the free-
+induction decay (FID) pulse sequence, which allow to
+measure the mean electric field components.
+Further,
+we show that it is possible to use variants of the Hahn-
+echo pulse sequence to additionally obtain the electric
+field’s standard deviation σE.
+From a semi-analytical
+ansatz we demonstrate exemplarily for a lithium ion bat-
+tery (LIB) that there is a direct relationship between the
+electric field’s standard deviation and the local ionic con-
+centration. A nanodiamond with a single NV-center can
+therefore work as a sensor which allows to simultaneously
+reconstruct the electric field distribution and to measure
+the ionic concentration with nm spatial resolution.
+II.
+ELECTRIC FIELD DISTRIBUTION IN
+LIQUID ELECTROLYTES
+Before introducing measurements of the electric field
+distribution by the NV-center, we would like to develop
+an analytic expression of the electric field induced inside
+the nanodiamond by the positive and negative ions of the
+electrolyte.
+The potential Φ at position r inside the nanodiamond
+due to a single charge q at position b, is described by
+arXiv:2301.04427v1  [quant-ph]  11 Jan 2023
+
+2
+FIG. 1. (a) Experimental setting. A nanodiamond which is dissolved in the liquid electrolyte of the battery is surrounded by
+positive (orange) and negative (blue) ions. Two perpendicular aligned gold wires allow to generate polarized microwave drives.
+(b) To work as a quantum sensor, the nanodiamond contains a vacancy (V) next to a nitrogen atom (red). (c) Standard deviation
+of Ez, calculated from 500 repeated sets of randomly placed ions of concentration c around the nanodiamond (rND = 100 nm)
+and inside a sphere of radius R. The relative permittivities are ϵND = 5.8 [22] and ϵe = 17.5 [23]. Solid lines are fits following
+Eq. (3) with A as a fit parameter. (d) Fit parameters A obtained from (c), compared to the theory value.
+Poisson’s equation
+∇2Φ (r) = −ρ (r)
+ϵ
+.
+(1)
+Here ϵ = ϵ0ϵi with i = e, ND, are the permittivities of, re-
+spectively, the electrolyte and the nanodiamond in terms
+of the vacuum permittivity ϵ0 and ρ is the charge density
+induced by q. The solution inside the nanodiamond, ΦND
+(see Methods for the detailed derivation), allows to ob-
+tain the electric field at the center of the nanodiamond,
+which is
+END =
+q
+4πϵ0
+3
+2ϵe + ϵND
+b
+b3 .
+(2)
+By considering the positions of ions of a molar concentra-
+tion c to be normally distributed within a sphere of radius
+R around a nanodiamond (radius rND), the standard de-
+viation of the electric field distribution at the center of
+the nanodiamond is
+σEz = A
+�
+c
+� 1
+rND
+− 1
+R
+�
+A =
+|q|
+ϵ0 (2ϵe + ϵND)
+�
+3NA
+4π .
+(3)
+To validate Eq. (3), we simulated the standard deviation
+of 500 sets of uniformly and randomly placed ions for dif-
+ferent molar ionic concentrations (see Fig. 1(c)). As it is
+the most widely used electrolyte of LIBs [24], we chose
+LiPF−
+6 with ϵe = 17.5 [23]. The total electric field was
+calculated as the linear sum of Eq. (2) for all randomly
+placed ions around a 200 nm spherical nanodiamond [25].
+As it can be seen from Fig. 1(d), the expected A value is
+in fair agreement with the simulations. From Eq. (3) it
+can be calculated that for R = 500 nm, the fluctuations
+will increase only by 3%, compared to σE (R = 400 nm).
+As σE therefore saturates for R ≳ 500 nm, this implies
+that electric field fluctuations only affect the nanodia-
+mond within sub-micrometer range and the system is
+limited by the confocal volume of the experimental setup,
+which typically is ∼ 1 µm3 [26, 27].
+III.
+SENSING OF STATIC ELECTRIC FIELDS
+INSIDE ELECTROLYTES
+An electric field E can in cylindrical coordinates be
+expressed by its axial component Ez, its transverse pro-
+jection E⊥ =
+�
+E2x + E2y and an angle φE, which defines
+the projections onto the x and y axis as Ex = E⊥ cos φE
+and Ey = E⊥ sin φE. The total Hamiltonian which de-
+scribes the NV-center in presence of electric and axial
+magnetic fields will in the following be denoted as ˆH0.
+By taking into account that the NV-center can be driven
+by two perpendicular microwave wires (see Fig. 1(a)) with
+amplitude Ω, frequency ωd and a phase φ between each
+other, the total ground state Hamiltonian in a frame ro-
+tating with ωd is ˆH = ˆH0 + ˆHd (see Methods), where
+ˆH0 = (∆ + ξz) ˆS2
+z + βz ˆSz − ξ⊥
+2
+�
+ˆS2
++eiφE + h.c.
+�
+ˆHd = Ω
+√
+2
+�
+ϵ−σ0,−1 + ϵ+σ†
+0,+1 + h.c.
+�
+.
+(4)
+Here ∆ = D − ωd is the detuning between the zero-
+field splitting, D = 2.87 GHz [28], and the microwave
+drive frequency.
+Si, i = x, y, z, are the spin-1 op-
+erators, which can be used to define ladder operators
+S± = Sx ± iSy. σ0,±1 = |0⟩ ⟨±1| are operators which
+describe transitions between |0⟩ and, respectively, |±1⟩.
+Frequency contributions generated by electric and axial
+magnetic fields are considered through ξz = d∥Ez and
+ξ⊥ = d⊥E⊥ (d∥ = 0.35 Hz cm/V, d⊥ = 17 Hz cm/V [29])
+and βz = γeBz (γe = 28 GHz/T [30]).
+The phase factors ϵ± =
+�
+1 − ie∓iφ�
+/2 which enter into
+Eq. (4), allow to describe the transitions which are caused
+by circularly (φ = ±π/2) or linearly (φ = 0) polarized
+microwave drives [31]. The time-evolution operators of
+ˆHd, ˆR (t) = e−i ˆ
+Hdt (see Methods), show that one can
+induce Rabi oscillations between |0⟩ and |1⟩ for right cir-
+cularly polarized drives and |0⟩ ↔ |−1⟩ for left circular
+polarizations.
+Linearly polarized drives allow to drive
+transitions between |0⟩ and both |±1⟩.
+
+MW
+Pos.Electrode3
+FIG. 2. (a) FID-variations to extract ξ⊥, φE and ξz through subsequent pulse sequences. Here Tπ (Tπ/2) is the duration of
+the microwave pulse such that a π-pulse (π/2-pulse) is performed. Subscripts ± denote circularly polarized drives which cause
+oscillations between |0⟩ and either |1⟩ or |−1⟩. Subscript 0 denotes linear polarization of the drive and the free evolution is
+described through ˆF. (b) FIDξ⊥ for different magnetic fields up to βz = 2.7 MHz, corresponding to Bz = 1 G. For βz = 0 the
+signal has the highest contrast with the lowest frequency of oscillation. (c) Fourier transform of FIDξ⊥,ξz with Ω = 10 MHz
+and Ex,y,z = 10 V/µm. Only for T ∗
+2 > 10 µs the peaks at ξ⊥ ± ξz = 2.4 ± 0.04 MHz and 2ξ⊥ can be resolved.
+In absence of microwave drives, the |±1⟩ states are
+symmetrically mixed by ξ⊥ and axial electric fields ef-
+fectively shift |0⟩ from |±1⟩, which can be seen from
+ˆF (τ) = e−i ˆ
+H0τ (see Methods). As axial and transverse
+electric fields thus act differently on the |ms = 0, ±1⟩
+states of the NV-center, one can derive variations of the
+Free Induction Decay (FID), which allow to extract these
+electric field components.
+A.
+Measurement of electric field components
+The FID consists of two microwave pulses separated
+by a free evolution period τ. Electric field contributions
+ξ⊥, φE and ξz can be sensed through FID-variations,
+as shown in Fig. 2(a). The NV-center can be initialized
+into its |0⟩ state via excitation with green laser light, fol-
+lowed by intersystem-crossing [32]. This state can then
+be driven to −i |1⟩ through a right-polarized π-pulse, de-
+noted as ˆR (Tπ)+, and will be influenced by both axial
+magnetic as well as transverse electric fields. The latter
+induce mixing with |−1⟩. By using a microwave π-pulse
+with the same polarization as the initial one, the trans-
+ferred population from |1⟩ to |−1⟩ can be obtained from
+the FID-signal
+FIDξ⊥ (τ) = | ⟨0| ˆR (Tπ)+ ˆF (τ) ˆR (Tπ)+ |0⟩ |2
+= cos2
+�
+τ
+�
+β2z + ξ2
+⊥
+�
++
+β2
+z
+β2z + ξ2
+⊥
+sin2
+�
+τ
+�
+β2z + ξ2
+⊥
+�
+,
+(5)
+which is a measure of the population which has been
+transferred from |1⟩ to |−1⟩. In Fig. 2(b) one can see this
+FID-signal as a function of the free evolution time τ for
+βz values up to 2.8 MHz, which corresponds to Bz = 1 G.
+Besides having a decreased contrast for βz ̸= 0, the
+frequency
+�
+β2z + ξ2
+⊥ of the FID-oscillations depends on
+both axial magnetic and transverse electric fields. It is
+therefore strongly recommended to perform the measure-
+ments in a magnetically shielded environment, for exam-
+ple by a µ-metal as in Ref. [33]. In the following it will
+be assumed that all measurement are performed without
+any magnetic field being present.
+The transverse electric field components are uniquely
+defined through φE, as ξx
+=
+ξ⊥ cos φE and ξy
+=
+ξ⊥ sin φE. A superposition state −eiπ/4 (|1⟩ + |−1⟩) /
+√
+2
+generated through a linearly polarized π-pulse (consid-
+ered via ˆR (Tπ)0, see Methods) will additionally to ξ⊥
+also be affected by φE as this phase differs in its sign
+for |1⟩ and |−1⟩ (see Methods). If either |1⟩ or |−1⟩ is
+projected to |0⟩ through the final microwave pulse, one
+obtains an FID-signal, which both depends on ξ⊥ and
+φE,
+FIDφE,ξ⊥ (τ) = 1
+2 (1 − sin (2τξ⊥) sin φE) .
+(6)
+One can obtain φE as the relative fraction between the
+value of the FID-signal at τ = 0 and its first maxima at
+2τξ⊥ = π/2,
+FIDφE,ξ⊥
+�
+τ = π
+2
+1
+2ξ⊥
+�
+FIDφE,ξ⊥ (τ = 0)
+= 1 − sin φE .
+(7)
+By using FIDξ⊥ and FIDξ⊥,φE, it is therefore possible to
+not only determine the electric field’s transverse compo-
+nent, but also to obtain the projection onto the x and y
+axes, which are determined through φE.
+Axial electric field contributions ξz cause a Stark
+shift between |0⟩ and |±1⟩.
+A superposition state
+(|0⟩ − i |−1⟩) /
+√
+2 generated by a circularly polarized
+π/2-pulse (see Fig. 2(a)) will therefore be affected both
+by ξz and ξ⊥. If the final microwave π/2-pulse has the
+same polarization as the initial one, an FID-signal is ob-
+tained which depends both on ξ⊥ and ξz,
+FIDξz,ξ⊥ (τ) = 1
+4
+�
+1 − 2 cos (τξ⊥) cos (τξz) + cos2 (τξ⊥)
+�
+,
+(8)
+if the NV-center was driven with ωd = D. The Fourier
+
+4
+transform of Eq. (8) (see Methods),
+�
+FID (ω > 0) = π
+4
+�1
+2δ (2ξ⊥ − ω)
+− δ (ξ⊥ + ξz − ω) − δ (ξ⊥ − ξz − ω)
+�
+, (9)
+shows, that ξz can be measured if it is possible to spec-
+trally resolve ξ⊥ ± ξz.
+To study this, we numerically
+[34, 35] simulated FIDξz,ξ⊥ and included dephasing at
+rates 1/T ∗
+2 through a Lindblad operator
+�
+1/T ∗
+2 Sz for
+T ∗
+2 in the range up to 15 µs (see Fig. 2(c)). One can re-
+solve ξ⊥±ξz for nanodiamonds with T ∗
+2 > 10 µs, which is
+higher than the value of typical nanodiamonds [36]. For
+a nanodiamond with T ∗
+2 ≈ 15 µs it would be possible to
+distinguish between ξ⊥ and ξz and therefore to determine
+the projection of the electric field onto the symmetry axis
+of the NV-center.
+IV.
+INFLUENCE OF FLUCTUATING
+ELECTRIC FIELDS
+It can be assumed that the ions surrounding the nan-
+odiamond will not stay static for the timescales in which
+measurements are performed but will be subject to, for
+instance, drift and diffusion.
+These fluctuations will
+affect the electric field inside the nanodiamond.
+Due
+to the limited T ∗
+2 of nanodiamonds, the FID pulse se-
+quences as introduced before will be mainly suitable for
+the measurement of the average electric fields (see Meth-
+ods).
+The coherence time of a nanodiamond can be
+significantly prolonged if instead of an FID, a Hahn-
+Echo pulse sequence is used [25].
+As it is shown in
+Fig. 3(a), we propose a modified version of the Hahn-
+Echo, where after the first free evolution interval, a π-
+pulse with right-circular polarization is performed, be-
+fore the spin is allowed to precess freely during a sec-
+ond free evolution interval τ. Before being read out, a
+right-circularly polarized π-pulse is applied, which leads
+to a signal Hahn (τ) = (1 − cos (2τξ⊥))2 /4. Simulations
+of this Hahn-Echo variation show that the averages (see
+Methods for an example) can be fit by
+⟨Hahn (τ)⟩ = 1
+4
+�
+1 − cos (2τξ⊥) e−τ/T2�2
+.
+(10)
+Here T2 is the sum of the intrinsic spin coherence time
+T2,int. = 100 µs [25] and a contribution due to the fluc-
+tuating electric fields,
+1
+T2
+=
+1
+T2,int.
++
+1
+T2,E
+.
+(11)
+In Fig. 3(b), one can see T2 as a function of the electric
+field’s standard deviation σE, where solid lines are T2,E =
+αEm/σ2
+E in terms of a fit parameters α. The total spin
+coherence time is therefore strongly affected by σE and
+the mean electric field value Em. If the mean transverse
+electric field has been sensed by the FID sequence as
+shown in Eq. (5), it is therefore possible to derive the
+electric field’s standard deviation, which together with
+ξ⊥, φE and ξz defines the electric field distribution. As
+there is a direct relationship between σE and the local
+ionic concentration (see Fig. 1(c)), the proposed Hahn-
+echo pulse sequence additionally allows to use the NV-
+center inside the nanodiamond as a local concentration
+sensor.
+FIG. 3.
+(a) Hahn-echo pulse sequence used to simulate
+Eq. (10).
+(b) Total T2 for numerically [34, 35] simulated
+Hahn-Echoes with T2,int = 100 µs, with the electric field com-
+ponents sampled from a normal distribution with mean Em
+and standard deviation σE.
+For the simulations a drive of
+Ω = 10 MHz was used. Solid lines are fits of αEm/σ2
+E. Every
+trajectory was obtained from 1000 individual simulations. Er-
+ror bars of one standard deviation are smaller than the data
+points.
+V.
+CONCLUSION AND OUTLOOK
+In conclusion we have shown here a full reconstruc-
+tion of the mean electric field generated in a liquid elec-
+trolyte, through the spin control of a quantum sensor
+immersed in the electrolyte. We have found exact ex-
+pressions correlating the electric field components with
+the free-induction decay of the sensor spin, and the de-
+pendence of the variance on the spin-echo measurements.
+Together we were able to deduce the electric field distri-
+bution and also measure the local ionic concentration, a
+key parameter in characterizing the performance of the
+liquid electrolyte for battery applications. We envisage
+that with improved modeling of the electric field distribu-
+tion in liquid electrolytes and using better quantum con-
+trol methods, for example using correlation spectroscopy
+[37], we could enhance the sensitivity of the sensor to the
+local electric-field environment, allowing for an in-situ
+monitoring of the battery using the liquid electrolyte.
+ACKNOWLEDGMENTS
+R. N. would like to acknowledge financial support by
+the Federal Ministry of Education and Research (BMBF)
+
+5
+project QMNDQCNet and DFG (Project No. 507241320
+and 46256793).
+S. V. K. and D. D. would like to
+acknowledge the funding support from BMBF (Grant
+No. 16KIS1590K). A. F. is the incumbent of the Elaine
+Blond Career Development Chair and acknowledges sup-
+port from Israel Science Foundation (ISF grants 963/19
+and 418/20) as well as the Abramson Family Center for
+Young Scientists and the Willner Family Leadership In-
+stitute for the Weizmann Institute of Science.
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+
+1
+Quantum sensing of electric field distributions of liquid electrolytes with NV-centers
+in nanodiamonds - Supplementary Information
+I.
+ELECTRIC FIELD AT CENTER OF NANODIAMOND
+In the following we would like to deduce the electric field of a single point charge q at a distance b from the origin
+of the nanodiamond with radius rND by following Ref. [S1]. Poisson’s equation describes the electrostatic potential Φ,
+∇2Φ (r) = −ρ (r)
+ϵ
+,
+(S1)
+where ϵ = ϵ0ϵi, i = e, ND is the permittivity of, respectively, the electrolyte and the nanodiamond in terms of the
+vacuum permittivity ϵ0. By exploiting azimuthal symmetry of the problem, the above expression reduces to Laplace’s
+equation for r ̸= b, which in spherical coordinates with |r| = r and θ the angle spanned by r and b is
+∇2Φ (r, θ) = 1
+r2
+∂
+∂r
+�
+r2 ∂Φ
+∂r
+�
++
+1
+r2 sin θ
+∂
+∂θ
+�
+sin θ∂Φ
+∂θ
+�
+= 0 .
+(S2)
+The general solution of this partial differential equation can be expressed in terms of the associated Legendre poly-
+nomials Pl of order l and in terms of two constants Al and Cl as [S1, S2]
+Φ (r, θ) =
+∞
+�
+l=0
+�
+Alrl + Cl
+1
+rl+1
+�
+Pl (cos θ) .
+(S3)
+As the potential inside the nanodiamond must be finite at r = 0, Cl needs to vanish and one therefore has
+ΦND (r, θ) =
+∞
+�
+l=0
+AlrlPl (cos θ) .
+(S4)
+By using that 1/|r − b| = �∞
+l=0
+�
+rl
+</rl+1
+>
+�
+Pl (cos θ) [S1, S2] with r≷ being the greater (smaller) of |r| and |b|, one can
+derive the potential in the electrolyte without discontinuity, i.e. without nanodiamond, to be
+˜Φe (r, θ) =
+q
+4πϵ0ϵe
+∞
+�
+l=0
+rl
+<
+rl+1
+>
+Pl (cos θ) .
+(S5)
+The general solution would then be given as a superposition of this expression with Eq. (S3), i.e. Φe = ˜Φe + Φ, which
+reads
+Φe (r, θ) =
+∞
+�
+l=0
+�
+Cl
+1
+rl+1 +
+q
+4πϵ0ϵe
+rl
+<
+rl+1
+>
+�
+Pl (cos θ) ,
+(S6)
+where it was used that in this case Al = 0 to ensure a vanishing potential at infinite distances to the origin, i.e.
+Φe → 0 for r → ∞. The constants Al and Cl, which enter into, respectively, Eq. (S4) and Eq. (S6), can be determined
+by requiring continuity at the interface between electrolyte and nanodiamond,
+�
+ϵeEe − ϵNDEND�
+· nND = 0
+(S7)
+�
+Ee − END�
+× nND ,
+(S8)
+where nND = r/r is the unit vector normal to the surface of the nanodiamond.
+These boundary conditions are
+satisfied, if
+Al =
+q
+4πϵ0ϵe
+1
+bl+1
+ϵe (2l + 1)
+ϵNDl + ϵe (l + 1)
+(S9)
+Cl =
+q
+4πϵ0ϵe
+lr2l+1
+ND
+bl+1
+ϵe − ϵND
+ϵNDl + ϵe (l + 1) .
+(S10)
+
+2
+The electrostatic potential inside the nanodiamond therefore is
+ΦND (r, θ) =
+q
+4πϵ0ϵe
+∞
+�
+l=0
+1
+bl+1
+ϵe (2l + 1)
+ϵNDl + ϵe (l + 1)rlPl (cos θ)
+(S11)
+and the electric field at the center, i.e. for r = 0, can be calculated as
+E (r = 0, θ) =
+q
+4πϵ0
+3
+2ϵe + ϵND
+b
+b3 ,
+(S12)
+if it is used that in cartesian coordinates one has ez = cos θer − sin θeθ with ez the azimuthally symmetric unit vector
+and er and eθ the radial and altitudinal unit vectors.
+A.
+Electric field variance
+The probability of an ion to be located at b witin a sphere of radius R around the nanodiamond is
+p (b) =
+� 3
+4π
+1
+R3−r3
+ND ,
+rND ≤ b ≤ R
+0,
+otherwise.
+(S13)
+It can be easily verified that this distribution is normalized, i.e.
+�
+R3 d3b p (b) = 1. Direct calculation reveals ⟨Ez⟩ = 0
+and therefore
+σ2
+Ez,ion = ⟨E2
+z⟩
+=
+9q2
+(4πϵ0)2
+1
+(2ϵe + ϵND)2
+1
+R3 − r3
+ND
+� 1
+rND
+− 1
+R
+�
+.
+(S14)
+Under the assumption that the electric fields generated by the single ions are uncorrelated, the total fluctuations
+are given by multiplying the above expression with the number of ions inside the sphere. The standard deviation
+σ2
+Ez = cNAV σ2
+Ez,ion of the electric field components with NA Avogadro’s number, c the molar ionic concentration
+and V the volume in which the ions reside therefore is
+σEz =
+|q|
+ϵ0 (2ϵe + ϵND)
+�
+3NA
+4π
+�
+c
+� 1
+rNV
+− 1
+R
+�
+.
+(S15)
+From this it can be seen that the expected electric field fluctuations increase with the molar concentration, i.e.
+σEz ∝ √c.
+II.
+HAMILTONIAN IN ROTATING FRAME
+As derived by Doherty et al. in Ref. [S3], the Hamiltonian of the NV-center in presence of axial magnetic fields Bz
+and electric field components Ei with i = x, y, z and ℏ = 1 is
+ˆHNV =
+�
+D + d∥Ez
+� ˆS2
+z + γeBz ˆSz
++ d⊥
+�
+Ex
+�
+ˆS2
+y − ˆS2
+x
+�
++ Ey
+�
+ˆSx ˆSy + ˆSy ˆSx
+��
+,
+(S16)
+with γe = 2.8 MHz/G the NV’s gyromagnetic ratio [S4] and d∥ = 0.35 Hz · cm/V and d⊥ = 17 Hz · cm/V the axial
+and transverse dipole moments [S5]. By rewriting this Hamiltonian in terms of its frequency contributions βz = γeBz,
+ξz = d∥Ez and ξ⊥ = d⊥
+�
+E2x + E2y and by introducing the electric field polarization φE, which defines the transverse
+electric field projections via ξx = ξ⊥ cos φE and ξy = ξ⊥ sin φE, Eq. (S16) can be rewritten as
+ˆHNV = (D + ξz) ˆS2
+z + βz ˆSz − ξ⊥
+2
+�
+eiφE ˆS2
++ + h.c.
+�
+,
+(S17)
+where ˆS± = ˆSx ± i ˆSy are spin-1 ladder-operators and h.c. means the hermitian conjugate.
+
+3
+The NV-center can be driven by perpendicular (compared to the NV’s symmetry axis) microwave magnetic fields
+of amplitude Ω = γeBd and frequency ωd.
+To exert polarized drives onto the NV-center, two wires which are
+perpendicular to each other (see Fig. 1(a) main text) are operated with a phase φ between each other. This drive can
+be described by an Hamiltonian [S6]
+ˆHd (t) = Ω
+�
+ˆSx cos (ωdt) + ˆSy cos (ωdt + φ)
+�
+.
+(S18)
+Defining phase-factors ϵ± (φ) =
+�
+1 − ie∓iφ�
+/2, similarly to Ref.
+[S6], allows to compactly account for different
+polarizations as ϵ+ = 1 only if φ = −π/2 (i.e. right-circular polarization) and ϵ− = 1 for left-circular polarized
+microwave fields (φ = +π/2). By transforming ˆHNV + ˆHd (t) into a frame oscillating with ωd through the unitary
+U = eiωdS2
+z, one can derive the Hamiltonian under the rotating-wave approximation, which is
+ˆH = ˆH0 + ˆHd
+ˆH0 = (∆ + ξz) ˆS2
+z + βz ˆSz − ξ⊥
+2
+�
+eiφE ˆS2
++ + h.c.
+�
+ˆHd = Ω
+√
+2 (ϵ− |0⟩ ⟨−1| + ϵ+ |1⟩ ⟨0| + h.c.) .
+(S19)
+A.
+Derivation of time-evolution operators
+To allow for the efficient calculation of pulse-sequences, time evolution operators of the free evolution ˆF (τ) and the
+drive ˆR (T) will be derived in the following.
+1.
+Free Evolution
+A possible set of eigenstates of ˆH0 is {|0⟩ , |+⟩ , |−⟩} with
+|+⟩ = cos θ
+2eiφE/2 |1⟩ + sin θ
+2e−iφE/2 |−1⟩
+|−⟩ = sin θ
+2eiφE/2 |1⟩ − cos θ
+2e−iφE/2 |−1⟩ ,
+(S20)
+where tan θ = −ξ⊥/βz, with corresponding eigenenergies ω0 = 0 and ω± = ∆ + ξz ±
+�
+β2z + ξ2
+⊥. The time evolution
+operator of ˆH0 is ˆF (τ) = �
+i={0,±} e−iωiτ |i⟩ ⟨i|, where the sum is performed over all eigenstates of ˆH0. In the basis
+of {|0⟩ , |±1⟩} this is
+ˆF (τ) = |0⟩ ⟨0| + e−iτ(∆+ξz)�
+iξ⊥
+x sin (τx)
+�
+eiφE |1⟩ ⟨−1| + h.c.
+�
++
+�
+cos (τx) − iβz
+x sin (τx)
+�
+|1⟩ ⟨1|
++
+�
+cos (τx) + iβz
+x sin (τx)
+�
+|−1⟩ ⟨−1|
+�
+.
+(S21)
+Here the frequency of oscillation has been defined as x =
+�
+β2z + ξ2
+⊥.
+2.
+Microwave Drive
+To derive operators which describe the action of the microwave pulses, it will be assumed that these pulses
+exceed all other frequency scales in magnitude, i.e.
+Ω ≫ ∆, βz, ξz, ξ⊥, such that
+ˆH ≈
+Ω
+√
+2
+ˆ�
+Hd with ˆ�
+Hd =
+
+4
+(ϵ− |0⟩ ⟨−1| + ϵ+ |1⟩ ⟨0| + h.c.). By noting that ˆ�
+H
+3
+d = ˆ�
+Hd, the time evolution
+ˆR (t) = e−it ˆ
+Hd =
+∞
+�
+k=0
+�
+−itΩ
+√
+2
+�n
+n!
+� ˆ�
+Hd
+�n
+,
+(S22)
+can be calculated as
+ˆR (t) = |1⟩ ⟨1|
+�
+1 − |ϵ+|2�
++ |−1⟩ ⟨−1|
+�
+1 − |ϵ−|2�
+− ϵ+ϵ− |1⟩ ⟨−1| − ϵ∗
++ϵ∗
+− |−1⟩ ⟨1|
++ cos
+� tΩ
+√
+2
+� �
+|0⟩ ⟨0| + |ϵ+|2 |1⟩ ⟨1| + |ϵ−|2 |−1⟩ ⟨−1| + ϵ+ϵ− |1⟩ ⟨−1| + ϵ∗
++ϵ∗
+− |−1⟩ ⟨1|
+�
+− i sin
+� tΩ
+√
+2
+�
+(ϵ− |0⟩ ⟨−1| + ϵ+ |1⟩ ⟨0| + h.c.) .
+(S23)
+Depending on the polarization, one can induce Rabi oscillations between |0⟩ and either |−1⟩ for φ = π/2 (denoted as
+ˆR+) or |+1⟩ (φ = −π/2, ˆR−),
+ˆR (t)± = |∓1⟩ ⟨∓1| + cos
+� Ωt
+√
+2
+� �
+|0⟩ ⟨0| + |±1⟩ ⟨±1|
+�
+− i sin
+� Ωt
+√
+2
+� �
+|0⟩ ⟨±1| + h.c.
+�
+.
+(S24)
+The system can be driven to both |±1⟩, if a linearly polarized drive is used,
+R (t)0 = 1
+2 (|1⟩ ⟨1| + |−1⟩ ⟨−1| + i |1⟩ ⟨−1| − i |−1⟩ ⟨1|)
++ cos
+� tΩ
+√
+2
+� �
+|0⟩ ⟨0|
++ 1
+2 (|1⟩ ⟨1| + |−1⟩ ⟨−1| − i |1⟩ ⟨−1| + i |−1⟩ ⟨1|)
+�
+− 1 + i
+2
+sin
+� tΩ
+√
+2
+�
+(|0⟩ ⟨−1| + |1⟩ ⟨0| + h.c.) .
+(S25)
+The last expression can similarly be compactly written by noting that (1 ± i) /2 = e±iπ/4/
+√
+2. These operators can
+then be used to describe the action of (polarized) π- and π/2-pulses onto the |ms = 0, ±1⟩-states of the NV-center.
+III.
+FOURIER TRANSFORMATION OF FID-SIGNAL
+Some arbitrary signals f and ˜f in time- and frequency-domain are connected to each other as
+˜f (ω) = FT [f (τ)] =
+� +∞
+−∞
+dτ f (τ) e−iωτ
+FT−1 �
+˜f (ω)
+�
+= 1
+2π
+� +∞
+−∞
+dω ˜f (ω) eiωτ .
+(S26)
+To simplify the calculation of the Fourier transformed FID-signal, one can rewrite FIDξ⊥,φE (Eq. (6) main text) as
+FIDξz,ξ⊥ (τ) = 1
+4
+�3
+2 + 1
+2 cos (2τξ⊥) − cos (τ [ξ⊥ + ξz])
+− cos (τ [ξ⊥ − ξz])
+�
+.
+(S27)
+From Eq. (S26), one sees that FT [cos (τx)] = π [δ (x − ω) + δ (x + ω)] and therefore
+�
+FID (ω) = π
+4
+�3
+2δ (ω) + 1
+2 [δ (2ξ⊥ − ω) + δ (2ξ⊥ + ω)]
+− [δ (ξ⊥ + ξz − ω) + δ (ξ⊥ + ξz + ω)]
+− [δ (ξ⊥ − ξz − ω) + δ (ξ⊥ − ξz + ω)]
+�
+.
+(S28)
+
+5
+IV.
+SIMULATED PULSE SEQUENCES FOR NORMALLY DISTRIBUTED ELECTRIC FIELDS
+FIG. S1. Simulated expected FID-values of FIDξ⊥ (Eq. (5) main text), calculated from 500 individual FID-simulations with
+drive amplitude of Ω = 10 MHz, intrinsic T ∗
+2,int. and electric field components sampled from a normal distribution with mean
+Em and standard deviation σE. Dephasing is considered through a Lindblad-Operator
+�
+1/T ∗
+2,int.Sz. For both mean electric
+field values of (a) 1.0 V/µm and (b) 4.0 V/µm, it is not possible to resolve ξ⊥.
+To understand how fluctuating electric fields alter the FID-signal, we numerically [S7, S8] simulated FIDξ⊥ (Eq. (5)
+main text) for normally distributed electric fields. Hereby, at every timestep at which the time-evolution is calcuated,
+the electric field components are passed from a beforehand sampled normal distribution with mean Em and standard
+deviation σE. It can be seen from Fig. S1 that the average FIDξ⊥ signal decays rapidly to its steady-state value of
+1/2, which is due to the short T ∗
+2 time of 1 µs. For this reason it is proposed to use the Hahn-Echo pulse sequence for
+measurements of strongly fluctuating electric fields.
+0
+50
+100
+150
+200
+τ in µs
+0.0
+0.2
+0.4
+0.6
+0.8
+⟨Hahn (τ)⟩
+Sim.
+Fit
+FIG. S2.
+Example of the average Hahn-echo signal, which was obtained numerically from 1000 individual simulations of
+the pulse sequence shown in Fig. 3(a) (main text) with a mean electric field value of Em = 1.0 V/µm, standard deviation
+σE = 0.75 V/µm, drive amplitude Ω = 10 MHz and intrisic T2,int. = 100 µs together with the fit following Eq. (10) (main text).
+The total T2 value obtained from this fit is T2 = (39.87 ± 0.86) µs.
+As described in the main text, the numerically obtained Hahn-echo trajectories (see Fig. S2 for an example) are well
+fitted by ⟨Hahn (τ)⟩ = 1
+4
+�
+1 − cos (2τξ⊥) e−τ/T2�2. Here both the intrinsic T2,int. = 100 µs and T2,E due to fluctuating
+elecric fields contribute to the total T2 via
+1
+T2
+=
+1
+T2,int.
++
+1
+T2,E
+.
+(S29)
+The latter can be fitted in terms of Em and σE via
+T2,E = αEm
+σ2
+E
+.
+(S30)
+The values of the fit parameter α can be found in Fig. S3.
+
+6
+1
+2
+Em in V/µm
+30
+35
+40
+α
+FIG. S3. Fit parameter α, obtained by numerically fitting Eq. (S29) and Eq. (S30) with T2,int. = 100 µs to the data from Fig. 3
+(main text).
+[S1] R. Messina, “Image charges in spherical geometry: Application to colloidal systems,” The Journal of Chemical
+Physics, vol. 117, no. 24, pp. 11062-11074, Dec. 2002.
+[S2] J. D. Jackson, “Klassische Elektrodynamik,” De Gruyter, Dec. 2006.
+[S3] M. W. Doherty, F. Dolde, H. Fedder, F. Jelezko, J. Wrachtrup, N. B. Manson, and L. C. L. Hollenberg, “Theory
+of the ground-state spin of the NV center in diamond,” Physical Review B, vol. 85, no. 20, p. 205203, May
+2012.
+[S4] E. Abe and K. Sasaki, “Tutorial: Magnetic resonance with nitrogen-vacancy centers in diamond - microwave
+engineering, materials science, and magnetometry,” Journal of Applied Physics, vol. 123, no. 16, p. 161101,
+Apr. 2018.
+[S5] E. Van Oort and M. Glasbeek, “Electric-field induced modulation of spin echoes of N-V centers in diamond,”
+Chemical Physics Letters, vol. 168, no. 6, pp. 529-532, May 1990.
+[S6] P. London, P. Balasubramanian, B. Naydenov, L. P. McGuiness, and F. Jelezko, “Strong driving of a single spin
+using arbitrarily polarized fields,” Physical Review A, vol. 90, no. 1, p. 012302, July 2014.
+[S7] J. Johansson, P. Nation, and F. Nori, “QuTiP: An open-source Python framework for the dynamics of open
+quantum systems,” Computer Physics Communications, vol. 183, no. 8, pp. 1760-1772, Aug. 2012.
+[S8] J. Johansson, P. Nation, and F. Nori, “QuTiP 2: A Python framework for the dynamics of open quantum
+systems,” Computer Physics Communications, vol. 184, no. 4, pp. 1234-1240, Apr. 2013.
+
diff --git a/8tE3T4oBgHgl3EQfSAk3/content/tmp_files/load_file.txt b/8tE3T4oBgHgl3EQfSAk3/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9072744033207d8aabd14ee591dcb515ad6e05ab
--- /dev/null
+++ b/8tE3T4oBgHgl3EQfSAk3/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf,len=859
+page_content='Quantum sensing of electric field distributions of liquid electrolytes with NV-centers  in nanodiamonds  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Hollendonner,1, 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Sharma,2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Dasari,3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Finkler,4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Kusminskiy,2, 5 and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Nagy1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' ∗  1Friedrich-Alexander-University Erlangen-Nuremberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 91058 Erlangen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Germany  2Max Planck Institute for the Science of Light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 91058 Erlangen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Germany  33rd Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' IQST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' and Research Center SCoPE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  University of Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 70569 Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Germany  4Department of Chemical and Biological Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Weizmann Institute of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Rehovot 7610001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Israel  5Institute for Theoretical Solid State Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' RWTH Aachen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 52074 Aachen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Germany  (Dated: January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 2023)  To use batteries as large-scale energy storage systems it is necessary to measure and understand  their degradation in-situ and in-operando.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' As a battery’s degradation is often the result of molecular  processes inside the electrolyte, a sensing platform which allows to measure the ions with a high  spatial resolution is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Primary candidates for such a platform are NV-centers in diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  We propose to use a single NV-center to deduce the electric field distribution generated by the  ions inside the electrolyte through microwave pulse sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' We show that the electric field can  be reconstructed with great accuracy by using a protocol which includes different variations of the  Free Induction Decay to obtain the mean electric field components and a modified Hahn-echo pulse  sequence to measure the electric field’s standard deviation σE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' From a semi-analytical ansatz we find  that for a lithium ion battery there is a direct relationship between σE and the ionic concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Our results show that it is therefore possible to use NV-centers as sensors to measure both the  electric field distribution and the local ionic concentration inside electrolytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  INTRODUCTION  Rechargeable batteries play an important role for our  society and are a key ingredient for the transition to-  wards renewable energy sources [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' As the production  of batteries is accompanied with a considerable use of re-  sources, recyclable [4] batteries with a long lifetime are  needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The latter is limited by degradation mechanisms,  such as the formation of solid-electrolyte interfaces [5] or  lithium-plating [6] which can reduce the battery’s capac-  ity with increasing cell age [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' As these processes happen  on a molecular level within nanometer scales [5], a sensor  which is capable of monitoring the ionic concentration  in-situ and in-operando with high spatial and temporal  resolutions is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Even though MRI allows to recon-  struct transport properties [8, 9] of a battery, tools which  allow to perform measurements inside the electrolyte are  still absent [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  It has been demonstrated that nitrogen-vacancy (NV)  centers in diamond (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 1(b)) are high-resolution  quantum sensors, which can detect oscillating or fluctu-  ating [10–13] magnetic fields with nano- [14, 15] and even  subpico-Tesla [16] sensitivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Besides this, NV-centers  have great ability for the detection of electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' They  can not only detect DC [17, 18] or AC [19] electric fields  with remarkable precision, but are additionally capable  of detecting single fundamental charges [20] even within  the diamond lattice [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' This electric field sensitivity was  used by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' [22] to show that, based on theoretical con-  ∗ roland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='nagy@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='de  siderations, bulk NV-centers can work as electrochemical  sensors if they are in contact with an electrolyte solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Here we show that nanodiamonds equipped with sin-  gle NV-centers can act as in-situ electric field sensors  inside liquid electrolytes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' By exploiting how  transverse and axial electric fields act on the NV-center’s  ground state spin states, we find variations of the free-  induction decay (FID) pulse sequence, which allow to  measure the mean electric field components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Further,  we show that it is possible to use variants of the Hahn-  echo pulse sequence to additionally obtain the electric  field’s standard deviation σE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  From a semi-analytical  ansatz we demonstrate exemplarily for a lithium ion bat-  tery (LIB) that there is a direct relationship between the  electric field’s standard deviation and the local ionic con-  centration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' A nanodiamond with a single NV-center can  therefore work as a sensor which allows to simultaneously  reconstruct the electric field distribution and to measure  the ionic concentration with nm spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  ELECTRIC FIELD DISTRIBUTION IN  LIQUID ELECTROLYTES  Before introducing measurements of the electric field  distribution by the NV-center, we would like to develop  an analytic expression of the electric field induced inside  the nanodiamond by the positive and negative ions of the  electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  The potential Φ at position r inside the nanodiamond  due to a single charge q at position b, is described by  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='04427v1  [quant-ph]  11 Jan 2023  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (a) Experimental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' A nanodiamond which is dissolved in the liquid electrolyte of the battery is surrounded by  positive (orange) and negative (blue) ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Two perpendicular aligned gold wires allow to generate polarized microwave drives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (b) To work as a quantum sensor, the nanodiamond contains a vacancy (V) next to a nitrogen atom (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (c) Standard deviation  of Ez, calculated from 500 repeated sets of randomly placed ions of concentration c around the nanodiamond (rND = 100 nm)  and inside a sphere of radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The relative permittivities are ϵND = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='8 [22] and ϵe = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='5 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Solid lines are fits following  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (3) with A as a fit parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (d) Fit parameters A obtained from (c), compared to the theory value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Poisson’s equation  ∇2Φ (r) = −ρ (r)  ϵ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (1)  Here ϵ = ϵ0ϵi with i = e, ND, are the permittivities of, re-  spectively, the electrolyte and the nanodiamond in terms  of the vacuum permittivity ϵ0 and ρ is the charge density  induced by q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The solution inside the nanodiamond, ΦND  (see Methods for the detailed derivation), allows to ob-  tain the electric field at the center of the nanodiamond,  which is  END =  q  4πϵ0  3  2ϵe + ϵND  b  b3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (2)  By considering the positions of ions of a molar concentra-  tion c to be normally distributed within a sphere of radius  R around a nanodiamond (radius rND), the standard de-  viation of the electric field distribution at the center of  the nanodiamond is  σEz = A  �  c  � 1  rND  − 1  R  �  A =  |q|  ϵ0 (2ϵe + ϵND)  �  3NA  4π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (3)  To validate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (3), we simulated the standard deviation  of 500 sets of uniformly and randomly placed ions for dif-  ferent molar ionic concentrations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' As it is  the most widely used electrolyte of LIBs [24], we chose  LiPF−  6 with ϵe = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='5 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The total electric field was  calculated as the linear sum of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (2) for all randomly  placed ions around a 200 nm spherical nanodiamond [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  As it can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 1(d), the expected A value is  in fair agreement with the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (3) it  can be calculated that for R = 500 nm, the fluctuations  will increase only by 3%, compared to σE (R = 400 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  As σE therefore saturates for R ≳ 500 nm, this implies  that electric field fluctuations only affect the nanodia-  mond within sub-micrometer range and the system is  limited by the confocal volume of the experimental setup,  which typically is ∼ 1 µm3 [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  SENSING OF STATIC ELECTRIC FIELDS  INSIDE ELECTROLYTES  An electric field E can in cylindrical coordinates be  expressed by its axial component Ez, its transverse pro-  jection E⊥ =  �  E2x + E2y and an angle φE, which defines  the projections onto the x and y axis as Ex = E⊥ cos φE  and Ey = E⊥ sin φE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The total Hamiltonian which de-  scribes the NV-center in presence of electric and axial  magnetic fields will in the following be denoted as ˆH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  By taking into account that the NV-center can be driven  by two perpendicular microwave wires (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 1(a)) with  amplitude Ω, frequency ωd and a phase φ between each  other, the total ground state Hamiltonian in a frame ro-  tating with ωd is ˆH = ˆH0 + ˆHd (see Methods), where  ˆH0 = (∆ + ξz) ˆS2  z + βz ˆSz − ξ⊥  2  �  ˆS2  +eiφE + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  �  ˆHd = Ω  √  2  �  ϵ−σ0,−1 + ϵ+σ†  0,+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (4)  Here ∆ = D − ωd is the detuning between the zero-  field splitting, D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='87 GHz [28], and the microwave  drive frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Si, i = x, y, z, are the spin-1 op-  erators, which can be used to define ladder operators  S± = Sx ± iSy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' σ0,±1 = |0⟩ ⟨±1| are operators which  describe transitions between |0⟩ and, respectively, |±1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Frequency contributions generated by electric and axial  magnetic fields are considered through ξz = d∥Ez and  ξ⊥ = d⊥E⊥ (d∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='35 Hz cm/V, d⊥ = 17 Hz cm/V [29])  and βz = γeBz (γe = 28 GHz/T [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  The phase factors ϵ± =  �  1 − ie∓iφ�  /2 which enter into  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (4), allow to describe the transitions which are caused  by circularly (φ = ±π/2) or linearly (φ = 0) polarized  microwave drives [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The time-evolution operators of  ˆHd, ˆR (t) = e−i ˆ  Hdt (see Methods), show that one can  induce Rabi oscillations between |0⟩ and |1⟩ for right cir-  cularly polarized drives and |0⟩ ↔ |−1⟩ for left circular  polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Linearly polarized drives allow to drive  transitions between |0⟩ and both |±1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  MW  Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='Electrode3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (a) FID-variations to extract ξ⊥, φE and ξz through subsequent pulse sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Here Tπ (Tπ/2) is the duration of  the microwave pulse such that a π-pulse (π/2-pulse) is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Subscripts ± denote circularly polarized drives which cause  oscillations between |0⟩ and either |1⟩ or |−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Subscript 0 denotes linear polarization of the drive and the free evolution is  described through ˆF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (b) FIDξ⊥ for different magnetic fields up to βz = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='7 MHz, corresponding to Bz = 1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' For βz = 0 the  signal has the highest contrast with the lowest frequency of oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (c) Fourier transform of FIDξ⊥,ξz with Ω = 10 MHz  and Ex,y,z = 10 V/µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Only for T ∗  2 > 10 µs the peaks at ξ⊥ ± ξz = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='04 MHz and 2ξ⊥ can be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  In absence of microwave drives, the |±1⟩ states are  symmetrically mixed by ξ⊥ and axial electric fields ef-  fectively shift |0⟩ from |±1⟩, which can be seen from  ˆF (τ) = e−i ˆ  H0τ (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' As axial and transverse  electric fields thus act differently on the |ms = 0, ±1⟩  states of the NV-center, one can derive variations of the  Free Induction Decay (FID), which allow to extract these  electric field components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Measurement of electric field components  The FID consists of two microwave pulses separated  by a free evolution period τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Electric field contributions  ξ⊥, φE and ξz can be sensed through FID-variations,  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The NV-center can be initialized  into its |0⟩ state via excitation with green laser light, fol-  lowed by intersystem-crossing [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' This state can then  be driven to −i |1⟩ through a right-polarized π-pulse, de-  noted as ˆR (Tπ)+, and will be influenced by both axial  magnetic as well as transverse electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The latter  induce mixing with |−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' By using a microwave π-pulse  with the same polarization as the initial one, the trans-  ferred population from |1⟩ to |−1⟩ can be obtained from  the FID-signal  FIDξ⊥ (τ) = | ⟨0| ˆR (Tπ)+ ˆF (τ) ˆR (Tπ)+ |0⟩ |2  = cos2  �  τ  �  β2z + ξ2  ⊥  �  +  β2  z  β2z + ξ2  ⊥  sin2  �  τ  �  β2z + ξ2  ⊥  �  ,  (5)  which is a measure of the population which has been  transferred from |1⟩ to |−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 2(b) one can see this  FID-signal as a function of the free evolution time τ for  βz values up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='8 MHz, which corresponds to Bz = 1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Besides having a decreased contrast for βz ̸= 0, the  frequency  �  β2z + ξ2  ⊥ of the FID-oscillations depends on  both axial magnetic and transverse electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' It is  therefore strongly recommended to perform the measure-  ments in a magnetically shielded environment, for exam-  ple by a µ-metal as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' In the following it will  be assumed that all measurement are performed without  any magnetic field being present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  The transverse electric field components are uniquely  defined through φE, as ξx  =  ξ⊥ cos φE and ξy  =  ξ⊥ sin φE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' A superposition state −eiπ/4 (|1⟩ + |−1⟩) /  √  2  generated through a linearly polarized π-pulse (consid-  ered via ˆR (Tπ)0, see Methods) will additionally to ξ⊥  also be affected by φE as this phase differs in its sign  for |1⟩ and |−1⟩ (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' If either |1⟩ or |−1⟩ is  projected to |0⟩ through the final microwave pulse, one  obtains an FID-signal, which both depends on ξ⊥ and  φE,  FIDφE,ξ⊥ (τ) = 1  2 (1 − sin (2τξ⊥) sin φE) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (6)  One can obtain φE as the relative fraction between the  value of the FID-signal at τ = 0 and its first maxima at  2τξ⊥ = π/2,  FIDφE,ξ⊥  �  τ = π  2  1  2ξ⊥  �  FIDφE,ξ⊥ (τ = 0)  = 1 − sin φE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (7)  By using FIDξ⊥ and FIDξ⊥,φE, it is therefore possible to  not only determine the electric field’s transverse compo-  nent, but also to obtain the projection onto the x and y  axes, which are determined through φE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Axial electric field contributions ξz cause a Stark  shift between |0⟩ and |±1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  A superposition state  (|0⟩ − i |−1⟩) /  √  2 generated by a circularly polarized  π/2-pulse (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 2(a)) will therefore be affected both  by ξz and ξ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' If the final microwave π/2-pulse has the  same polarization as the initial one, an FID-signal is ob-  tained which depends both on ξ⊥ and ξz,  FIDξz,ξ⊥ (τ) = 1  4  �  1 − 2 cos (τξ⊥) cos (τξz) + cos2 (τξ⊥)  �  ,  (8)  if the NV-center was driven with ωd = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The Fourier  4  transform of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (8) (see Methods),  �  FID (ω > 0) = π  4  �1  2δ (2ξ⊥ − ω)  − δ (ξ⊥ + ξz − ω) − δ (ξ⊥ − ξz − ω)  �  , (9)  shows, that ξz can be measured if it is possible to spec-  trally resolve ξ⊥ ± ξz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  To study this, we numerically  [34, 35] simulated FIDξz,ξ⊥ and included dephasing at  rates 1/T ∗  2 through a Lindblad operator  �  1/T ∗  2 Sz for  T ∗  2 in the range up to 15 µs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' One can re-  solve ξ⊥±ξz for nanodiamonds with T ∗  2 > 10 µs, which is  higher than the value of typical nanodiamonds [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' For  a nanodiamond with T ∗  2 ≈ 15 µs it would be possible to  distinguish between ξ⊥ and ξz and therefore to determine  the projection of the electric field onto the symmetry axis  of the NV-center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  INFLUENCE OF FLUCTUATING  ELECTRIC FIELDS  It can be assumed that the ions surrounding the nan-  odiamond will not stay static for the timescales in which  measurements are performed but will be subject to, for  instance, drift and diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  These fluctuations will  affect the electric field inside the nanodiamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Due  to the limited T ∗  2 of nanodiamonds, the FID pulse se-  quences as introduced before will be mainly suitable for  the measurement of the average electric fields (see Meth-  ods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  The coherence time of a nanodiamond can be  significantly prolonged if instead of an FID, a Hahn-  Echo pulse sequence is used [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  As it is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 3(a), we propose a modified version of the Hahn-  Echo, where after the first free evolution interval, a π-  pulse with right-circular polarization is performed, be-  fore the spin is allowed to precess freely during a sec-  ond free evolution interval τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Before being read out, a  right-circularly polarized π-pulse is applied, which leads  to a signal Hahn (τ) = (1 − cos (2τξ⊥))2 /4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Simulations  of this Hahn-Echo variation show that the averages (see  Methods for an example) can be fit by  ⟨Hahn (τ)⟩ = 1  4  �  1 − cos (2τξ⊥) e−τ/T2�2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (10)  Here T2 is the sum of the intrinsic spin coherence time  T2,int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' = 100 µs [25] and a contribution due to the fluc-  tuating electric fields,  1  T2  =  1  T2,int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  +  1  T2,E  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (11)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 3(b), one can see T2 as a function of the electric  field’s standard deviation σE, where solid lines are T2,E =  αEm/σ2  E in terms of a fit parameters α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The total spin  coherence time is therefore strongly affected by σE and  the mean electric field value Em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' If the mean transverse  electric field has been sensed by the FID sequence as  shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (5), it is therefore possible to derive the  electric field’s standard deviation, which together with  ξ⊥, φE and ξz defines the electric field distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' As  there is a direct relationship between σE and the local  ionic concentration (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 1(c)), the proposed Hahn-  echo pulse sequence additionally allows to use the NV-  center inside the nanodiamond as a local concentration  sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (a) Hahn-echo pulse sequence used to simulate  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (b) Total T2 for numerically [34, 35] simulated  Hahn-Echoes with T2,int = 100 µs, with the electric field com-  ponents sampled from a normal distribution with mean Em  and standard deviation σE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  For the simulations a drive of  Ω = 10 MHz was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Solid lines are fits of αEm/σ2  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Every  trajectory was obtained from 1000 individual simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Er-  ror bars of one standard deviation are smaller than the data  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  CONCLUSION AND OUTLOOK  In conclusion we have shown here a full reconstruc-  tion of the mean electric field generated in a liquid elec-  trolyte, through the spin control of a quantum sensor  immersed in the electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' We have found exact ex-  pressions correlating the electric field components with  the free-induction decay of the sensor spin, and the de-  pendence of the variance on the spin-echo measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Together we were able to deduce the electric field distri-  bution and also measure the local ionic concentration, a  key parameter in characterizing the performance of the  liquid electrolyte for battery applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' We envisage  that with improved modeling of the electric field distribu-  tion in liquid electrolytes and using better quantum con-  trol methods, for example using correlation spectroscopy  [37], we could enhance the sensitivity of the sensor to the  local electric-field environment, allowing for an in-situ  monitoring of the battery using the liquid electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' would like to acknowledge financial support by  the Federal Ministry of Education and Research (BMBF)  5  project QMNDQCNet and DFG (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 507241320  and 46256793).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' would like to  acknowledge the funding support from BMBF (Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 16KIS1590K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' is the incumbent of the Elaine  Blond Career Development Chair and acknowledges sup-  port from Israel Science Foundation (ISF grants 963/19  and 418/20) as well as the Abramson Family Center for  Young Scientists and the Willner Family Leadership In-  stitute for the Weizmann Institute of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
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+page_content=' Stolkin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Walton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
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+page_content='  1  Quantum sensing of electric field distributions of liquid electrolytes with NV-centers  in nanodiamonds - Supplementary Information  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  ELECTRIC FIELD AT CENTER OF NANODIAMOND  In the following we would like to deduce the electric field of a single point charge q at a distance b from the origin  of the nanodiamond with radius rND by following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' [S1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Poisson’s equation describes the electrostatic potential Φ,  ∇2Φ (r) = −ρ (r)  ϵ  ,  (S1)  where ϵ = ϵ0ϵi, i = e, ND is the permittivity of, respectively, the electrolyte and the nanodiamond in terms of the  vacuum permittivity ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' By exploiting azimuthal symmetry of the problem, the above expression reduces to Laplace’s  equation for r ̸= b, which in spherical coordinates with |r| = r and θ the angle spanned by r and b is  ∇2Φ (r, θ) = 1  r2  ∂  ∂r  �  r2 ∂Φ  ∂r  �  +  1  r2 sin θ  ∂  ∂θ  �  sin θ∂Φ  ∂θ  �  = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S2)  The general solution of this partial differential equation can be expressed in terms of the associated Legendre poly-  nomials Pl of order l and in terms of two constants Al and Cl as [S1, S2]  Φ (r, θ) =  ∞  �  l=0  �  Alrl + Cl  1  rl+1  �  Pl (cos θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S3)  As the potential inside the nanodiamond must be finite at r = 0, Cl needs to vanish and one therefore has  ΦND (r, θ) =  ∞  �  l=0  AlrlPl (cos θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S4)  By using that 1/|r − b| = �∞  l=0  �  rl  </rl+1  >  �  Pl (cos θ) [S1, S2] with r≷ being the greater (smaller) of |r| and |b|, one can  derive the potential in the electrolyte without discontinuity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' without nanodiamond, to be  ˜Φe (r, θ) =  q  4πϵ0ϵe  ∞  �  l=0  rl  <  rl+1  >  Pl (cos θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S5)  The general solution would then be given as a superposition of this expression with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (S3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Φe = ˜Φe + Φ, which  reads  Φe (r, θ) =  ∞  �  l=0  �  Cl  1  rl+1 +  q  4πϵ0ϵe  rl  <  rl+1  >  �  Pl (cos θ) ,  (S6)  where it was used that in this case Al = 0 to ensure a vanishing potential at infinite distances to the origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Φe → 0 for r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The constants Al and Cl, which enter into, respectively, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (S4) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (S6), can be determined  by requiring continuity at the interface between electrolyte and nanodiamond,  �  ϵeEe − ϵNDEND�  nND = 0  (S7)  �  Ee − END�  × nND ,  (S8)  where nND = r/r is the unit vector normal to the surface of the nanodiamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  These boundary conditions are  satisfied, if  Al =  q  4πϵ0ϵe  1  bl+1  ϵe (2l + 1)  ϵNDl + ϵe (l + 1)  (S9)  Cl =  q  4πϵ0ϵe  lr2l+1  ND  bl+1  ϵe − ϵND  ϵNDl + ϵe (l + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S10)  2  The electrostatic potential inside the nanodiamond therefore is  ΦND (r, θ) =  q  4πϵ0ϵe  ∞  �  l=0  1  bl+1  ϵe (2l + 1)  ϵNDl + ϵe (l + 1)rlPl (cos θ)  (S11)  and the electric field at the center, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' for r = 0, can be calculated as  E (r = 0, θ) =  q  4πϵ0  3  2ϵe + ϵND  b  b3 ,  (S12)  if it is used that in cartesian coordinates one has ez = cos θer − sin θeθ with ez the azimuthally symmetric unit vector  and er and eθ the radial and altitudinal unit vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Electric field variance  The probability of an ion to be located at b witin a sphere of radius R around the nanodiamond is  p (b) =  � 3  4π  1  R3−r3  ND ,  rND ≤ b ≤ R  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S13)  It can be easily verified that this distribution is normalized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  �  R3 d3b p (b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Direct calculation reveals ⟨Ez⟩ = 0  and therefore  σ2  Ez,ion = ⟨E2  z⟩  =  9q2  (4πϵ0)2  1  (2ϵe + ϵND)2  1  R3 − r3  ND  � 1  rND  − 1  R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S14)  Under the assumption that the electric fields generated by the single ions are uncorrelated, the total fluctuations  are given by multiplying the above expression with the number of ions inside the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The standard deviation  σ2  Ez = cNAV σ2  Ez,ion of the electric field components with NA Avogadro’s number, c the molar ionic concentration  and V the volume in which the ions reside therefore is  σEz =  |q|  ϵ0 (2ϵe + ϵND)  �  3NA  4π  �  c  � 1  rNV  − 1  R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S15)  From this it can be seen that the expected electric field fluctuations increase with the molar concentration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  σEz ∝ √c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  HAMILTONIAN IN ROTATING FRAME  As derived by Doherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' [S3], the Hamiltonian of the NV-center in presence of axial magnetic fields Bz  and electric field components Ei with i = x, y, z and ℏ = 1 is  ˆHNV =  �  D + d∥Ez  � ˆS2  z + γeBz ˆSz  + d⊥  �  Ex  �  ˆS2  y − ˆS2  x  �  + Ey  �  ˆSx ˆSy + ˆSy ˆSx  ��  ,  (S16)  with γe = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='8 MHz/G the NV’s gyromagnetic ratio [S4] and d∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='35 Hz · cm/V and d⊥ = 17 Hz · cm/V the axial  and transverse dipole moments [S5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' By rewriting this Hamiltonian in terms of its frequency contributions βz = γeBz,  ξz = d∥Ez and ξ⊥ = d⊥  �  E2x + E2y and by introducing the electric field polarization φE, which defines the transverse  electric field projections via ξx = ξ⊥ cos φE and ξy = ξ⊥ sin φE, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (S16) can be rewritten as  ˆHNV = (D + ξz) ˆS2  z + βz ˆSz − ξ⊥  2  �  eiφE ˆS2  + + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  �  ,  (S17)  where ˆS± = ˆSx ± i ˆSy are spin-1 ladder-operators and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' means the hermitian conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  3  The NV-center can be driven by perpendicular (compared to the NV’s symmetry axis) microwave magnetic fields  of amplitude Ω = γeBd and frequency ωd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  To exert polarized drives onto the NV-center, two wires which are  perpendicular to each other (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 1(a) main text) are operated with a phase φ between each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' This drive can  be described by an Hamiltonian [S6]  ˆHd (t) = Ω  �  ˆSx cos (ωdt) + ˆSy cos (ωdt + φ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S18)  Defining phase-factors ϵ± (φ) =  �  1 − ie∓iφ�  /2, similarly to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  [S6], allows to compactly account for different  polarizations as ϵ+ = 1 only if φ = −π/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' right-circular polarization) and ϵ− = 1 for left-circular polarized  microwave fields (φ = +π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' By transforming ˆHNV + ˆHd (t) into a frame oscillating with ωd through the unitary  U = eiωdS2  z, one can derive the Hamiltonian under the rotating-wave approximation, which is  ˆH = ˆH0 + ˆHd  ˆH0 = (∆ + ξz) ˆS2  z + βz ˆSz − ξ⊥  2  �  eiφE ˆS2  + + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  �  ˆHd = Ω  √  2 (ϵ− |0⟩ ⟨−1| + ϵ+ |1⟩ ⟨0| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S19)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Derivation of time-evolution operators  To allow for the efficient calculation of pulse-sequences, time evolution operators of the free evolution ˆF (τ) and the  drive ˆR (T) will be derived in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Free Evolution  A possible set of eigenstates of ˆH0 is {|0⟩ , |+⟩ , |−⟩} with  |+⟩ = cos θ  2eiφE/2 |1⟩ + sin θ  2e−iφE/2 |−1⟩  |−⟩ = sin θ  2eiφE/2 |1⟩ − cos θ  2e−iφE/2 |−1⟩ ,  (S20)  where tan θ = −ξ⊥/βz, with corresponding eigenenergies ω0 = 0 and ω± = ∆ + ξz ±  �  β2z + ξ2  ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' The time evolution  operator of ˆH0 is ˆF (τ) = �  i={0,±} e−iωiτ |i⟩ ⟨i|, where the sum is performed over all eigenstates of ˆH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' In the basis  of {|0⟩ , |±1⟩} this is  ˆF (τ) = |0⟩ ⟨0| + e−iτ(∆+ξz)�  iξ⊥  x sin (τx)  �  eiφE |1⟩ ⟨−1| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  �  +  �  cos (τx) − iβz  x sin (τx)  �  |1⟩ ⟨1|  +  �  cos (τx) + iβz  x sin (τx)  �  |−1⟩ ⟨−1|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S21)  Here the frequency of oscillation has been defined as x =  �  β2z + ξ2  ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Microwave Drive  To derive operators which describe the action of the microwave pulses, it will be assumed that these pulses  exceed all other frequency scales in magnitude, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Ω ≫ ∆, βz, ξz, ξ⊥, such that  ˆH ≈  Ω  √  2  ˆ�  Hd with ˆ�  Hd =  4  (ϵ− |0⟩ ⟨−1| + ϵ+ |1⟩ ⟨0| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' By noting that ˆ�  H  3  d = ˆ�  Hd, the time evolution  ˆR (t) = e−it ˆ  Hd =  ∞  �  k=0  �  −itΩ  √  2  �n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  � ˆ�  Hd  �n  ,  (S22)  can be calculated as  ˆR (t) = |1⟩ ⟨1|  �  1 − |ϵ+|2�  + |−1⟩ ⟨−1|  �  1 − |ϵ−|2�  − ϵ+ϵ− |1⟩ ⟨−1| − ϵ∗  +ϵ∗  − |−1⟩ ⟨1|  + cos  � tΩ  √  2  � �  |0⟩ ⟨0| + |ϵ+|2 |1⟩ ⟨1| + |ϵ−|2 |−1⟩ ⟨−1| + ϵ+ϵ− |1⟩ ⟨−1| + ϵ∗  +ϵ∗  − |−1⟩ ⟨1|  �  − i sin  � tΩ  √  2  �  (ϵ− |0⟩ ⟨−1| + ϵ+ |1⟩ ⟨0| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S23)  Depending on the polarization, one can induce Rabi oscillations between |0⟩ and either |−1⟩ for φ = π/2 (denoted as  ˆR+) or |+1⟩ (φ = −π/2, ˆR−),  ˆR (t)± = |∓1⟩ ⟨∓1| + cos  � Ωt  √  2  � �  |0⟩ ⟨0| + |±1⟩ ⟨±1|  �  − i sin  � Ωt  √  2  � �  |0⟩ ⟨±1| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S24)  The system can be driven to both |±1⟩, if a linearly polarized drive is used,  R (t)0 = 1  2 (|1⟩ ⟨1| + |−1⟩ ⟨−1| + i |1⟩ ⟨−1| − i |−1⟩ ⟨1|)  + cos  � tΩ  √  2  � �  |0⟩ ⟨0|  + 1  2 (|1⟩ ⟨1| + |−1⟩ ⟨−1| − i |1⟩ ⟨−1| + i |−1⟩ ⟨1|)  �  − 1 + i  2  sin  � tΩ  √  2  �  (|0⟩ ⟨−1| + |1⟩ ⟨0| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S25)  The last expression can similarly be compactly written by noting that (1 ± i) /2 = e±iπ/4/  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' These operators can  then be used to describe the action of (polarized) π- and π/2-pulses onto the |ms = 0, ±1⟩-states of the NV-center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  FOURIER TRANSFORMATION OF FID-SIGNAL  Some arbitrary signals f and ˜f in time- and frequency-domain are connected to each other as  ˜f (ω) = FT [f (τ)] =  � +∞  −∞  dτ f (τ) e−iωτ  FT−1 �  ˜f (ω)  �  = 1  2π  � +∞  −∞  dω ˜f (ω) eiωτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S26)  To simplify the calculation of the Fourier transformed FID-signal, one can rewrite FIDξ⊥,φE (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (6) main text) as  FIDξz,ξ⊥ (τ) = 1  4  �3  2 + 1  2 cos (2τξ⊥) − cos (τ [ξ⊥ + ξz])  − cos (τ [ξ⊥ − ξz])  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S27)  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (S26), one sees that FT [cos (τx)] = π [δ (x − ω) + δ (x + ω)] and therefore  �  FID (ω) = π  4  �3  2δ (ω) + 1  2 [δ (2ξ⊥ − ω) + δ (2ξ⊥ + ω)]  − [δ (ξ⊥ + ξz − ω) + δ (ξ⊥ + ξz + ω)]  − [δ (ξ⊥ − ξz − ω) + δ (ξ⊥ − ξz + ω)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S28)  5  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  SIMULATED PULSE SEQUENCES FOR NORMALLY DISTRIBUTED ELECTRIC FIELDS  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Simulated expected FID-values of FIDξ⊥ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (5) main text), calculated from 500 individual FID-simulations with  drive amplitude of Ω = 10 MHz, intrinsic T ∗  2,int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' and electric field components sampled from a normal distribution with mean  Em and standard deviation σE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Dephasing is considered through a Lindblad-Operator  �  1/T ∗  2,int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' For both mean electric  field values of (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='0 V/µm and (b) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='0 V/µm, it is not possible to resolve ξ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  To understand how fluctuating electric fields alter the FID-signal, we numerically [S7, S8] simulated FIDξ⊥ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (5)  main text) for normally distributed electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Hereby, at every timestep at which the time-evolution is calcuated,  the electric field components are passed from a beforehand sampled normal distribution with mean Em and standard  deviation σE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' S1 that the average FIDξ⊥ signal decays rapidly to its steady-state value of  1/2, which is due to the short T ∗  2 time of 1 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' For this reason it is proposed to use the Hahn-Echo pulse sequence for  measurements of strongly fluctuating electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  0  50  100  150  200  τ in µs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='8  ⟨Hahn (τ)⟩  Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Fit  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  Example of the average Hahn-echo signal, which was obtained numerically from 1000 individual simulations of  the pulse sequence shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 3(a) (main text) with a mean electric field value of Em = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='0 V/µm, standard deviation  σE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='75 V/µm, drive amplitude Ω = 10 MHz and intrisic T2,int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' = 100 µs together with the fit following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (10) (main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  The total T2 value obtained from this fit is T2 = (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='86) µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  As described in the main text, the numerically obtained Hahn-echo trajectories (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' S2 for an example) are well  fitted by ⟨Hahn (τ)⟩ = 1  4  �  1 − cos (2τξ⊥) e−τ/T2�2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Here both the intrinsic T2,int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' = 100 µs and T2,E due to fluctuating  elecric fields contribute to the total T2 via  1  T2  =  1  T2,int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  +  1  T2,E  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S29)  The latter can be fitted in terms of Em and σE via  T2,E = αEm  σ2  E  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  (S30)  The values of the fit parameter α can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content='  6  1  2  Em in V/µm  30  35  40  α  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' Fit parameter α, obtained by numerically fitting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (S29) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' (S30) with T2,int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' = 100 µs to the data from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
+page_content=' 3  (main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE3T4oBgHgl3EQfSAk3/content/2301.04427v1.pdf'}
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+3D ZEROS IN ELECTROMAGNETIC FIELDS
+Alex J. Vernon1†, Mark R. Dennis2‡, and Francisco J. Rodr´ıguez-Fortu˜no1∗
+1Department of Physics and London Centre for Nanotechnology, King’s College London,
+Strand, London WC2R 2LS, UK
+2School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK
+†alexander.vernon@kcl.ac.uk
+‡m.r.dennis@bham.ac.uk
+∗francisco.rodriguez fortuno@kcl.ac.uk
+Abstract. We present a study of 3D electromagnetic field zeros, uncovering their re-
+markable characteristic features and propose a classifying framework. These are a spe-
+cial case of general dark spots in optical fields, which sculpt light’s spatial structure into
+matter-moving, information-rich vortices, escape the diffraction limit for single-molecule
+imaging, and can trap particles for nanoscale manipulation. Conventional dark spots
+are two-dimensional in two aspects: localised in a plane and having a non-zero out-of-
+plane field component. We focus on non-paraxial fields, where three-dimensional dark
+spots can exist non-stably at fully localised points, making distinct imprints in the flux
+of energy and momentum, and in the light’s polarisation texture. With this work, we
+hope to enhance current dark spot applications, or inspire new ones impossible with
+lower-dimensional zeros.
+1. Introduction
+An optical vortex is the name commonly given to a zero in a complex scalar field, such
+as a component of the electric E or magnetic H field. Vortices in these components occur
+naturally in general 3D monochromatic interference [1], where they are infinitely thin con-
+tinuous strands either extending infinitely through space, or coiled into knotted, un-knotted
+or linked closed loops [2]–[5]. On a vortex strand, the phase of the complex scalar field (with
+zero real and imaginary parts) is undefined creating circulation in the phase of the rest of the
+field. This phase increases in a clockwise or anti-clockwise sense by an integer multiple of 2π
+along any closed loop containing one vortex line. Vortex lines in optics have direct analogues
+in acoustics and water waves, and as a type of topological defect, are related to vortices in
+(super)fluids [6] and in Bose-Einstein condensates [7], and even cosmic strings [8]. Strong
+research interest in optical vortices over the past 30 years, combined with the availability
+of instruments and the flexibility in generating [9]–[12] and structuring [13] vortex-carrying
+beams, has positioned optics to act as a sandbox for exploring topological phenomena that
+appear more broadly across physics.
+When considering the full 3D vector characteristics of an optical field, vortex lines in
+individual field components like Ex, Ey, and Ez are basis-dependent and not so physically
+meaningful. By picturing these different scalar vortex threads permeating the vector field,
+we can appreciate how unlikely it is that the optical field is zero at a point (i.e. E = 0, all
+1
+arXiv:2301.03540v1  [physics.optics]  9 Jan 2023
+
+2
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+three components simultaneously zero) in typical 3D interference (the vortex line in each of
+the three field components would meet at such a zero point, requiring the manipulation of
+three extra parameters beyond the spatial x, y, z). Despite the rarity of zeros in the wild, a
+lower-dimensional version can be readily manufactured in optical beams, and is remarkably
+well-studied. Paraxial doughnut beams have an axial zero in the transverse field surrounded
+by a bright ring, and are used in modern spectroscopy techniques [14], [15] because of the
+zero’s immunity to the diffraction limit. The transverse field effectively consists of one or
+two scalar components with the vortex line along the beam axis, causing the real part of
+the local wavevector to curl around the axis and imbue the beam with intrinsic orbital an-
+gular momentum. The longitudinal field, meanwhile, is non-zero (albeit very small due to
+paraxiality) in the centre of the beam which, therefore, is better imagined not as an exact
+axial zero, but as a dim line of linear polarisation (an L line) polarised parallel to the beam
+direction. This, and its confinement in only two dimensions, stretching along the third, is
+why we refer to the almost-dark centre of the doughnut beam as a two-dimensional zero.
+Its topological index is straightforward to define by counting how many multiples of 2π the
+phases of the transverse components climb through over an enclosing circuit. The intrinsic
+orbital angular momentum carried by doughnut beams is the key property of the spatial
+structure of light that can rotate matter [16], [17] and store information [18]–[20].
+Surprisingly, the fully localised, three-dimensional optical field zero, E = 0, has been left
+largely unexplored. This is probably due to its unstable nature—a perturbation will destroy
+the zero point (i.e. cause the vortices in the three components no longer to coincide). Never-
+theless, such a point is theoretically possible and can be artificially synthesised [21], but very
+little is understood about how it is imprinted into the surrounding field, and there is no clas-
+sifying topological index like the topological charge of a 2D vortex. The 3D electromagnetic
+field zero is the focus of this work. A zero in the E-field alone has codimension 6, requiring
+that the six total degrees of freedom of two real, three-dimensional vectors (the real and
+imaginary parts of the three components E) are suppressed simultaneously. This means 3D
+zeros exist stably in a six-dimensional parameter space, and is why optical field zeros are
+not natural in random interference patterns spanning only three spatial dimensions, being
+hidden by instability. Instead, 3D zeros must be revealed by tuning an additional three
+parameters (this is discussed in [22] for a zero in two electric field components). Some of
+these parameters could be the polarisation components of a plane wave, for example, and in
+fact, 3D zeros can be very easily manufactured and controlled in pure plane wave interfer-
+ence or near fields with a simple technique [21], and their higher dimensional confinement
+could provide a greater degree of precision in dark spot spectroscopy. Due to their electric
+field dependence, the zero in E is coupled to a collection of singularities, each with its own
+topological signature, in various physical quantities associated with the light field includ-
+ing the complex Poynting vector, canonical momentum, spin momentum and spin angular
+momentum. Learning how energy flow and momentum circulate around a 3D vortex could
+inspire applications which would be otherwise unfeasible using typical lower dimensional
+zeros. Alternatively, the magnetic field H may vanish at a point, or more extremely, both
+E and H might simultaneously vanish, giving a true electromagnetic null with codimension
+12. Here, we report the key features of a 3D field electric or magnetic field zero, including
+the way that polarisation singularities are forced to intersect and the flux of the complex
+Poynting vector and canonical and spin momentum. With these findings, for the first time,
+we propose a framework to classify the physically realisable varieties of 3D field zero.
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+3
+2. Results
+To contextualise our study, we begin with some brief intuition on the special features
+which we might expect to find near to a 3D zero.
+If either E or H is zero at a point r0, then of that field, say E, the flux of energy, canon-
+ical momentum, spin angular momentum (and other quantities) are zero too. Since these
+fluxes are vector quantities, their direction is singular at r0 and an imprint is made in the
+surrounding space where they are well-defined. In three spatial dimensions, even if these
+fluxes are divergence-less, there is more than one possible (topologically unique) imprint
+which can be left by and characterise the zero in E. The electric field spin is particularly
+interesting, because its zeros (in non-paraxial fields) are co-dimension 2 objects—meaning
+they are one-dimensional continuous lines, defining the threads of pure linear electric polar-
+isation. This continuity should require at least one zero-spin line, an L line, to pass through
+r0. A similar argument can be made for lines of pure circular electric polarisation, except
+that C lines are defined by a complex quadratic equation, E · E = 0, equivalent to a real
+quartic equation, |E · E|2 = 0, which has either zero, two or four real roots. It turns out,
+as we will show, that a given number of C lines and L lines must always intersect in a 3D
+electric field zero. Before reporting these and other findings in detail from mathematical
+argument and analytical simulations in section 2.3 and beyond, the next two subsections 2.1
+and 2.2 provide an overview of polarisation singularities and set out our way of classifying
+3D field zeros using dyadics associated with the field.
+2.1. Overview of Polarisation Singularities in Paraxial and Non-Paraxial Fields.
+L lines and C lines are called polarisation singularities and are the vector version of scalar
+vortex lines in wave fields, existing in light [23]–[25], acoustic and water waves [26] (both
+acoustic and water waves have a vector nature [27], [28]) where some property of the general
+polarisation ellipse is not defined. In 3D fields, polarisation singularities are often described
+as the underlying skeleton which embeds highly complex topologies into the field’s polar-
+isation texture [29], [30].
+Polarisation singularities have been studied in full 3D and in
+paraxial fields [31], where in paraxial fields (considering only the two transverse field com-
+ponents), polarisation is circular at points and linear along lines. Propagating the paraxial
+field (maintaining the transverse polarisation) draws out the C points and L lines in the
+transverse plane into C lines and L surfaces in three dimensions.
+A polarisation ellipse has orthogonal semi-major and semi-minor axes, telling us which
+way the ellipse is oriented. But because a polarisation circle has no semi-major or semi-
+minor axes, at a C point, the orientation of the circle is undefined causing neighbouring
+polarisation ellipses (almost circular) to rotate when tracked along a C point-enclosing loop.
+The ellipse major axis is described throughout space with a line field, in that the axis is
+oriented some way in space but does not point one way or another—an ellipse looks identi-
+cal to its 180 degree rotated self. This means that along the enclosing circuit, the rotating
+ellipses turn continuously through an integer multiple of π radians, rather than 2π, which
+is why C points are assigned a half-integer index. When the field is fully three dimensional
+and the polarisation ellipse is free to tilt in any Cartesian direction, circular polarisation still
+occurs along one-dimensional threads (C lines which are no longer straight as in the paraxial
+case) but the surrounding polarisation ellipses also twist, so that their major axes sweep out
+M¨obius strips [32]–[34]. Analogues of C lines exist in polychromatic fields, shaping the rest
+of the field into other remarkable topological structures [35].
+
+4
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+L lines/L surfaces in paraxial fields (ignoring longitudinal fields) separate regions of left
+and right handed polarisation ellipses.
+In non-paraxial fields, L lines are strictly one-
+dimensional lines (not surfaces) and complement C lines in shaping the surrounding po-
+larisation structure.
+This reduction of dimension to the L entity occurs because, to be
+linearly polarised, the real and imaginary parts of the field (say E = p + iq) need to be
+(anti)parallel (not necessarily equal). If E is paraxial and linearly polarised, then in the
+transverse plane, the ratio of the x components of p and q must equal the ratio of their y
+components—a single condition, dissolving only one degree of freedom of one vector relative
+to the other. If E is non-paraxial, then an extra condition accounting for the ratio of the
+z components of p and q must be satisfied for linear polarisation [23]. Between paraxial
+and full 3D fields, the linear polarisation object’s codimension, which is the dimension of
+the electric spin angular momentum field SE minus the dimension of the L line/L surface
+which lies in SE, increases from one to two. The spin angular momentum of the field is zero
+when linearly polarised, meaning the direction of the normal to the field oscillations cannot
+be defined. Drawing a circuit around an L line, the spin vector rotates through 2π radians
+in a clockwise or anti-clockwise sense and defines the L line’s topological index.
+The characteristics of scalar vortices and C lines and L lines are visualised in Fig. 1.
+2.2. Indexing Point-like Singularities. Polarisation singularities occur equally often
+among the general polarisation ellipses in E and H fields, and need not coincide with each
+other. Phase singularities, C lines and L lines are all indexed by looking at the circulation or
+rotation of a scalar or vector quantity around a loop enclosing the singularity of interest [36].
+All three of these singularities are threads in 3D fields, but the winding number concept can
+be generalised to higher dimensional singularities and calculated for point-like, 3D vector
+singularities via the topological degree. Instead of integrating a quantity associated with
+a line singularity around a 1D closed circuit, for isolated singular points in 3D, we should
+integrate an appropriate quantity over a closed surface enclosing the point singularity. For
+a vector V(rS) on a surface S (rS ∈ S) in 3D real space, for example, the topological de-
+gree of rS �→ V (the mapping from the real space surface rS to V) is a calculation of the
+integer number of times that every possible direction of V is realised (on a sphere) on all
+the points rS on the surface S. As with other kinds of topological singularities in physical
+fields, the easiest realised topological degrees (winding numbers) are ±1. Mathematically, a
+0, ±1 topological degree is the integral of the determinant of the dyadic D(V) of V over S
+divided by A, the area of S,
+(1)
+deg(V) = 1
+A
+�
+S
+det(D(V))dS.
+The dyadic D(V), also called the Jacobian matrix of V, contains the first order spatial
+derivatives of each component of V.
+The sign of the determinant of D(V) equals the
+product of the signs of its eigenvalues. For 3D vectors where D(V) is a 3 × 3 matrix, it
+is possible for drastically different behaviour of V to be hidden under the same topological
+degree. For example, if V(r = 0) = 0 (meaning the direction of V is singular at the origin)
+and we assume that a linear map from an origin-enclosing surface to V has a topological
+degree of −1, then D(V) at r = 0 could have signed eigenvalues (in any order) of + + −
+or − − −. Physically, the origin could either be a saddle point or a sink for V with no
+distinction in topological degree because both + + − and − − − eigenvalues multiply to a
+negative sign. Rather than calculating the topological degree, to try to classify the flux of
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+5
+C-l
+in
+e
+L
+-l
+in
+e
+S
+c
+a
+l
+a
+r
+V
+o
+r
+t
+e
+x
+±
+2
+l
+π
+on-plane view
+on-plane view
+Figure 1. Visualisation of scalar and polarisation singularities in a non-
+paraxial electromagnetic field. Scalar vortices (black line) exist in complex
+scalar fields, such as the components of E, where the scalar field is zero
+and its phase is undefined, forming 1D threads in the interference of three
+or more plane waves. Around a scalar vortex line, the phase of the field
+increases by an integer l multiple of 2π in a clockwise or anticlockwise
+sense. Singular lines exist in the complex vector characteristic of E and
+H fields, called polarisation singularities, which include C lines (lines of
+circular polarisation) and L lines (lines of linear polarisation). In a circuit
+around a point on a C line (blue line), in the plane of the polarisation
+circle at that point, nearby polarisation ellipses rotate through an integer
+multiple of π radians. Around an L line (green line), the normal to nearby
+polarisation ellipses rotates by an integer multiple of 2π radians.
+energy and canonical momentum through a 3D optical field zero, we use the signs of the
+eigenvalues of their first order dyadics evaluated in the position of the field zero.
+We use the ideas discussed here to report our findings in the following sub-sections,
+beginning with the six possible ways that C lines and L lines can intersect in a 3D zero.
+2.3. Polarisation Singularities at a 3D Electric Field Zero. We will focus on a 3D
+electric field zero in a position r0, that is E(r0) = 0, and study the nearby strands of circular
+
+6
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+and linear electric polarisation. Identical arguments to those given here could be made for
+magnetic field zeros (H(r0) = 0) and magnetic polarisation singularities, or for simultaneous
+electric and magnetic field zeros (E(r0) = H(r0) = 0) and polarisation singularities of either
+E or H. Any smooth function of r is nearly linear over small distances, which means all
+fundamental behaviour of the electric field in the immediate vicinity of the zero is captured by
+its Jacobian, JE = D(E), a complex 3×3 matrix containing all first-order spatial derivatives
+of Ex, Ey and Ez, evaluated at r0,
+(2)
+JE = D(E) =
+�
+�
+�
+∂Ex
+∂x
+∂Ex
+∂y
+∂Ex
+∂z
+∂Ey
+∂x
+∂Ey
+∂y
+∂Ey
+∂z
+∂Ez
+∂x
+∂Ez
+∂y
+∂Ez
+∂z
+�
+�
+� = (∇ ⊗ E)T .
+The Jacobian of the magnetic field at r0, JH, can be defined similarly. In free space, JE
+and JH are always traceless because E and H are divergence-free. Maxwell’s equations also
+require that if E(r0) = 0, then JH must be symmetric at r0 and vice versa for H(r0) = 0.
+We make a first-order approximation of the electric field vector near r0 with,
+(3)
+˜E = JEv,
+where v = r−r0.
+Nearby C lines emerge in our approximated field wherever ˜E · ˜E = 0, which we may
+calculate using (3) and separate into real and imaginary parts,
+(4)
+˜E · ˜E = (JEv) · (JEv)
+= vT Mv + ivT Nv,
+where M = Re{JT
+EJE} and N = Im{JT
+EJE}. The two terms in equation (4) are quadric
+surfaces connecting constant valued real and imaginary parts of ˜E · ˜E, and the real and
+imaginary surfaces described by setting (4) equal to zero cross in real space where ˜E is
+circularly polarised. The real 3 × 3 matrices M and N are symmetric and always have real
+eigenvalues. Normally, these eigenvalues have signs + + − or − − + (in any order) so that
+the surfaces vT Mv = 0 and vT Nv = 0 are both double cones, vertices touching at v = 0
+as shown in Fig. 2(a). The cones have an elliptical cross section whose ellipticity is constant
+with distance from v = 0 in the linear approximation. Because two ellipses can intersect
+at either zero, two or four points (as shown in the lower part of Fig. 2(a)), there must be
+either zero, two or four C lines passing through the electric field zero. If one matrix, say
+M, is positive or negative definite (all positive or all negative eigenvalues), Re{˜E · ˜E} will
+solely increase or decrease in all outward directions from v = 0. Then, the constant-valued
+surface vT Mv = C becomes an ellipsoid, and vT Mv = 0 is satisfied only at v = 0 so that
+no C lines pass through the 3D vortex.
+To reveal the number of L lines that extend through the 3D electric field zero, we must
+calculate the electric field spin, given by,
+(5)
+SE ∝ Im{E∗ × E} = 2Re{E} × Im{E}.
+When the electric field is linearly polarised (SE = 0), the real and imaginary parts of E
+must be (anti)parallel. Under the approximation (3), this means,
+(6)
+Re{JE}v = λIm{JE}v,
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+7
+b
+a
+L-line
+C-line
+cone cross section
+on unit sphere
+Re{E · E} = 0
+E = 0
+Im{E · E} = 0
+2
+0
+4
+Figure 2. Electric polarisation singularities passing through a 3D electric
+field zero at a position r0. (a) Visualisation of why zero, two or four C
+lines must pass through r0. In a first-order approximation, the surfaces
+Re{E · E} = 0 (red) and Im{E · E} = 0 (blue) are double cones, and where
+they intersect, C lines exist. Two double cones intersect along two or four
+lines, or do not intersect at all, which is easy to see by considering the cones’
+cross sections on the unit sphere: ellipses which cross at zero, two or four
+points. (b) Six different examples of electric field zeros created at a position
+r0 (red circle), one per unique combination of C lines and L lines meeting
+there. The C lines are marked by blue regions where E · E ≈ 0 and the L
+lines by the green regions where Im{E∗ × E} ≈ 0. Each field zero is created
+in analytical simulations by designing the polarisation of ten plane waves
+with random wavevectors, wavelength 500 nm, to interfere destructively at
+r0. The plane waves have different polarisations and wavevectors for each
+example zero in (b).
+where λ is a positive or negative scalar. The directions of the L lines crossing through v = 0
+are given by the three eigenvectors of the matrix Im{JE}−1Re{JE}. Since this matrix is
+real-valued, either all three of these eigenvectors are real, corresponding to three L lines, or
+only one of them is real and is accompanied by a conjugate pair of complex eigenvectors. In
+that case, just one L line passes through the 3D zero because v cannot point in a complex
+direction.
+Summarising, either zero, two or four C lines and either one or three L lines always meet
+at r0 in a 3D electric field zero E(r0) = 0. An identical conclusion can be drawn for C lines
+and L lines of the magnetic field for the case of H(r0) = 0. In Fig. 2(b), an example of
+each of the six possible C line/L line combinations through a 3D zero is presented, the zeros
+
+8
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+created in the interference of ten plane waves. Each zero is enforced by separate ensembles
+of ten plane waves with random wavevector directions that are deliberately polarised to
+destructively interfere at a single point.
+2.4. Energy Flux Singularity. The flow of energy in a light field is described by the
+complex Poynting vector,
+1
+2E∗ × H.
+The real part of this vector (often itself called the
+‘Poynting vector’) corresponds to the time-averaged power transfer (sometimes known as
+active power) in the field, while reactive power (associated with oscillations in the transfer
+of power) is accounted for by the less-used imaginary part. We refer to these two real vectors
+as Pr and Pi,
+(7)
+Pr = 1
+2Re{E∗ × H}
+(8)
+Pi = 1
+2Im{E∗ × H}
+When either E or H is zero at a point r0, the complex Poynting vector vanishes, and its
+real and imaginary parts circulate in the space around the zero according to their first-order
+derivatives at r0. The real part Pr is divergence-less in free space where there is no absorption
+or energy generation, and must therefore be organised into a vector saddle point at r0. An
+example flow of active power around a 3D electric field zero created at r0 (E(r0) = 0,
+H(r0) ̸= 0) is given in the top row of panels in Fig. 3, where Pr is plotted on the xy, xz, and
+yz planes coinciding at r0. Although there is no net flow of active power in or out of the zero,
+Pr streamlines can be arranged in two topologically different ways depending on whether
+the signs of the eigenvalues of its first-order dyadic, Im{(JT
+E − JE)J∗
+E} (written electrically
+without prefactors), are + + − or + − −, corresponding to two possible topological orders
+of −1 or +1. One might notice that the imaginary Poynting vector Pi, which is plotted on
+the same planes for the same free space electric field zero at r0 in the lower row of panels of
+Fig. 3, is not divergence-free—in fact, it is physically possible for a source, sink or saddle of
+Pi to exist there, depending on whether E or H is zero. To see why, we first note that using
+Maxwell’s equations in free space (see supplemental information), the imaginary Poynting
+vector can be decomposed into a sum of two terms, one polarisation-independent and one
+polarisation-dependent, each containing electric and magnetic contributions,
+(9)
+Pi = − c2
+2ω ϵ0Re{(JT
+E − JE)E∗}
+= c2
+2ω µ0Re{(JT
+H − JH)H∗}
+= c2
+4ω
+�
+−1
+2ϵ0∇(E∗ · E) + 1
+2µ0∇(H∗ · H)
+�
++ c2
+4ω Re{ϵ0JEE∗ − µ0JHH∗}.
+The first term in Eq. (9) represents the difference in gradient of the electric and magnetic
+energy density of the light field, while polarisation-dependent behaviour of Pi derives from
+the second term since JEE∗ and JHH∗ contain inter-component multiplication. In certain
+cases such as a uniformly polarised standing wave, the second term is zero and the gradient
+of the difference in electric and magnetic energy density determines the direction of reactive
+power flow. Because E∗ · E = |E|2 is a positive real quantity, a 3D zero in E is a source for
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+9
+Pi
+Pr
+x
+y
+0.08λ
+x
+z
+y
+z
+y
+x
+x
+y
+z
+z
+Figure 3. Flow of the real (Pr, red) and imaginary (Pi, teal) parts of the
+Poynting vector, 1
+2E∗ × H, on the xy, xz and yz planes coinciding with an
+electric field zero at position r0 (blue circle). The real Poynting vector is
+divergence-free, meaning a vector saddle point of Pr is set up at r0. The
+imaginary Poynting vector is not necessarily divergence-free and can be or-
+ganised in a sink at r0 when E(r0) = 0 (a source is not possible unless
+the magnetic field is zero). Results are generated by designing the polari-
+sation of ten plane waves with random propagation directions to interfere
+completely at r0.
+the vector ∇(E∗ · E) (and likewise for H). Depending on how the polarisation-independent
+and polarisation-dependent terms combine in Eq. (9), the imaginary Poynting vector could
+have non-zero divergence at r0. Note that there is a difference in sign between the electric
+and magnetic terms in Eq. (9), meaning Pi behaves differently for E(r0) = 0, H(r0) ̸= 0
+and H(r0) = 0, E(r0) ̸= 0 and E(r0) = H(r0) = 0 3D zeros. To understand the flow of
+Pi through an optical field zero, we assume a non-dual electric field zero (E(r0) = 0 and
+H(r0) ̸= 0) and make a first-order approximation of Pi, this time referring to the relevant
+linear transformation matrix as the first-order dyadic of the imaginary Poynting vector,
+D(Pi), which is defined identically to JE in Eq. (2) with Pi and its components in place of
+E. Our approximate imaginary Poynting vector is,
+(10)
+˜Pi = D(Pi)v
+
+10
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+where v = r − r0. The dyadic D(Pi) = (∇ ⊗ Pi)T evaluated at r0 is, using the electric
+representation of Pi in Eq. (9) (top line),
+(11)
+D(Pi) = − c2
+2ω ϵ0Re{(JT
+E − JE)J∗
+E}.
+There are no second order derivatives of E in Eq. (11) because E(r0) = 0. Surprisingly,
+D(Pi) cannot have three positive eigenvalues, as justified in the supplemental information.
+The result is that at a 3D electric field zero, Pi is organised into one of two types of sad-
+dle with topological degree 1 or −1, or a sink with topological degree −1, never a source.
+When H(r0) = 0 and E(r0) ̸= 0, the opposite is true because of the dual-asymmetry of the
+imaginary Poynting vector: Pi can form a saddle or source at r0 but not a sink.
+2.5. Orbital Current Singularity. When divided by c2, the real Poynting vector Eq. (7)
+turns into a momentum density, the kinetic momentum density, which, using Maxwell’s
+equations for time-harmonic fields, can be split in to a well-known sum of separate orbit
+and spin contributions [37], [38]. For instance, by substituting (with prefactors) the curl of
+E for H, the kinetic momentum density can be written as,
+(12)
+Π =
+1
+2c2 Re{E∗ × H}
+= 1
+2ω ϵ0Im{E∗ · (∇)E} + 1
+2ω ϵ0∇ × 1
+2Im{E∗ × E},
+where A · (∇)B = Ax∇Bx + Ay∇By + Az∇Bz = JT
+BA, with JB being the Jacobian
+of B defined identically to Eq. (2) (the decomposition is explained in more detail in the
+supplemental information). The first decomposed term is po
+E, the orbital contribution to
+the kinetic momentum density, called the canonical momentum density, imparted by the
+electric field only,
+(13)
+po
+E = 1
+2ω ϵ0Im{E∗ · (∇)E} = 1
+2ω ϵ0Im{JT
+EE∗}.
+Eq. (12) can also be written purely in terms of H and by averaging these equivalent repre-
+sentations of Π, the dual-symmetric canonical momentum density is obtained,
+(14)
+po = 1
+4ω Im{ϵ0E∗ · (∇)E + µ0H∗ · (∇)H}.
+This momentum density definition contains both the electric and magnetic field’s influence,
+and produces the total orbital angular momentum of the field within a volume when r×po is
+integrated. Naturally, the electric and magnetic contributions to (14) become zero whenever
+E = 0 and H = 0. This means that, in a 3D electric field zero positioned at r0, the direction
+of the electric contribution po
+E is undefined and should circulate around r0 in some fashion.
+Of course, while the total canonical momentum density at r0 is not zero when only E = 0,
+we could draw the same conclusions we make here for Eq. (14) rather than Eq. (13) near a
+dual 3D vortex (E(r0) = H(r0) = 0). Note that by normalising E, the argument to Im{}
+in Eq. (13) defines the local electric wavevector [25],
+(15)
+ke
+loc = −ie∗ · (∇)e,
+where e =
+E
+√
+E∗·E. The real part of ke
+loc is the local phase gradient of the electric field, while
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+11
+Re{kloc} on yz plane
+xplane = -25 nm
+xplane = -95 nm
+xplane = 0 nm
+xplane = +25 nm
+|Re{kloc}| < 0.1*k
+r0
+Figure 4. Vortex pseudo-line (red) of the real electric local wavevector,
+Re{ke
+loc}, passing though an electric field zero at position r0 (blue circle),
+that is E(r0) = 0 and H(r0) ̸= 0. The red line indicates regions of space
+where |Re{ke
+loc}| < 0.1k, where k = 2π
+λ and λ = 500 nm. The line is roughly
+oriented along the x axis and the electric local wavevector is plotted on four
+different yz planes. The three planes which coincide with the line are −25
+nm, 0 nm, +25 nm in the x direction away from r0, showing clear vortex-
+like circulation of momentum around the axis of the red line. On the fourth
+plane −95 nm away from r0, the vortex-like circulation of Re{ke
+loc} has lost
+some definition, highlighting that Re{ke
+loc} is not exactly zero along a line,
+and only appears line-like near to the E field zero (the only location where
+Re{ke
+loc} is exactly zero is at r0, because Re{ke
+loc} vanishes at points, not
+along lines). Results are generated from interference of ten plane waves
+with random wavevectors, wavelength λ = 500 nm, deliberately polarised
+to create a 3D electric field zero at r0.
+
+12
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+Im{ke
+loc} points in the direction of decreasing electric field intensity. A three-dimensional,
+real vector, Re{kE
+loc} (and therefore canonical momentum density) can vanish at localised
+points in space with non-zero electric field, where a saddle-like circulation of Re{kE
+loc} sur-
+rounds [39], similar to the top row of panels in Fig. 3. But when the electric field vanishes
+and the direction of Re{kE
+loc} is automatically undefined, a different behaviour emerges.
+To understand why, we once again make a first-order approximation, this time of po
+E,
+capturing the electric canonical momentum very near to a 3D electric field zero at r0 in its
+dyadic D(po
+E),
+(16)
+˜po
+E = D(po
+E)v,
+where v = r − r0. The dyadic D(po
+E) = (∇ ⊗ po
+E)T at a general point in space is given by,
+(17)
+D(po
+E) = 1
+4ω ϵ0Im{JT
+EJ∗
+E} + 1
+4ω ϵ0Im{E∗
+xHess(Ex) + E∗
+yHess(Ey) + E∗
+zHess(Ez)},
+where Hess(A) is the Hessian matrix of the scalar field A,
+(18)
+Hess(A) =
+�
+�
+�
+∂2A
+∂x2
+∂2A
+∂x∂y
+∂2A
+∂x∂z
+∂2A
+∂y∂x
+∂2A
+∂y2
+∂2A
+∂y∂z
+∂2A
+∂z∂x
+∂2A
+∂z∂y
+∂2A
+∂z2
+�
+�
+� .
+As E approaches zero, the trace-less matrix D(po
+E) is dominated by the first term in Eq. (17)
+and if evaluated at a location r0 where E(r0) = 0, the linear approximation of po
+E responds
+only to the properties of the matrix in the first term of Eq. (17), Im{JT
+EJ∗
+E}. This is an
+anti-symmetric matrix which always has one zero and two purely imaginary eigenvalues,
+meaning that in the direction of the one real eigenvector of D(po
+E) at r0, the approximated
+electric canonical momentum does not increase at all, producing a zero-momentum line. The
+imaginary eigenvalues of D(po
+E) twists po
+E into a surrounding vortex-like structure. This
+special type of vector field singularity is called a circulation. Fundamentally, the canonical
+momentum should only be zero at confined points in general 3D fields, so this apparent
+vortex line is only preserved locally to the electric field zero at r0, dissolving with distance
+as higher-order derivatives of po
+E become significant (it is, in fact, just a very elongated null
+point of po
+E). The direction of the vortex pseudo-line in the vicinity of the electric field zero
+is also given by the curl of the orbital current,
+(19) D = ∇×po
+E ∝ Re{∇Ex}×Im{∇Ex}+Re{∇Ey}×Im{∇Ey}+Re{∇Ez}×Im{∇Ez}.
+We visualise this feature in Fig. 4, where a 3D electric field zero is created at a point r0
+by deliberately polarising ten plane waves, each with random wavevectors, to destructively
+interfere at r0. The real part of the electric local wavevector, Re{ke
+loc}, the real part of
+Eq. (15), is calculated and the region of space where |Re{ke
+loc}| < 0.1k (k = 2π
+λ ) is revealed
+by a red line approximately 0.1λ in length. The electric local wavevector is proportional to
+po
+E and shows the direction of canonical momentum carried by the electric field. This red
+line is not continuous; Re{ke
+loc} actually vanishes only at r0 but it increases in magnitude
+so slowly in a certain direction (the direction of the real eigenvector of Im{JT
+EJ∗
+E}) that a
+line-like structure of |Re{ke
+loc}| ≈ 0 exists very near to r0, stirring the electric canonical
+momentum into a local vortex. This is shown by the four yz planes on which Re{ke
+loc} is
+plotted in Fig. 4. The real part of the electric local wavevector forms a swirl around the
+red line, a swirl losing definition if the plotting plane is too far from r0. This remarkable
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+13
+structure always appears when all three electric field components are zero together at a point.
+2.6. Spin Current. In the decomposition of the kinetic momentum density, Eq. (12), the
+second term is called the spin momentum. It is proportional (and should not be confused
+with) the curl of the spin angular momentum of the electric, magnetic or electromagnetic field
+depending on the representation. Like before, we will focus on the electric representation of
+the decomposed kinetic momentum density, referring to the electric spin momentum with
+ps
+E,
+(20)
+ps
+E = 1
+2ω ϵ0∇ × 1
+2Im{E∗ × E} = − 1
+2ω ϵ0Im{JEE∗}.
+The electric spin momentum is a divergence-free vector whose dyadic D(ps
+E) has three non-
+zero eigenvalues when evaluated in the position of an electric field zero, organising ps
+E into
+one of two types of 3D vector saddle point, just like the real Poynting vector in Fig. 3.
+Expressing, in Eq. (20), the electric spin momentum with the electric field Jacobian reveals
+that only a difference in sign and orientation of JE separates ps
+E from the electric canon-
+ical momentum po
+E, given by Eq. (13). This means that, in a dual electric-magnetic zero,
+E(r0) = H(r0) = 0, where JE is symmetric from Maxwell’s equations, the spin and canoni-
+cal momentum dyadics are equal and opposite, D(ps
+E) = −D(po
+E) (this also means that the
+dyadic of the real Poynting vector is zero). In a first-order approximation of both ps
+E and
+po
+E near r0 in this case, a zero-line exists in exactly the same place for both vectors, and
+around it, ps
+E and po
+E have vortex-like circulation with opposite handedness to each other.
+2.7. Spin Angular Momentum. The dual spin angular momentum, created by the rota-
+tion of the electric and magnetic field vectors, is given by [40],
+(21)
+S = 1
+4ω Im{ϵ0E∗ × E + µ0H∗ × H}.
+The electric and magnetic parts individually describe the ellipticity of the electric and mag-
+netic polarisation ellipses, pointing in the perpendicular direction to the ellipse plane. Once
+more for simplicity, we will focus on the singularity in the electric field spin angular momen-
+tum, SE =
+1
+4ωIm{ϵ0E∗ × E}, left in a 3D electric field zero positioned at r0. The total spin
+angular momentum, Eq. (21), is not zero if only E(r0) = 0, but we could draw similar con-
+clusions for S as we do here for SE when the electric and magnetic fields are simultaneously
+zero at r0.
+Decomposing SE using Maxwell’s equations, we can write its first-order dyadic at r0 in
+terms of the light field Jacobian matrices (see supplemental material),
+(22)
+D(SE) =
+1
+4ω2 ϵ0Re{(JT
+H − JH)J∗
+E}.
+We note that Eq. (22), describing the spatial derivatives of the electric field spin only, de-
+pends on the magnetic field Jacobian matrix JH, which is automatically symmetric whenever
+E = 0 from Maxwell’s equations. The consequence is that JT
+H − JH = 0 and all elements
+of D(SE) at r0 are zero when E(r0) = 0. Higher-order derivatives of SE (Hessian matrices
+for each component) need to be calculated to fully understand the flux of the electric spin
+angular momentum in the neighbourhood of a 3D zero in E.
+
+14
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+Matrix at r0
+Characteristic
+JE
+3D complex Jacobian of the electric field at r0 (Eq. (2))
+1.
+Im{JE}−1Re{JE}
+Number of real eigenvalues is the number of L lines passing through r0
+2.
+Re{JT
+EJE}
+Eigenvectors are the principle axes of the double cone Re{E · E} = 0.
+Number of intersections of this double cone with that of matrix 3 are the number of C lines.
+3.
+Im{JT
+EJE}
+Eigenvectors are the principle axes of the double cone Im{E · E} = 0.
+Number of intersections of this double cone with that of matrix 2 are the number of C lines.
+4.
+Im{JT
+EJ∗
+E}
+Direction of real eigenvector (there is only one) is the axis of the electric local wavevector vortex.
+Imaginary eigenvectors give the handedness of momentum circulation.
+5.
+−Im{JEJ∗
+E}
+Proportional to first-order dyadic of spin current (Eq. (20)).
+Eigenvalue signs give the type of minimum at r0
+6.
+Im{(JT
+E − JE)J∗
+E}
+Proportional to first-order dyadic of real Poynting vector (active power flow).
+Eigenvalue signs give the type of minimum at r0
+7.
+−Re{(JT
+E − JE)J∗
+E}
+Proportional to first-order dyadic of imaginary Poynting vector (reactive power flow).
+Eigenvalue signs give the type of minimum at r0
+Table 1. Summary of the seven dyadics (numbered) which classify the
+vector field singularities organised by a 3D electric field zero.
+2.8. Summary Table. Here, in Table 1, we summarise the seven dyadics which classify
+the number of crossing C lines and L lines, the flux of the real and imaginary parts of the
+Poynting vector, the spin current, and the orientation of the canonical momentum vortex
+pseudo-line existing at a 3D electric field zero, E(r0) = 0 while H(r0) ̸= 0. To characterise
+a magnetic field zero, the matrices can be written magnetically by substituting JE for JH
+(and changing the ‘−’ sign in front of matrix 7 to a ‘+’), in which case matrices 1, 2, and
+3 characterise magnetic polarisation singularities, and matrix 4 and 5 the magnetic local
+wavevector and spin current respectively. In the case of a dual 3D zero, E(r0) = H(r0) = 0,
+matrices 6 and 7 are zero because both JE and JH are symmetric.
+3. Discussion
+Three-dimensional optical field zeros are co-dimension 6 entities which, unlike axial zeros
+in beams, are completely localised, the optical field growing brighter in all outward directions.
+Although they rarely occur naturally in light (requiring three additional parameters beyond
+spatial x, y, z due to their codimension), 3D zeros can be deliberately created in plane wave
+interference or in the near fields of light-scattering matter [21] to reveal the unusual features
+they imprint in the light field’s energy, wavevector and polarisation structures. Both with
+mathematical argument and by creating field zeros in plane wave interference, we showed
+that whenever the electric or magnetic field is zero at a point r0, then some combination of
+zero, two or four C lines, lines of pure circular polarisation, and one or three L lines, lines of
+pure linear polarisation of the field in question, intersect at r0. Likewise, an imprint is made
+at r0 in the surrounding flux of the parts of the complex Poynting vector 1
+2E∗ × H, the local
+wavevector, the spin momentum and spin angular momentum, each organised in a vector
+source, sink or saddle point. The signs of the eigenvalues of the first-order dyadics of each
+quantity at r0 reveal this. Of particular interest is the canonical momentum: while typically
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+15
+vanishing at confined points in space, a zero in E or H at r0 twists the canonical momentum
+imparted by that null-containing field into a sub-wavelength, vortex-like structure around
+an axis with an easily calculated direction. We say it is a sub-wavelength object because,
+although it resembles the twisted vortex structures of well-known doughnut beams, it is
+not preserved with increasing distance from r0.
+In the combination of the way energy
+flows through r0 and the number of intersecting polarisation singularities, any 3D field zero
+inscribes one of a discrete number of topologically unique signatures in the electromagnetic
+field. We identify seven dyadics whose spectra could classify all physically possible imprints
+of 3D optical field zeros.
+It is tempting to speculate that a surface enclosing an electric or magnetic field point
+zero might, in addition to the quantities already identified, possess a nonzero topological
+Chern number due to a nontrivial geometric phase 2-form (Berry curvature) resulting from
+the neighbouring polarisation pattern. The appropriate expression for the geometric phase
+2-form is the curl of the local wavevector Eq. (15),
+(23)
+V = ∇ × ke
+loc
+Near an electric field zero, V is anti-symmetric; integrating over a small sphere centred on
+the field zero gives zero. We showed that in its neighbourhood, a 3D zero in E constructs
+a local wavevector vortex with an identifiable axis along which |V| is very large.
+It is
+interesting that even when the complete vector characteristics of light are considered, a
+linear momentum vortex line still persists when all three field components are zero at a
+confined point. This vector field vortex is an analogue to a phase vortex in a complex scalar
+field, with a key difference being that the vector field vortex line is not continuous. Although
+the electromagnetic zero has some topological effects as we described in this paper, it is not
+so strong as to endow a surface around it with a nonzero Chern number.
+We have shown that, despite being unstable to perturbation, 3D zeros of the electric and
+electromagnetic field have topological properties generalising those of scalar vortices and
+polarisation singularities. Further studies might indicate how these properties behave under
+perturbation. We hope that by highlighting the unusual properties of 3D field zeros, we
+can inspire new applications that may be otherwise unachievable with traditionally used,
+lower-dimensional dark spots, such as those in beams or simple standing waves.
+4. Methods
+3D electric field zeros were created in analytical simulations of ten monochromatic inter-
+fering plane waves. In all simulations, ten random wavevectors (all of the same magnitude
+k = 2π
+λ ) were generated, and for each, two orthogonal polarisation basis vectors were defined,
+representing the two electric field degrees of freedom of a plane wave propagating in that
+direction. The ten plane waves were then polarised deliberately to destructively interfere
+and leave a 3D electric field zero at a single confined point, r0, following the procedure given
+in [21]. Let eikj·rˆej,1 and eikj·rˆej,2 be the two orthogonal polarisation states (degrees of free-
+dom) of the electric field of the jth plane wave with unit amplitude at the origin (j ranges
+from 1 to 10, kj is the jth plane wave’s random wavevector with magnitude |kj| =
+2π
+λ ,
+and ˆej,1 and ˆej,2 are two orthogonal unit vectors satisfying ˆej,1 · ˆej,2 = 0, ˆej,1 · kj = 0,
+ˆej,2 · kj = 0). In total, we have twenty available polarisation degrees of freedom, and by
+propagating each plane wave, we can calculate the electric field that each individual degree
+of freedom develops in the position of a desired electric field zero, r0. Now, we multiply
+
+16
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+each degree of freedom by a complex scalar, so that the jth plane wave has components
+xj,1eikj·rˆej,1 and xj,2eikj·rˆej,2. Adding together all scaled degrees of freedom, evaluated at
+r = r0, we have a linear system of three equations, one per component of the total field
+at r0, with complex variables xj,1 and xj,2 representing the amplitude of the orthogonal
+components of the jth plane wave phasor. Setting to zero all three total electric field com-
+ponents at r0, we may solve the system of equations to find the polarisation components of
+each plane wave required for complete destructive interference at r0. Since only three scalar
+conditions are enforced (Ex = 0, Ey = 0 and Ez = 0 for the total field at r0) by twenty
+degrees of freedom, the system is under-determined and seventeen possible solutions exist
+for a 3D zero at r0. Any one of these solutions may be chosen to realise the zero, or, as we
+do, the solutions may be combined in a linear sum with random complex amplitudes. A 3D
+zero could be produced with as few as four plane waves (in fact, a zero could be enforced by
+only two plane waves, but it would not be three-dimensional), though the total field would
+appear less random.
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+Bliokh, K. Y., Bekshaev, A. Y. & Nori, F. Optical momentum and angular momen-
+tum in complex media: from the Abraham–Minkowski debate to unusual properties of
+surface plasmon-polaritons. New Journal of Physics 19 (2017).
+39.
+Berry, M. V. & Shukla, P. Geometry of 3D monochromatic light: local wavevectors,
+phases, curl forces, and superoscillations. Journal of Optics 21 (2019).
+40.
+Bekshaev, A. Y., Bliokh, K. Y. & Nori, F. Transverse Spin and Momentum in Two-
+Wave Interference. Physical Review X 5 (2015).
+5. Acknowledgements
+We would like to thank Sinuh´e Perea-Puente for a mathematical proof. This work was sup-
+ported by European Research Council Starting Grant ERC2016-STG-714151-PSINFONI.
+6. Author Contribution
+A.J.V. conducted mathematical analyses and simulations; M.R.D. gave direction to and
+supervised the research; F.J.R-F. supervised the research. All authors wrote the manuscript;
+A.J.V. wrote the first draft.
+7. Competing Interests
+The Authors declare no competing interests.
+
diff --git a/9dE1T4oBgHgl3EQf8AVQ/content/tmp_files/load_file.txt b/9dE1T4oBgHgl3EQf8AVQ/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf,len=654
+page_content='3D ZEROS IN ELECTROMAGNETIC FIELDS  Alex J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Vernon1†, Mark R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Dennis2‡, and Francisco J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Rodr´ıguez-Fortu˜no1∗  1Department of Physics and London Centre for Nanotechnology, King’s College London,  Strand, London WC2R 2LS, UK  2School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK  †alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='vernon@kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='uk  ‡m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='dennis@bham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='uk  ∗francisco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='rodriguez fortuno@kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='uk  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' We present a study of 3D electromagnetic field zeros, uncovering their re-  markable characteristic features and propose a classifying framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' These are a spe-  cial case of general dark spots in optical fields, which sculpt light’s spatial structure into  matter-moving, information-rich vortices, escape the diffraction limit for single-molecule  imaging, and can trap particles for nanoscale manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Conventional dark spots  are two-dimensional in two aspects: localised in a plane and having a non-zero out-of-  plane field component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' We focus on non-paraxial fields, where three-dimensional dark  spots can exist non-stably at fully localised points, making distinct imprints in the flux  of energy and momentum, and in the light’s polarisation texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' With this work, we  hope to enhance current dark spot applications, or inspire new ones impossible with  lower-dimensional zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Introduction  An optical vortex is the name commonly given to a zero in a complex scalar field, such  as a component of the electric E or magnetic H field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Vortices in these components occur  naturally in general 3D monochromatic interference [1], where they are infinitely thin con-  tinuous strands either extending infinitely through space, or coiled into knotted, un-knotted  or linked closed loops [2]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' On a vortex strand, the phase of the complex scalar field (with  zero real and imaginary parts) is undefined creating circulation in the phase of the rest of the  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This phase increases in a clockwise or anti-clockwise sense by an integer multiple of 2π  along any closed loop containing one vortex line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Vortex lines in optics have direct analogues  in acoustics and water waves, and as a type of topological defect, are related to vortices in  (super)fluids [6] and in Bose-Einstein condensates [7], and even cosmic strings [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Strong  research interest in optical vortices over the past 30 years, combined with the availability  of instruments and the flexibility in generating [9]–[12] and structuring [13] vortex-carrying  beams, has positioned optics to act as a sandbox for exploring topological phenomena that  appear more broadly across physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  When considering the full 3D vector characteristics of an optical field, vortex lines in  individual field components like Ex, Ey, and Ez are basis-dependent and not so physically  meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' By picturing these different scalar vortex threads permeating the vector field,  we can appreciate how unlikely it is that the optical field is zero at a point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' E = 0, all  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='03540v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='optics]  9 Jan 2023  2  3D ZEROS IN ELECTROMAGNETIC FIELDS  three components simultaneously zero) in typical 3D interference (the vortex line in each of  the three field components would meet at such a zero point, requiring the manipulation of  three extra parameters beyond the spatial x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Despite the rarity of zeros in the wild, a  lower-dimensional version can be readily manufactured in optical beams, and is remarkably  well-studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Paraxial doughnut beams have an axial zero in the transverse field surrounded  by a bright ring, and are used in modern spectroscopy techniques [14], [15] because of the  zero’s immunity to the diffraction limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The transverse field effectively consists of one or  two scalar components with the vortex line along the beam axis, causing the real part of  the local wavevector to curl around the axis and imbue the beam with intrinsic orbital an-  gular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The longitudinal field, meanwhile, is non-zero (albeit very small due to  paraxiality) in the centre of the beam which, therefore, is better imagined not as an exact  axial zero, but as a dim line of linear polarisation (an L line) polarised parallel to the beam  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This, and its confinement in only two dimensions, stretching along the third, is  why we refer to the almost-dark centre of the doughnut beam as a two-dimensional zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Its topological index is straightforward to define by counting how many multiples of 2π the  phases of the transverse components climb through over an enclosing circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The intrinsic  orbital angular momentum carried by doughnut beams is the key property of the spatial  structure of light that can rotate matter [16], [17] and store information [18]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Surprisingly, the fully localised, three-dimensional optical field zero, E = 0, has been left  largely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This is probably due to its unstable nature—a perturbation will destroy  the zero point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' cause the vortices in the three components no longer to coincide).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Never-  theless, such a point is theoretically possible and can be artificially synthesised [21], but very  little is understood about how it is imprinted into the surrounding field, and there is no clas-  sifying topological index like the topological charge of a 2D vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The 3D electromagnetic  field zero is the focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' A zero in the E-field alone has codimension 6, requiring  that the six total degrees of freedom of two real, three-dimensional vectors (the real and  imaginary parts of the three components E) are suppressed simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This means 3D  zeros exist stably in a six-dimensional parameter space, and is why optical field zeros are  not natural in random interference patterns spanning only three spatial dimensions, being  hidden by instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Instead, 3D zeros must be revealed by tuning an additional three  parameters (this is discussed in [22] for a zero in two electric field components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Some of  these parameters could be the polarisation components of a plane wave, for example, and in  fact, 3D zeros can be very easily manufactured and controlled in pure plane wave interfer-  ence or near fields with a simple technique [21], and their higher dimensional confinement  could provide a greater degree of precision in dark spot spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Due to their electric  field dependence, the zero in E is coupled to a collection of singularities, each with its own  topological signature, in various physical quantities associated with the light field includ-  ing the complex Poynting vector, canonical momentum, spin momentum and spin angular  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Learning how energy flow and momentum circulate around a 3D vortex could  inspire applications which would be otherwise unfeasible using typical lower dimensional  zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Alternatively, the magnetic field H may vanish at a point, or more extremely, both  E and H might simultaneously vanish, giving a true electromagnetic null with codimension  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Here, we report the key features of a 3D field electric or magnetic field zero, including  the way that polarisation singularities are forced to intersect and the flux of the complex  Poynting vector and canonical and spin momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' With these findings, for the first time,  we propose a framework to classify the physically realisable varieties of 3D field zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  3D ZEROS IN ELECTROMAGNETIC FIELDS  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Results  To contextualise our study, we begin with some brief intuition on the special features  which we might expect to find near to a 3D zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  If either E or H is zero at a point r0, then of that field, say E, the flux of energy, canon-  ical momentum, spin angular momentum (and other quantities) are zero too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Since these  fluxes are vector quantities, their direction is singular at r0 and an imprint is made in the  surrounding space where they are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In three spatial dimensions, even if these  fluxes are divergence-less, there is more than one possible (topologically unique) imprint  which can be left by and characterise the zero in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The electric field spin is particularly  interesting, because its zeros (in non-paraxial fields) are co-dimension 2 objects—meaning  they are one-dimensional continuous lines, defining the threads of pure linear electric polar-  isation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This continuity should require at least one zero-spin line, an L line, to pass through  r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' A similar argument can be made for lines of pure circular electric polarisation, except  that C lines are defined by a complex quadratic equation, E · E = 0, equivalent to a real  quartic equation, |E · E|2 = 0, which has either zero, two or four real roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' It turns out,  as we will show, that a given number of C lines and L lines must always intersect in a 3D  electric field zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Before reporting these and other findings in detail from mathematical  argument and analytical simulations in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='3 and beyond, the next two subsections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='1  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='2 provide an overview of polarisation singularities and set out our way of classifying  3D field zeros using dyadics associated with the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Overview of Polarisation Singularities in Paraxial and Non-Paraxial Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  L lines and C lines are called polarisation singularities and are the vector version of scalar  vortex lines in wave fields, existing in light [23]–[25], acoustic and water waves [26] (both  acoustic and water waves have a vector nature [27], [28]) where some property of the general  polarisation ellipse is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In 3D fields, polarisation singularities are often described  as the underlying skeleton which embeds highly complex topologies into the field’s polar-  isation texture [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Polarisation singularities have been studied in full 3D and in  paraxial fields [31], where in paraxial fields (considering only the two transverse field com-  ponents), polarisation is circular at points and linear along lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Propagating the paraxial  field (maintaining the transverse polarisation) draws out the C points and L lines in the  transverse plane into C lines and L surfaces in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  A polarisation ellipse has orthogonal semi-major and semi-minor axes, telling us which  way the ellipse is oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' But because a polarisation circle has no semi-major or semi-  minor axes, at a C point, the orientation of the circle is undefined causing neighbouring  polarisation ellipses (almost circular) to rotate when tracked along a C point-enclosing loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  The ellipse major axis is described throughout space with a line field, in that the axis is  oriented some way in space but does not point one way or another—an ellipse looks identi-  cal to its 180 degree rotated self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This means that along the enclosing circuit, the rotating  ellipses turn continuously through an integer multiple of π radians, rather than 2π, which  is why C points are assigned a half-integer index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' When the field is fully three dimensional  and the polarisation ellipse is free to tilt in any Cartesian direction, circular polarisation still  occurs along one-dimensional threads (C lines which are no longer straight as in the paraxial  case) but the surrounding polarisation ellipses also twist, so that their major axes sweep out  M¨obius strips [32]–[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Analogues of C lines exist in polychromatic fields, shaping the rest  of the field into other remarkable topological structures [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  4  3D ZEROS IN ELECTROMAGNETIC FIELDS  L lines/L surfaces in paraxial fields (ignoring longitudinal fields) separate regions of left  and right handed polarisation ellipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  In non-paraxial fields, L lines are strictly one-  dimensional lines (not surfaces) and complement C lines in shaping the surrounding po-  larisation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  This reduction of dimension to the L entity occurs because, to be  linearly polarised, the real and imaginary parts of the field (say E = p + iq) need to be  (anti)parallel (not necessarily equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' If E is paraxial and linearly polarised, then in the  transverse plane, the ratio of the x components of p and q must equal the ratio of their y  components—a single condition, dissolving only one degree of freedom of one vector relative  to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' If E is non-paraxial, then an extra condition accounting for the ratio of the  z components of p and q must be satisfied for linear polarisation [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Between paraxial  and full 3D fields, the linear polarisation object’s codimension, which is the dimension of  the electric spin angular momentum field SE minus the dimension of the L line/L surface  which lies in SE, increases from one to two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The spin angular momentum of the field is zero  when linearly polarised, meaning the direction of the normal to the field oscillations cannot  be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Drawing a circuit around an L line, the spin vector rotates through 2π radians  in a clockwise or anti-clockwise sense and defines the L line’s topological index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  The characteristics of scalar vortices and C lines and L lines are visualised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Indexing Point-like Singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Polarisation singularities occur equally often  among the general polarisation ellipses in E and H fields, and need not coincide with each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Phase singularities, C lines and L lines are all indexed by looking at the circulation or  rotation of a scalar or vector quantity around a loop enclosing the singularity of interest [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  All three of these singularities are threads in 3D fields, but the winding number concept can  be generalised to higher dimensional singularities and calculated for point-like, 3D vector  singularities via the topological degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Instead of integrating a quantity associated with  a line singularity around a 1D closed circuit, for isolated singular points in 3D, we should  integrate an appropriate quantity over a closed surface enclosing the point singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' For  a vector V(rS) on a surface S (rS ∈ S) in 3D real space, for example, the topological de-  gree of rS �→ V (the mapping from the real space surface rS to V) is a calculation of the  integer number of times that every possible direction of V is realised (on a sphere) on all  the points rS on the surface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' As with other kinds of topological singularities in physical  fields, the easiest realised topological degrees (winding numbers) are ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Mathematically, a  0, ±1 topological degree is the integral of the determinant of the dyadic D(V) of V over S  divided by A, the area of S,  (1)  deg(V) = 1  A  �  S  det(D(V))dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  The dyadic D(V), also called the Jacobian matrix of V, contains the first order spatial  derivatives of each component of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  The sign of the determinant of D(V) equals the  product of the signs of its eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' For 3D vectors where D(V) is a 3 × 3 matrix, it  is possible for drastically different behaviour of V to be hidden under the same topological  degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' For example, if V(r = 0) = 0 (meaning the direction of V is singular at the origin)  and we assume that a linear map from an origin-enclosing surface to V has a topological  degree of −1, then D(V) at r = 0 could have signed eigenvalues (in any order) of + + −  or − − −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Physically, the origin could either be a saddle point or a sink for V with no  distinction in topological degree because both + + − and − − − eigenvalues multiply to a  negative sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Rather than calculating the topological degree, to try to classify the flux of  3D ZEROS IN ELECTROMAGNETIC FIELDS  5  C-l  in  e  L  l  in  e  S  c  a  l  a  r  V  o  r  t  e  x  ±  2  l  π  on-plane view  on-plane view  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Visualisation of scalar and polarisation singularities in a non-  paraxial electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Scalar vortices (black line) exist in complex  scalar fields, such as the components of E, where the scalar field is zero  and its phase is undefined, forming 1D threads in the interference of three  or more plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Around a scalar vortex line, the phase of the field  increases by an integer l multiple of 2π in a clockwise or anticlockwise  sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Singular lines exist in the complex vector characteristic of E and  H fields, called polarisation singularities, which include C lines (lines of  circular polarisation) and L lines (lines of linear polarisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In a circuit  around a point on a C line (blue line), in the plane of the polarisation  circle at that point, nearby polarisation ellipses rotate through an integer  multiple of π radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Around an L line (green line), the normal to nearby  polarisation ellipses rotates by an integer multiple of 2π radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  energy and canonical momentum through a 3D optical field zero, we use the signs of the  eigenvalues of their first order dyadics evaluated in the position of the field zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  We use the ideas discussed here to report our findings in the following sub-sections,  beginning with the six possible ways that C lines and L lines can intersect in a 3D zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Polarisation Singularities at a 3D Electric Field Zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' We will focus on a 3D  electric field zero in a position r0, that is E(r0) = 0, and study the nearby strands of circular  6  3D ZEROS IN ELECTROMAGNETIC FIELDS  and linear electric polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Identical arguments to those given here could be made for  magnetic field zeros (H(r0) = 0) and magnetic polarisation singularities, or for simultaneous  electric and magnetic field zeros (E(r0) = H(r0) = 0) and polarisation singularities of either  E or H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Any smooth function of r is nearly linear over small distances, which means all  fundamental behaviour of the electric field in the immediate vicinity of the zero is captured by  its Jacobian, JE = D(E), a complex 3×3 matrix containing all first-order spatial derivatives  of Ex, Ey and Ez, evaluated at r0,  (2)  JE = D(E) =  �  �  �  ∂Ex  ∂x  ∂Ex  ∂y  ∂Ex  ∂z  ∂Ey  ∂x  ∂Ey  ∂y  ∂Ey  ∂z  ∂Ez  ∂x  ∂Ez  ∂y  ∂Ez  ∂z  �  �  � = (∇ ⊗ E)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  The Jacobian of the magnetic field at r0, JH, can be defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In free space, JE  and JH are always traceless because E and H are divergence-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Maxwell’s equations also  require that if E(r0) = 0, then JH must be symmetric at r0 and vice versa for H(r0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  We make a first-order approximation of the electric field vector near r0 with,  (3)  ˜E = JEv,  where v = r−r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Nearby C lines emerge in our approximated field wherever ˜E · ˜E = 0, which we may  calculate using (3) and separate into real and imaginary parts,  (4)  ˜E · ˜E = (JEv) · (JEv)  = vT Mv + ivT Nv,  where M = Re{JT  EJE} and N = Im{JT  EJE}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The two terms in equation (4) are quadric  surfaces connecting constant valued real and imaginary parts of ˜E · ˜E, and the real and  imaginary surfaces described by setting (4) equal to zero cross in real space where ˜E is  circularly polarised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The real 3 × 3 matrices M and N are symmetric and always have real  eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Normally, these eigenvalues have signs + + − or − − + (in any order) so that  the surfaces vT Mv = 0 and vT Nv = 0 are both double cones, vertices touching at v = 0  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The cones have an elliptical cross section whose ellipticity is constant  with distance from v = 0 in the linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Because two ellipses can intersect  at either zero, two or four points (as shown in the lower part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' 2(a)), there must be  either zero, two or four C lines passing through the electric field zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' If one matrix, say  M, is positive or negative definite (all positive or all negative eigenvalues), Re{˜E · ˜E} will  solely increase or decrease in all outward directions from v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Then, the constant-valued  surface vT Mv = C becomes an ellipsoid, and vT Mv = 0 is satisfied only at v = 0 so that  no C lines pass through the 3D vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  To reveal the number of L lines that extend through the 3D electric field zero, we must  calculate the electric field spin, given by,  (5)  SE ∝ Im{E∗ × E} = 2Re{E} × Im{E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  When the electric field is linearly polarised (SE = 0), the real and imaginary parts of E  must be (anti)parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Under the approximation (3), this means,  (6)  Re{JE}v = λIm{JE}v,  3D ZEROS IN ELECTROMAGNETIC FIELDS  7  b  a  L-line  C-line  cone cross section  on unit sphere  Re{E · E} = 0  E = 0  Im{E · E} = 0  2  0  4  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Electric polarisation singularities passing through a 3D electric  field zero at a position r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (a) Visualisation of why zero, two or four C  lines must pass through r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In a first-order approximation, the surfaces  Re{E · E} = 0 (red) and Im{E · E} = 0 (blue) are double cones, and where  they intersect, C lines exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Two double cones intersect along two or four  lines, or do not intersect at all, which is easy to see by considering the cones’  cross sections on the unit sphere: ellipses which cross at zero, two or four  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (b) Six different examples of electric field zeros created at a position  r0 (red circle), one per unique combination of C lines and L lines meeting  there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The C lines are marked by blue regions where E · E ≈ 0 and the L  lines by the green regions where Im{E∗ × E} ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Each field zero is created  in analytical simulations by designing the polarisation of ten plane waves  with random wavevectors, wavelength 500 nm, to interfere destructively at  r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The plane waves have different polarisations and wavevectors for each  example zero in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  where λ is a positive or negative scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The directions of the L lines crossing through v = 0  are given by the three eigenvectors of the matrix Im{JE}−1Re{JE}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Since this matrix is  real-valued, either all three of these eigenvectors are real, corresponding to three L lines, or  only one of them is real and is accompanied by a conjugate pair of complex eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In  that case, just one L line passes through the 3D zero because v cannot point in a complex  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Summarising, either zero, two or four C lines and either one or three L lines always meet  at r0 in a 3D electric field zero E(r0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' An identical conclusion can be drawn for C lines  and L lines of the magnetic field for the case of H(r0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' 2(b), an example of  each of the six possible C line/L line combinations through a 3D zero is presented, the zeros  8  3D ZEROS IN ELECTROMAGNETIC FIELDS  created in the interference of ten plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Each zero is enforced by separate ensembles  of ten plane waves with random wavevector directions that are deliberately polarised to  destructively interfere at a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Energy Flux Singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The flow of energy in a light field is described by the  complex Poynting vector,  1  2E∗ × H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  The real part of this vector (often itself called the  ‘Poynting vector’) corresponds to the time-averaged power transfer (sometimes known as  active power) in the field, while reactive power (associated with oscillations in the transfer  of power) is accounted for by the less-used imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' We refer to these two real vectors  as Pr and Pi,  (7)  Pr = 1  2Re{E∗ × H}  (8)  Pi = 1  2Im{E∗ × H}  When either E or H is zero at a point r0, the complex Poynting vector vanishes, and its  real and imaginary parts circulate in the space around the zero according to their first-order  derivatives at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The real part Pr is divergence-less in free space where there is no absorption  or energy generation, and must therefore be organised into a vector saddle point at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' An  example flow of active power around a 3D electric field zero created at r0 (E(r0) = 0,  H(r0) ̸= 0) is given in the top row of panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' 3, where Pr is plotted on the xy, xz, and  yz planes coinciding at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Although there is no net flow of active power in or out of the zero,  Pr streamlines can be arranged in two topologically different ways depending on whether  the signs of the eigenvalues of its first-order dyadic, Im{(JT  E − JE)J∗  E} (written electrically  without prefactors), are + + − or + − −, corresponding to two possible topological orders  of −1 or +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' One might notice that the imaginary Poynting vector Pi, which is plotted on  the same planes for the same free space electric field zero at r0 in the lower row of panels of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' 3, is not divergence-free—in fact, it is physically possible for a source, sink or saddle of  Pi to exist there, depending on whether E or H is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' To see why, we first note that using  Maxwell’s equations in free space (see supplemental information), the imaginary Poynting  vector can be decomposed into a sum of two terms, one polarisation-independent and one  polarisation-dependent, each containing electric and magnetic contributions,  (9)  Pi = − c2  2ω ϵ0Re{(JT  E − JE)E∗}  = c2  2ω µ0Re{(JT  H − JH)H∗}  = c2  4ω  �  −1  2ϵ0∇(E∗ · E) + 1  2µ0∇(H∗ · H)  �  + c2  4ω Re{ϵ0JEE∗ − µ0JHH∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (9) represents the difference in gradient of the electric and magnetic  energy density of the light field, while polarisation-dependent behaviour of Pi derives from  the second term since JEE∗ and JHH∗ contain inter-component multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In certain  cases such as a uniformly polarised standing wave, the second term is zero and the gradient  of the difference in electric and magnetic energy density determines the direction of reactive  power flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Because E∗ · E = |E|2 is a positive real quantity, a 3D zero in E is a source for  3D ZEROS IN ELECTROMAGNETIC FIELDS  9  Pi  Pr  x  y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='08λ  x  z  y  z  y  x  x  y  z  z  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Flow of the real (Pr, red) and imaginary (Pi, teal) parts of the  Poynting vector, 1  2E∗ × H, on the xy, xz and yz planes coinciding with an  electric field zero at position r0 (blue circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The real Poynting vector is  divergence-free, meaning a vector saddle point of Pr is set up at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The  imaginary Poynting vector is not necessarily divergence-free and can be or-  ganised in a sink at r0 when E(r0) = 0 (a source is not possible unless  the magnetic field is zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Results are generated by designing the polari-  sation of ten plane waves with random propagation directions to interfere  completely at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  the vector ∇(E∗ · E) (and likewise for H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Depending on how the polarisation-independent  and polarisation-dependent terms combine in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (9), the imaginary Poynting vector could  have non-zero divergence at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Note that there is a difference in sign between the electric  and magnetic terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (9), meaning Pi behaves differently for E(r0) = 0, H(r0) ̸= 0  and H(r0) = 0, E(r0) ̸= 0 and E(r0) = H(r0) = 0 3D zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' To understand the flow of  Pi through an optical field zero, we assume a non-dual electric field zero (E(r0) = 0 and  H(r0) ̸= 0) and make a first-order approximation of Pi, this time referring to the relevant  linear transformation matrix as the first-order dyadic of the imaginary Poynting vector,  D(Pi), which is defined identically to JE in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (2) with Pi and its components in place of  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Our approximate imaginary Poynting vector is,  (10)  ˜Pi = D(Pi)v  10  3D ZEROS IN ELECTROMAGNETIC FIELDS  where v = r − r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The dyadic D(Pi) = (∇ ⊗ Pi)T evaluated at r0 is, using the electric  representation of Pi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (9) (top line),  (11)  D(Pi) = − c2  2ω ϵ0Re{(JT  E − JE)J∗  E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  There are no second order derivatives of E in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (11) because E(r0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Surprisingly,  D(Pi) cannot have three positive eigenvalues, as justified in the supplemental information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  The result is that at a 3D electric field zero, Pi is organised into one of two types of sad-  dle with topological degree 1 or −1, or a sink with topological degree −1, never a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  When H(r0) = 0 and E(r0) ̸= 0, the opposite is true because of the dual-asymmetry of the  imaginary Poynting vector: Pi can form a saddle or source at r0 but not a sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Orbital Current Singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' When divided by c2, the real Poynting vector Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (7)  turns into a momentum density, the kinetic momentum density, which, using Maxwell’s  equations for time-harmonic fields, can be split in to a well-known sum of separate orbit  and spin contributions [37], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' For instance, by substituting (with prefactors) the curl of  E for H, the kinetic momentum density can be written as,  (12)  Π =  1  2c2 Re{E∗ × H}  = 1  2ω ϵ0Im{E∗ · (∇)E} + 1  2ω ϵ0∇ × 1  2Im{E∗ × E},  where A · (∇)B = Ax∇Bx + Ay∇By + Az∇Bz = JT  BA, with JB being the Jacobian  of B defined identically to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (2) (the decomposition is explained in more detail in the  supplemental information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The first decomposed term is po  E, the orbital contribution to  the kinetic momentum density, called the canonical momentum density, imparted by the  electric field only,  (13)  po  E = 1  2ω ϵ0Im{E∗ · (∇)E} = 1  2ω ϵ0Im{JT  EE∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (12) can also be written purely in terms of H and by averaging these equivalent repre-  sentations of Π, the dual-symmetric canonical momentum density is obtained,  (14)  po = 1  4ω Im{ϵ0E∗ · (∇)E + µ0H∗ · (∇)H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  This momentum density definition contains both the electric and magnetic field’s influence,  and produces the total orbital angular momentum of the field within a volume when r×po is  integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Naturally, the electric and magnetic contributions to (14) become zero whenever  E = 0 and H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This means that, in a 3D electric field zero positioned at r0, the direction  of the electric contribution po  E is undefined and should circulate around r0 in some fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Of course, while the total canonical momentum density at r0 is not zero when only E = 0,  we could draw the same conclusions we make here for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (14) rather than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (13) near a  dual 3D vortex (E(r0) = H(r0) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Note that by normalising E, the argument to Im{}  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (13) defines the local electric wavevector [25],  (15)  ke  loc = −ie∗ · (∇)e,  where e =  E  √  E∗·E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The real part of ke  loc is the local phase gradient of the electric field, while  3D ZEROS IN ELECTROMAGNETIC FIELDS  11  Re{kloc} on yz plane  xplane = -25 nm  xplane = -95 nm  xplane = 0 nm  xplane = +25 nm  |Re{kloc}| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='1*k  r0  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Vortex pseudo-line (red) of the real electric local wavevector,  Re{ke  loc}, passing though an electric field zero at position r0 (blue circle),  that is E(r0) = 0 and H(r0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The red line indicates regions of space  where |Re{ke  loc}| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='1k, where k = 2π  λ and λ = 500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The line is roughly  oriented along the x axis and the electric local wavevector is plotted on four  different yz planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The three planes which coincide with the line are −25  nm, 0 nm, +25 nm in the x direction away from r0, showing clear vortex-  like circulation of momentum around the axis of the red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' On the fourth  plane −95 nm away from r0, the vortex-like circulation of Re{ke  loc} has lost  some definition, highlighting that Re{ke  loc} is not exactly zero along a line,  and only appears line-like near to the E field zero (the only location where  Re{ke  loc} is exactly zero is at r0, because Re{ke  loc} vanishes at points, not  along lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Results are generated from interference of ten plane waves  with random wavevectors, wavelength λ = 500 nm, deliberately polarised  to create a 3D electric field zero at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  12  3D ZEROS IN ELECTROMAGNETIC FIELDS  Im{ke  loc} points in the direction of decreasing electric field intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' A three-dimensional,  real vector, Re{kE  loc} (and therefore canonical momentum density) can vanish at localised  points in space with non-zero electric field, where a saddle-like circulation of Re{kE  loc} sur-  rounds [39], similar to the top row of panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' But when the electric field vanishes  and the direction of Re{kE  loc} is automatically undefined, a different behaviour emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  To understand why, we once again make a first-order approximation, this time of po  E,  capturing the electric canonical momentum very near to a 3D electric field zero at r0 in its  dyadic D(po  E),  (16)  ˜po  E = D(po  E)v,  where v = r − r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The dyadic D(po  E) = (∇ ⊗ po  E)T at a general point in space is given by,  (17)  D(po  E) = 1  4ω ϵ0Im{JT  EJ∗  E} + 1  4ω ϵ0Im{E∗  xHess(Ex) + E∗  yHess(Ey) + E∗  zHess(Ez)},  where Hess(A) is the Hessian matrix of the scalar field A,  (18)  Hess(A) =  �  �  �  ∂2A  ∂x2  ∂2A  ∂x∂y  ∂2A  ∂x∂z  ∂2A  ∂y∂x  ∂2A  ∂y2  ∂2A  ∂y∂z  ∂2A  ∂z∂x  ∂2A  ∂z∂y  ∂2A  ∂z2  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  As E approaches zero, the trace-less matrix D(po  E) is dominated by the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (17)  and if evaluated at a location r0 where E(r0) = 0, the linear approximation of po  E responds  only to the properties of the matrix in the first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (17), Im{JT  EJ∗  E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This is an  anti-symmetric matrix which always has one zero and two purely imaginary eigenvalues,  meaning that in the direction of the one real eigenvector of D(po  E) at r0, the approximated  electric canonical momentum does not increase at all, producing a zero-momentum line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The  imaginary eigenvalues of D(po  E) twists po  E into a surrounding vortex-like structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This  special type of vector field singularity is called a circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Fundamentally, the canonical  momentum should only be zero at confined points in general 3D fields, so this apparent  vortex line is only preserved locally to the electric field zero at r0, dissolving with distance  as higher-order derivatives of po  E become significant (it is, in fact, just a very elongated null  point of po  E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The direction of the vortex pseudo-line in the vicinity of the electric field zero  is also given by the curl of the orbital current,  (19) D = ∇×po  E ∝ Re{∇Ex}×Im{∇Ex}+Re{∇Ey}×Im{∇Ey}+Re{∇Ez}×Im{∇Ez}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  We visualise this feature in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' 4, where a 3D electric field zero is created at a point r0  by deliberately polarising ten plane waves, each with random wavevectors, to destructively  interfere at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The real part of the electric local wavevector, Re{ke  loc}, the real part of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (15), is calculated and the region of space where |Re{ke  loc}| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='1k (k = 2π  λ ) is revealed  by a red line approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='1λ in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The electric local wavevector is proportional to  po  E and shows the direction of canonical momentum carried by the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This red  line is not continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Re{ke  loc} actually vanishes only at r0 but it increases in magnitude  so slowly in a certain direction (the direction of the real eigenvector of Im{JT  EJ∗  E}) that a  line-like structure of |Re{ke  loc}| ≈ 0 exists very near to r0, stirring the electric canonical  momentum into a local vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This is shown by the four yz planes on which Re{ke  loc} is  plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The real part of the electric local wavevector forms a swirl around the  red line, a swirl losing definition if the plotting plane is too far from r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This remarkable  3D ZEROS IN ELECTROMAGNETIC FIELDS  13  structure always appears when all three electric field components are zero together at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Spin Current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In the decomposition of the kinetic momentum density, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (12), the  second term is called the spin momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' It is proportional (and should not be confused  with) the curl of the spin angular momentum of the electric, magnetic or electromagnetic field  depending on the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Like before, we will focus on the electric representation of  the decomposed kinetic momentum density, referring to the electric spin momentum with  ps  E,  (20)  ps  E = 1  2ω ϵ0∇ × 1  2Im{E∗ × E} = − 1  2ω ϵ0Im{JEE∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  The electric spin momentum is a divergence-free vector whose dyadic D(ps  E) has three non-  zero eigenvalues when evaluated in the position of an electric field zero, organising ps  E into  one of two types of 3D vector saddle point, just like the real Poynting vector in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Expressing, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (20), the electric spin momentum with the electric field Jacobian reveals  that only a difference in sign and orientation of JE separates ps  E from the electric canon-  ical momentum po  E, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This means that, in a dual electric-magnetic zero,  E(r0) = H(r0) = 0, where JE is symmetric from Maxwell’s equations, the spin and canoni-  cal momentum dyadics are equal and opposite, D(ps  E) = −D(po  E) (this also means that the  dyadic of the real Poynting vector is zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In a first-order approximation of both ps  E and  po  E near r0 in this case, a zero-line exists in exactly the same place for both vectors, and  around it, ps  E and po  E have vortex-like circulation with opposite handedness to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Spin Angular Momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The dual spin angular momentum, created by the rota-  tion of the electric and magnetic field vectors, is given by [40],  (21)  S = 1  4ω Im{ϵ0E∗ × E + µ0H∗ × H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  The electric and magnetic parts individually describe the ellipticity of the electric and mag-  netic polarisation ellipses, pointing in the perpendicular direction to the ellipse plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Once  more for simplicity, we will focus on the singularity in the electric field spin angular momen-  tum, SE =  1  4ωIm{ϵ0E∗ × E}, left in a 3D electric field zero positioned at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The total spin  angular momentum, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (21), is not zero if only E(r0) = 0, but we could draw similar con-  clusions for S as we do here for SE when the electric and magnetic fields are simultaneously  zero at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Decomposing SE using Maxwell’s equations, we can write its first-order dyadic at r0 in  terms of the light field Jacobian matrices (see supplemental material),  (22)  D(SE) =  1  4ω2 ϵ0Re{(JT  H − JH)J∗  E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  We note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (22), describing the spatial derivatives of the electric field spin only, de-  pends on the magnetic field Jacobian matrix JH, which is automatically symmetric whenever  E = 0 from Maxwell’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The consequence is that JT  H − JH = 0 and all elements  of D(SE) at r0 are zero when E(r0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Higher-order derivatives of SE (Hessian matrices  for each component) need to be calculated to fully understand the flux of the electric spin  angular momentum in the neighbourhood of a 3D zero in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  14  3D ZEROS IN ELECTROMAGNETIC FIELDS  Matrix at r0  Characteristic  JE  3D complex Jacobian of the electric field at r0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (2))  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Im{JE}−1Re{JE}  Number of real eigenvalues is the number of L lines passing through r0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Re{JT  EJE}  Eigenvectors are the principle axes of the double cone Re{E · E} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Number of intersections of this double cone with that of matrix 3 are the number of C lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Im{JT  EJE}  Eigenvectors are the principle axes of the double cone Im{E · E} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Number of intersections of this double cone with that of matrix 2 are the number of C lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Im{JT  EJ∗  E}  Direction of real eigenvector (there is only one) is the axis of the electric local wavevector vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Imaginary eigenvectors give the handedness of momentum circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  −Im{JEJ∗  E}  Proportional to first-order dyadic of spin current (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Eigenvalue signs give the type of minimum at r0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Im{(JT  E − JE)J∗  E}  Proportional to first-order dyadic of real Poynting vector (active power flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Eigenvalue signs give the type of minimum at r0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  −Re{(JT  E − JE)J∗  E}  Proportional to first-order dyadic of imaginary Poynting vector (reactive power flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Eigenvalue signs give the type of minimum at r0  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Summary of the seven dyadics (numbered) which classify the  vector field singularities organised by a 3D electric field zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Summary Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Here, in Table 1, we summarise the seven dyadics which classify  the number of crossing C lines and L lines, the flux of the real and imaginary parts of the  Poynting vector, the spin current, and the orientation of the canonical momentum vortex  pseudo-line existing at a 3D electric field zero, E(r0) = 0 while H(r0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' To characterise  a magnetic field zero, the matrices can be written magnetically by substituting JE for JH  (and changing the ‘−’ sign in front of matrix 7 to a ‘+’), in which case matrices 1, 2, and  3 characterise magnetic polarisation singularities, and matrix 4 and 5 the magnetic local  wavevector and spin current respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In the case of a dual 3D zero, E(r0) = H(r0) = 0,  matrices 6 and 7 are zero because both JE and JH are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Discussion  Three-dimensional optical field zeros are co-dimension 6 entities which, unlike axial zeros  in beams, are completely localised, the optical field growing brighter in all outward directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Although they rarely occur naturally in light (requiring three additional parameters beyond  spatial x, y, z due to their codimension), 3D zeros can be deliberately created in plane wave  interference or in the near fields of light-scattering matter [21] to reveal the unusual features  they imprint in the light field’s energy, wavevector and polarisation structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Both with  mathematical argument and by creating field zeros in plane wave interference, we showed  that whenever the electric or magnetic field is zero at a point r0, then some combination of  zero, two or four C lines, lines of pure circular polarisation, and one or three L lines, lines of  pure linear polarisation of the field in question, intersect at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Likewise, an imprint is made  at r0 in the surrounding flux of the parts of the complex Poynting vector 1  2E∗ × H, the local  wavevector, the spin momentum and spin angular momentum, each organised in a vector  source, sink or saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The signs of the eigenvalues of the first-order dyadics of each  quantity at r0 reveal this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Of particular interest is the canonical momentum: while typically  3D ZEROS IN ELECTROMAGNETIC FIELDS  15  vanishing at confined points in space, a zero in E or H at r0 twists the canonical momentum  imparted by that null-containing field into a sub-wavelength, vortex-like structure around  an axis with an easily calculated direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' We say it is a sub-wavelength object because,  although it resembles the twisted vortex structures of well-known doughnut beams, it is  not preserved with increasing distance from r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  In the combination of the way energy  flows through r0 and the number of intersecting polarisation singularities, any 3D field zero  inscribes one of a discrete number of topologically unique signatures in the electromagnetic  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' We identify seven dyadics whose spectra could classify all physically possible imprints  of 3D optical field zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  It is tempting to speculate that a surface enclosing an electric or magnetic field point  zero might, in addition to the quantities already identified, possess a nonzero topological  Chern number due to a nontrivial geometric phase 2-form (Berry curvature) resulting from  the neighbouring polarisation pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The appropriate expression for the geometric phase  2-form is the curl of the local wavevector Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' (15),  (23)  V = ∇ × ke  loc  Near an electric field zero, V is anti-symmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' integrating over a small sphere centred on  the field zero gives zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' We showed that in its neighbourhood, a 3D zero in E constructs  a local wavevector vortex with an identifiable axis along which |V| is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  It is  interesting that even when the complete vector characteristics of light are considered, a  linear momentum vortex line still persists when all three field components are zero at a  confined point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This vector field vortex is an analogue to a phase vortex in a complex scalar  field, with a key difference being that the vector field vortex line is not continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Although  the electromagnetic zero has some topological effects as we described in this paper, it is not  so strong as to endow a surface around it with a nonzero Chern number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  We have shown that, despite being unstable to perturbation, 3D zeros of the electric and  electromagnetic field have topological properties generalising those of scalar vortices and  polarisation singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Further studies might indicate how these properties behave under  perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' We hope that by highlighting the unusual properties of 3D field zeros, we  can inspire new applications that may be otherwise unachievable with traditionally used,  lower-dimensional dark spots, such as those in beams or simple standing waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Methods  3D electric field zeros were created in analytical simulations of ten monochromatic inter-  fering plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In all simulations, ten random wavevectors (all of the same magnitude  k = 2π  λ ) were generated, and for each, two orthogonal polarisation basis vectors were defined,  representing the two electric field degrees of freedom of a plane wave propagating in that  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' The ten plane waves were then polarised deliberately to destructively interfere  and leave a 3D electric field zero at a single confined point, r0, following the procedure given  in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Let eikj·rˆej,1 and eikj·rˆej,2 be the two orthogonal polarisation states (degrees of free-  dom) of the electric field of the jth plane wave with unit amplitude at the origin (j ranges  from 1 to 10, kj is the jth plane wave’s random wavevector with magnitude |kj| =  2π  λ ,  and ˆej,1 and ˆej,2 are two orthogonal unit vectors satisfying ˆej,1 · ˆej,2 = 0, ˆej,1 · kj = 0,  ˆej,2 · kj = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' In total, we have twenty available polarisation degrees of freedom, and by  propagating each plane wave, we can calculate the electric field that each individual degree  of freedom develops in the position of a desired electric field zero, r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Now, we multiply  16  3D ZEROS IN ELECTROMAGNETIC FIELDS  each degree of freedom by a complex scalar, so that the jth plane wave has components  xj,1eikj·rˆej,1 and xj,2eikj·rˆej,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Adding together all scaled degrees of freedom, evaluated at  r = r0, we have a linear system of three equations, one per component of the total field  at r0, with complex variables xj,1 and xj,2 representing the amplitude of the orthogonal  components of the jth plane wave phasor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Setting to zero all three total electric field com-  ponents at r0, we may solve the system of equations to find the polarisation components of  each plane wave required for complete destructive interference at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Since only three scalar  conditions are enforced (Ex = 0, Ey = 0 and Ez = 0 for the total field at r0) by twenty  degrees of freedom, the system is under-determined and seventeen possible solutions exist  for a 3D zero at r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Any one of these solutions may be chosen to realise the zero, or, as we  do, the solutions may be combined in a linear sum with random complex amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' A 3D  zero could be produced with as few as four plane waves (in fact, a zero could be enforced by  only two plane waves, but it would not be three-dimensional), though the total field would  appear less random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content='  Guo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content='  Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content=' Ultrasmall Optical Vortex Knots Generated  by Spin-Selective Metasurface Holograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Advanced Optical Materials 7 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content='  Li, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Optical vortex knots and links via holographic meta-  surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Advances in Physics: X 6 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content=', Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content='  Lim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content=' Field theory spin and  momentum in water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content='  Sugic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=', Droop, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=', Otte, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content=' Nature Commu-  nications 12 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content='  Larocque, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content='  Dennis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Polarization singularities in paraxial vector fields: morphology and statis-  tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content='  Freund, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Multitwist optical M¨obius strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content=' Optics  Letters 36 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content='  Bauer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=', Banzer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=', Karimi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content='  Pisanty, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content=' Knotting fractional-order  knots with the polarization state of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Nature Photonics 13 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content='  Berry, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Index formulae for singular lines of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+page_content='  Berry, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Optical currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Journal of Optics A: Pure and Applied Optics 11 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Bliokh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=', Bekshaev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' & Nori, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Optical momentum and angular momen-  tum in complex media: from the Abraham–Minkowski debate to unusual properties of  surface plasmon-polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' New Journal of Physics 19 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Berry, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' & Shukla, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Geometry of 3D monochromatic light: local wavevectors,  phases, curl forces, and superoscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Journal of Optics 21 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  Bekshaev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=', Bliokh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' & Nori, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Transverse Spin and Momentum in Two-  Wave Interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Physical Review X 5 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Acknowledgements  We would like to thank Sinuh´e Perea-Puente for a mathematical proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' This work was sup-  ported by European Research Council Starting Grant ERC2016-STG-714151-PSINFONI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Author Contribution  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' conducted mathematical analyses and simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' gave direction to and  supervised the research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='R-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' supervised the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' All authors wrote the manuscript;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' wrote the first draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
+page_content=' Competing Interests  The Authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQf8AVQ/content/2301.03540v1.pdf'}
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+Locomotion-Action-Manipulation:
+Synthesizing Human-Scene Interactions in Complex 3D Environments
+Jiye Lee
+Hanbyul Joo
+Seoul National University
+{kay2353,hbjoo}@snu.ac.kr
+Figure 1. Our system, LAMA, produces high-quality and realistic 3D human motions that include locomotion, scene interactions, and
+manipulations given a 3D environment and designated interaction cues.
+Abstract
+Synthesizing interaction-involved human motions has
+been challenging due to the high complexity of 3D environ-
+ments and the diversity of possible human behaviors within.
+We present LAMA, Locomotion-Action-MAnipulation, to
+synthesize natural and plausible long term human move-
+ments in complex indoor environments. The key motivation
+of LAMA is to build a unified framework to encompass a
+series of motions commonly observable in our daily lives,
+including locomotion, interactions with 3D scenes, and ma-
+nipulations of 3D objects. LAMA is based on a reinforce-
+ment learning framework coupled with a motion matching
+algorithm to synthesize locomotion and scene interaction
+seamlessly under common constraints and collision avoid-
+ance handling. LAMA also exploits a motion editing frame-
+work via manifold learning to cover possible variations
+in interaction and manipulation motions. We quantitatively
+and qualitatively demonstrate that LAMA outperforms ex-
+isting approaches in various challenging scenarios. Project
+page: https://lama-www.github.io/.
+1. Introduction
+In our daily lives, we can easily observe that humans do
+not live in isolation nor in voids, but continuously interact
+with a complex environment surrounded by many objects.
+Notably, humans perform such a diverse set of daily life
+actions effortlessly. Imagine that we visit a new indoor en-
+vironment (e.g., a hotel room) we have never been before.
+It is expected that we can still easily figure out how to move
+from rooms to rooms, how to sit on a chair, how to open the
+doors of closets, and so on. However, endowing machines
+or virtual humans with such abilities is still a largely unex-
+plored area, despite its importance.
+Synthesizing scene interactions within real-life 3D envi-
+ronments has been a challenging research problem due to
+its complexity and diversity. Human movements in real life
+consists of various types of behaviors, including locomotion
+with avoiding cluttered areas, diverse interactions with 3D
+scenes, and sophisticated object-manipulations. In particu-
+lar, the spatial constraint that arises from real-life 3D envi-
+ronments where many objects are cluttered makes motion
+synthesis highly constrained and complex, and various pos-
+sible arrangements of 3D environments make generalization
+difficult. As human-scene interactions cover a wide range of
+technical challenges, previous approaches have focused on
+sub-problems, such as (1) modeling static poses [17,24,49,
+64,69,71,72] or (2) human object interactions with a single
+target object or interaction type [10, 47, 53–55, 66, 67, 70].
+Recent methods [15,59,60] extend to synthesizing dynamic
+interaction motions in cluttered real-world 3D scenes. How-
+ever, the performance of these methods are fundamentally
+limited due to the lack of 3D ground truth data that contains
+both human motions and paired 3D environments.
+1
+arXiv:2301.02667v1  [cs.CV]  9 Jan 2023
+
+In this paper, we present LAMA, Locomotion-Action-
+MAnipulation, to synthesize natural and plausible long term
+human motions in complex indoor environments. The key
+motivation of LAMA is to build a unified framework to
+include locomotion, interactions with 3D scenes, and ma-
+nipulations of 3D objects, which are the series of motions
+commonly observable in our daily lives. LAMA is based
+on a reinforcement learning framework coupled with a mo-
+tion matching algorithm to synthesize locomotion and scene
+interaction seamlessly while adapting to complicated 3D
+scenes with collision avoidance handling. The reinforce-
+ment learning framework interprets the 3D information of
+the given scene and optimally traverses among the motion
+capture database via motion matching. As an advantage, our
+system does not require any “scene-paired” datasets where
+human movements are captured with the surrounding 3D
+environments simultaneously, which is rarely available. To
+further cover the numerous variations of interaction mo-
+tions, we also exploit an autoencoder based motion editing
+approach to learn the motion manifold space [20] in which
+the editing is performed. Through extensive quantitative
+and qualitative evaluations against existing approaches, we
+demonstrate that our method outperforms previous methods
+in various challenging scenarios.
+Our contributions are summarized as follows: (1) we
+present the first method to generate realistic long term mo-
+tions combined with locomotion, interaction with scene,
+and manipulation in complicated cluttered scenes; (2) we
+propose a novel, unified framework that synthesizes loco-
+motion and human-scene interactions in a seamless man-
+ner, by introducing scene interpretation terms to a reinforce-
+ment learning based approach to automatically generate op-
+timal transitions; and (3) our outputs show the state-of-the-
+art motion synthesis quality with longer duration (more than
+10 sec) than previous methods.
+2. Related Work
+Generating Human-Scene Interactions. Generating
+natural human motion has been a widely researched topic
+in the computer vision community. Early methods focus on
+synthesizing or predicting human movements by exploiting
+neural networks [11,13,35,35,38,46,56,58]. However, these
+approaches primarily address the synthesis of human mo-
+tion itself, without taking into account the surrounding 3D
+environments. Recent approaches begin to tackle modeling
+and synthesizing human interactions within 3D scenes, or
+with objects. Most of the researches focus on statically pos-
+ing humans within the given 3D environment [16,24,69,71],
+by generating human scene interaction poses from vari-
+ous types of input including object semantics [17], im-
+ages [21,23,64,65,68], and text descriptions [49,72].
+More recently, there have been approaches to synthesize
+dynamic human object interactions (e.g., sitting on chairs,
+Encoder
+Decoder
+Task-Adaptive Motion Editing
+Motion Generation
+Action 
+Controller
+3D Scene
+Interaction 
+Cue     
+Action 
+Posture
+Motion
+Synthesizer
+Optimization
+Figure 2. Overview of LAMA.
+carrying boxes). Starke et al. [53] introduce an autoregres-
+sive learning framework with object geometry-based envi-
+ronmental encodings to synthesize various human-object
+interactions. Later work [15, 70] extends this by synthe-
+sizing motions conditioned with variations of objects and
+contact points. Other approaches [47, 54, 55, 66, 67] focus
+on generating natural hand movements for manipulation,
+which is extended by including full body motions [54].
+Physics-based character control to synthesize human object
+interactions has been also explored in [8,10,39,47,66]. Al-
+though these approaches cover a wide range of human ob-
+ject interactions, most of them solely focus on the relation-
+ship between human and the target object without long-term
+navigation in cluttered 3D scenes.
+More recent approaches include generating natural hu-
+man scene interactions within a complex 3D scene clut-
+tered with many objects [6, 59–61], closely related to ours.
+These methods are trained using human motion datasets
+paired with 3D scenes, which require both ground truth mo-
+tions and simultaneously captured 3D scenes for supervi-
+sion. Due to such difficulties, some methods exploit syn-
+thetic datasets [6,61] or data fitted from depth videos [60].
+In previous approaches [15,59], navigation to move through
+cluttered environments is often performed by a separate
+module via a path planning algorithm (e.g., A∗ algorithm)
+by approximating the volume of a human as a cylinder. This
+path planning based methods approximate the spatial infor-
+mation of the scene and the human body and therefore have
+limitations under highly complex conditions.
+Motion Synthesis and Editing. Synthesizing natural
+human motions by leveraging motion capture data has also
+been a long-researched topic in computer graphics. Some
+approaches [26,37] construct motion graphs, where plausi-
+ble transitions are inserted as edges and motion synthesis is
+done by traversing through the constructed graph. Similar
+approaches [31, 51] connect motion patches to synthesize
+interactions in a virtual environment or multi-person inter-
+actions. Due to its versatility and simplicity, a number of
+variations have been made on the graph based approach,
+such as motion grammar [22] which enforces traversing
+rules in the motion graph. Motion matching [5, 9] can also
+be understood as a special case of motion graph traver-
+sal, where the plausible transitions are not precomputed but
+searched during runtime. Recent advances in deep learning
+allow to leverage motion capture data for motion manifold
+2
+
+rendel
+reset
+prev
+play
+nextlearning [19, 20, 52]. Autoregressive approaches based on
+variational autoencoders (VAE) [36, 46] and recurrent neu-
+ral networks [14,29,41] are also used to forecast future mo-
+tions based on past frames. These frameworks are general-
+ized to synthesizing a diverse set of motions including lo-
+comotion on terrains [19] mazes [36], action-specified mo-
+tions [46], and interaction-involved sports [29, 41]. Neural
+network-based methods are also reported to be successful
+in various motion editing tasks such as skeleton retarget-
+ing [2], style transfer [3,20], and inbetweening [14].
+Reinforcement learning (RL) has also been successful in
+combination with both data-driven and physics-based ap-
+proaches for synthesizing human motions. Combined with
+data-driven approaches, these RL frameworks serve as a
+control module that generates corresponding motions to a
+given user input by traversing through motion graphs [28],
+latent space [34, 36, 57], and precomputed transition ta-
+bles [30]. Deep reinforcement learning (DRL) has been
+widely used recently in physics simulation as well to syn-
+thesize physically plausible movements with a diverse set
+of motor skills [4,32,41,43–45,62].
+3. Method
+3.1. Overview
+Our system, dubbed as LAMA, outputs a sequence of
+human poses M = {mt}T
+t=1 by taking the 3D surrounding
+cues W and desired interaction cues Φ, as inputs:
+M = LAMA(W, Φ).
+(1)
+The output posture at time t, mt = (p0, r1, ..., rJ) ∈
+R3J+3, is represented by a concatenated vector of global
+root position p0 ∈ R3 and local joint orientations of J
+joints where each j-th joint is in angle-axis representations
+rj ∈ so(3). Throughout our system, the skeleton tree struc-
+ture and joint offsets are fixed and shown in Fig. 3 (a). We
+represent the 3D environments W = {wi} as a set of 3D
+object and environment meshes, including the background
+scene mesh and other object meshes targeted for manip-
+ulation. The interaction cues, Φ = [φ1, φ2, ...φn], are an
+ordered list of desired interaction inputs φi = {qj}j∈Ji
+where qj ∈ R3 indicates desired positions of j-th joint,
+and Ji is a set of specified joints for interaction (in prac-
+tice, few joints such as root 1 or end-effectors). Examples of
+the 3D environment W and interaction inputs φi are shown
+in Fig. 5 (a). Intuitively, φi specifies the expected positions
+of selected joints of the human character. Note that we do
+not specify the exact timing of the interaction, as the timing
+is automatically determined by our action controller. More
+details are addressed in Sec. 3.4.
+To synthesize locomotion, interaction, and manipulation
+together, LAMA is designed via a three-level system com-
+1For root, orientation in angle-axis representation is also included in φ.
+Figure 3. (a) Skeleton with joints and box nodes. (b) Automatically
+detected collision points (colored as red).
+posed of the action controller A and the motion synthesizer
+S, followed by a manifold-based motion editor E. By taking
+3D scene cues W and desired interaction cues Φ as input,
+the action controller A makes the use of a reinforcement
+learning (RL) framework by training the control policy π to
+sample an action at time t, π(at|st, W, Φ), where at con-
+tains the plausible next action cues including predicted ac-
+tion types and short-term future forecasting. st is the state
+cues to represent the current status of human characters in-
+cluding its body posture, surrounding scene occupancy, and
+current target interaction cue, which can be computed via
+a function ψ, st = ψ(mt−1, mt, W, Φ). Intuitively, action
+controller A predicts the plausible next action cues at by
+considering the current character-scene state st. The gener-
+ated action signals at from the action controller A is pro-
+vided as the input for the motion synthesizer S, which then
+determines the posture at the next time step mt+1, i. e.,
+S(mt, at) = mt+1. Afterwards, the character’s next state
+can be computed again via st+1 = ψ(mt, mt+1, W, Φ),
+which is input to the action controller recursively.
+Followed by the initial motion generation part from A
+and S, our system furthermore applies a motion editor
+E(M) = ˜M, where ˜M = { ˜Mt}T
+t=1 is the edited motions
+to further express the motions involving complex human-
+object interactions such as manipulation (e.g. moving ob-
+jects, opening doors). Fig. 2 shows the overview of LAMA.
+3.2. Scene-Aware Action Controller
+Based on reinforcement learning, our action controller
+A enables the character to perform locomotion and desired
+actions with fulfilling the interaction cues Φ and avoiding
+collisions in the 3D environment W. A is a trained control
+policy π where π(at|st, W, Φ). Different from previous
+approaches where navigation and scene-object interactions
+(e.g., sitting) are performed by separate modules [15, 59],
+our RL-based framework performs both in a unified way
+with a common objective by automatically determining the
+transition from navigation to specific actions. As a key ad-
+vantage, LAMA can be robustly generalized to challenging
+unseen 3D clutters in long-term human motion synthesis
+and also outperforms previous methods by avoiding colli-
+sions throughout the whole process, including navigation
+3
+
+Joint
+- Node
+Jointand interaction.
+State. The state st = ψ(mt−1, mt, W, Φ) at time t is
+a feature vector representing the current status of the hu-
+man character. st = (sbody
+t
+, sscene
+t
+, sinter
+t
+) is composed of
+body configuration sbody, 2D scene occupancy sscene, and
+desired current target interaction sinter. Body configuration
+sbody = {r, ˙r, θup, h, pe} includes r, ˙r ∈ RJ′×6 that are the
+joint rotations and velocities respectively for the J′ joints
+excluding the root in 6D representations [73], θup ∈ R that
+is the up vector of the root (represented by the angle w.r.t
+the Y-axis), h ∈ R that is the root height from the floor,
+and pe ∈ Re×3 that is the end-effector positions in person-
+centric coordinate (where e is the number of end-effectors).
+sscene = {gocc, groot} includes scene occupancy informa-
+tion in 2D floor plane, as shown in Fig. 4. gocc ∈ Rn2 rep-
+resents the 2D occupancy grid on the floor plane of neigh-
+boring n cells around the agent and groot ∈ R2 denote the
+current 2D global root position of the character in the dis-
+cretized grid plane. sinter is an element of Φ and represents
+the interaction cue the character is currently targeting, that
+is sinter = φi.
+Action. Given the current status of the character st,
+the control policy π outputs the feasible action at
+=
+(atype
+t
+, afuture
+t
+, aoffset
+t
+). atype
+t
+provides the probabilities
+of next action type among all possible actions, determining
+the transition timing between actions (e.g., from locomo-
+tion to sitting). afuture
+t
+predict future motion cues such as
+plausible root position for the next 10, 20, and 30 frames.
+aoffset
+t
+is intended to update the raw motion data searched
+from the motion database in motion synthesizer module S.
+Intuitively, our learned control policy generates an optimal
+posture offset aoffset
+t
+which is applied to the closest plau-
+sible raw posture in the database. This enables the character
+to perform more plausible scene-aware human poses, allow-
+ing our system to be generalized to any unseen 3D scenes
+given a limited amount of motion capture data. More details
+are addressed in Sec. 3.3.
+3.3. Motion Synthesizer
+Given the current motion output mt and actions sig-
+nals at from the action controller A as inputs, the mo-
+tion synthesizer produces the next plausible character pos-
+ture: S(mt, at) = mt+1. As the first step, motion synthe-
+sizer searches for the motion from a motion database that
+best matches the closest motion feature, then modifies the
+searched raw motion to be more suitable for the scene envi-
+ronment. To this end, motion synthesizer’s output mt+1 is
+in turn fed into the action controller recursively. We exploit
+a modified version of motion matching algorithm [5, 9, 18]
+for the first step of motion synthesis. In motion matching,
+motion synthesis is performed periodically by searching the
+most plausible next shot motion segments from a motion
+DB, and compositing them into a long connected sequence.
+Figure 4. Visual representation of the 2D occupancy grid near the
+root. Grid on the right represents top view of the grid. Blue in-
+dicates root position while gray represents the space is occupied.
+Occupied cells near the root are colored as black.
+Motion features. Motion feature represents the charac-
+teristic of each frame in the short motion segment and is
+computed as f(m) = {{pj}, { ˙pj}, θup, c, ofuture}. From
+a posture m, the positions and velocities pj, ˙pj
+∈ R3
+are extracted for the selected joints j ∈ {Head, Hand,
+Foot}, which are defined in a person-centric coordinate
+of m. θup ∈ R3 is the up-vector of the root joint, and
+c ∈ {0, 0.5, 1} indicates automatically computed foot con-
+tact cues of the left and right foot (0 for non-contact, 1 for
+contact, 0.5 for non-contact but close to the floor within
+a threshold). ofuture = {{pdt
+0 }, {rdt
+0 }} contains the cues
+for the short-term future postures, where pdt
+0 and rdt
+0 are
+the position and orientation of root joint at dt frames later
+from the current target frame. ofuture are computed in 2D
+XZ plane in person-centric coordinate of the current tar-
+get motion m, and thus pdt
+0 , rdt
+0
+∈ R2. The selected fu-
+ture frames are action-type specific, and for locomotion we
+extract 10, 20, and 30 frames in the future (at 30Hz) fol-
+lowing [9]. Intuitively, the motion feature extracts the target
+frame’s posture and temporal cues by considering neigh-
+boring frames 2. We pre-compute motion features for every
+frame of the motion clips in the motion database. Motion
+feature of the current state of the character, or the query
+feature, is also computed in the same way based on posture
+mt−1, mt and afuture
+t
+produced by the action controller,
+that is xt = f(mt−1, mt, atype
+t
+, afuture
+t
+). The component
+afuture
+t
+serves as ofuture in the query feature, which can be
+understood as the action controller providing cues for pre-
+dicted future postures.
+Motion searching and updating. The query motion fea-
+ture xt from the current character is computed as addressed
+above, and let the motion features in motion database de-
+noted as yk for the k-th clips in the DB. Motion searching
+finds the best matches in the motion database by computing
+the weighted euclidean distances between the query feature
+and DB features:
+k∗ = arg min
+k
+||wT
+f (xt − yk)||2,
+(2)
+where wf is a fixed weight vector to control the impor-
+2In practice, the input of feature extractor function f should take into
+account the motions of neighboring timesteps.
+4
+
+2D
+occupancy gridtance of feature elements. After finding the best match ˆmk∗
+from motion database, the motion synthesizer further up-
+dates it with the predicted motion offset aoffset
+t
+from at,
+that is τ( ˆmk∗+1, aoffset) = mt+1, where ˆmk∗+1 is the
+next plausible character posture and τ is an update function
+to update selected joints. In practice, the motion searching
+is performed periodically (e.g., every N-th frames) to make
+the synthesized motion temporally more coherent.
+3.4. Learning for Scene-Aware Action Controller
+In the reinforcement learning framework, the objective is
+to learn the optimal policy which maximizes the discounted
+cumulative reward. In our method, we design rewards to
+guide the agent to perform locomotion towards the target
+objects (e.g., sofa) and also perform desired interaction with
+the object (e.g., sitting). In particular, our RL-framework
+performs both navigation and interaction with common con-
+straints (e.g., smooth transitions, collision avoidance).
+Our reward function consist of the following terms:
+Rtotal = wtrRtr + wactRact + wregRreg,
+(3)
+where wtr, wact, and wreg are the weights to balance among
+reward terms. The trajectory reward Rtr is obtained when
+the character moves towards the desired interaction input φ
+while meeting the spatial constraints from the surrounding
+3D scene, described below:
+Rtr = rcoli · rpos · rroot, where
+(4)
+rcoli = exp
+�
+− 1
+σ2
+coli
+�
+b∈B
+wbρ(b, W)
+�
+,
+(5)
+rpos = exp
+�
+�−
+1
+σ2
+root
+�
+j∈J
+∥p0 − qj∥2
+�
+� ,
+(6)
+rvel =
+�
+1
+when ˙proot ≥ σth
+σvel∥ ˙p0∥2
+else.
+(7)
+The collision-avoidance reward rcoli penalizes collisions
+with 3D scenes. As depicted in Fig. 3 (a), body limbs in
+the skeletal structure are represented as a set of box-shaped
+nodes B with a fixed width, where each element b ∈ B is
+a 3D box representation of legs and arms (we exclude torso
+and head). The function ρ(b, W) detects the collision be-
+tween edges of a box-shaped node b with 3D scene meshes
+W and returns the number of intersection points. (Fig. 3
+(b)). wb is the weights to control importance of each limb b.
+The collision-avoidance reward is maximized when no pen-
+etration occurs, making the control policy π to find the opti-
+mal trajectory and pose offset to avoid physically implausi-
+ble collisions and penetrations. rpos are obtained when the
+agent moves to reach the targeting interaction cue φ, by en-
+couraging agent’s root position p0 to be closer to the target
+interaction cue {qj}. rvel encourages the character to move
+by penalizing when the root velocity ˙proot is less than a
+threshold σth. σcoli, σroot, and vel are weights to control
+the balance between terms.
+Action reward Ract enforces the synthesized motion to
+fulfill the given interaction cue φ = {qj}:
+Ract = rinter · r∆t · r∆v,
+where
+rinter = exp
+�
+�−
+1
+σ2
+inter
+�
+j∈J
+∥pj − qj∥2
+�
+� ,
+r∆t = exp
+�
+−σ2
+∆tCtr
+�
+,
+r∆v = exp
+�
+−σ2
+∆vCvel
+�
+,
+(8)
+where interaction reward term rinter is maximized when the
+performed action meets the positional constraints provided
+by interaction cues. Smoothness reward terms r∆t and r∆v
+minimizes the transition cost, which is based on the subpart
+of the feature distances defined in Eq. 2, where Ctr is the
+weighted feature distances of pj, θup, and c, and Cvel is
+from ˙p. These are intended to penalize the case where the
+character makes abrupt changes.
+Regularization reward Rreg penalizes the aoffset
+t
+exces-
+sively modifying the original posture brought from the mo-
+tion synthesizer, denoted as ˆmt, and maintains temporal
+consistency among frames.
+Rreg = exp
+�
+−
+1
+σ2reg
+�
+∥ ˆmt − mt∥2 + ∥mt − mt−1∥2��
+.
+It is reported that [33, 41] multiplying rewards with con-
+sistent goals are suitable for learning, as the reward is re-
+ceived when the conditions are simultaneously met. Fur-
+thermore, to accelerate learning, we use early termination
+conditions [43] and limited action transitions. The episode
+is terminated when the character moves out of the scene
+bounding box, or when the collision reward rcoli is under a
+certain threshold. Also, the action controller first checks in
+advance whether the action signal is valid when it makes
+transitions from locomotion to other actions. When the
+nearest feature distance of Eq. 2 in the motion synthesizer
+is over a certain threshold, the action controller discards the
+transition and continues navigating. The control policy is
+learned through Proximal Policy Optimization (PPO) algo-
+rithm [50].
+3.5. Task-Adaptive Motion Editing
+Interaction includes a massively diverse pool of mo-
+tions, and these variations cannot be fully handled by lim-
+ited amount of motion database. In order to cover such di-
+versity, we include a task-adaptive motion editing module
+in our motion synthesis framework. The goal of our edit-
+ing module E is (1) to edit motion M to fit into diverse
+5
+
+Figure 5. Visual representation of system input Φ, W and output
+motion sequence. On the left, interaction cues are shown as cyan
+spheres and arrows (indicating orientation). The right is the syn-
+thesized human motion ˜
+M.
+target object geometries (e.g., sitting on a chair with dif-
+ferent height), and (2) to generate additional hand move-
+ments for manipulation (e.g., grasping). In particular, in the
+case of manipulation, additional interaction cue φ can be
+provided to enforce an end-effector (e.g., a hand) to fol-
+low the desired trajectories to express the manipulation task
+on the target object, as shown in Fig 8 (left). The edited
+motion ˜M = E(M) should not only fulfill the sparsely
+given positional constraints, but also preserve the temporal
+consistency between frames and spatial correlations among
+joints in order to maintain its naturalness. We adopt the mo-
+tion manifold learning approach with convolutional autoen-
+coders [20] to compress motion to a latent vector within a
+motion manifold space. Motion editing is done by searching
+for an optimal latent vector among the manifold. For train-
+ing the autoencoder, motion sequence, which we denote as
+X converted from M, is represented as a time-series of hu-
+man postures by concatenating joint rotations in 6D repre-
+sentations [73], root height, root transform relative to the
+previous frame projected on the XZ plane, and foot contact
+labels. The encoder and decoder module are trained based
+on reconstruction loss, ||X − Ψ−1(Ψ (X)) ||2, where Ψ is
+the encoder and Ψ−1 is the decoder.
+The latent vector from the encoder z = Ψ(X) repre-
+sent the motion manifold space by preserving the spatio-
+temporal relationship among joints and frames within the
+motion sequence. As demonstrated in [20], editing motions
+in this manifold space ensures the edited motion to be re-
+alistic and temporally coherent. To this end, we find the
+optimal latent vector z∗ by minimizing a loss function L
+by constraining the outputs motions to follow the interac-
+tion constraint φ. We also include additional regularizers in
+L so that the output motion to maintain the foot locations
+and root trajectories to the original motions. See supp. mat.
+for more details on L. Finally, the edited motion ˜M can be
+computed via Ψ−1(z∗).
+4. Experiments
+We evaluate LAMA’s ability on synthesizing long-term
+motions with various human-scene and human-object inter-
+Method
+Plausibility
+Naturalness
+Slip
+Penetration
+FDtotal
+FDroot
+FDjoint
+Wang et al. [60]
+5.13
+3.88
+1.38
+0.45
+0.93
+Wang et al. [60]*
+24.8
+4.58
+1.44
+0.44
+1.00
+SAMP [15]
+10.5
+12.49
+1.25
+0.30
+0.95
+LAMA (ours)
+5.21
+1.52
+1.22
+0.31
+0.91
+Table 1. Baseline comparison Foot slip loss (cm, ↓) averaged over
+all frames. Penetration loss(percentage, ↓) is counted based on in-
+tersection points of the 3D environment and the skeleton. Natural-
+ness score is based on fr´echet distance (FD ↓). Wang et al. with an
+asterisk indicates without post-processing optimization.
+actions involved. We exploit an extensive set of quantitative
+metrics and perceptual study to evaluate the physical plau-
+sibility and naturalness of the synthesized motion.
+Dataset. For constructing the database for the motion
+synthesizer, motion capture data are selectively collected
+and refined from Ubisoft La Forge [14], COUCH [70],
+and SAMP [15]. All the data used in this system are mo-
+tion capture data (in bvh format) with no scene or ob-
+ject related information, and are retargeted into a unified
+skeletal structure with MotionBuilder. We use PROX [16]
+and Matterport3D [7] datasets for 3D environment and
+SAPIEN [63] object meshes for manipulation. Our code and
+pre-processed data will be publicly released.
+Implementation Details. The policy and the value net-
+work of the action controller module consists of 4 and
+2 fully connected layers of 256 nodes, respectively. The
+encoder and decoder of the task-adaptive motion editing
+module consist of three convolutional layers. Adam opti-
+mizer [25] is used for training and optimization. We use
+Nvidia RTX 3090 for training the action controller and the
+motion editing module. It takes 10 to 80 minutes to learn
+a single control policy, where the training time mainly de-
+pends on how difficult the interaction cues are to achieve.
+For optimization in the motion editing module, it takes 3 to
+4 minutes for 500 epochs. See supp. mat. for more detail.
+4.1. Experimental Setup
+Evaluation metrics. Quantifying motion synthesis quality
+is challenging due to the lack of ground-truth data or offi-
+cial evaluation metrics. We try to quantify them in terms of
+physical plausibility and naturalness.
+• Physical plausibility: We use contact and penetration
+metrics to evaluate the physical plausibility of the synthe-
+sized motions. Contact loss penalizes the foot movement
+when the foot is in contact. Since foot contact is a critical
+element in dynamics, contact-based metric is closely related
+in determining the physical plausibility of motions. Pene-
+tration loss (“Penetration” in Table 1) measures implausible
+cases when the body penetrates the objects in the scene. We
+compute penetration metric by counting frames where the
+6
+
+interaction cue Φ2
+interaction cue ΦFigure 6. Comparison with LAMA (left) and LAMA without col-
+lision reward (right). As shown in the right, without collision re-
+ward the character fails to avoid collisions with obstacles (marked
+as red).
+intersection points (Sec. 3.4) goes over a certain threshold. 3
+• Naturalness: We measure the naturalness of the synthe-
+sized motions by measuring the Fr´echet distance, as re-
+ported in [15, 35, 40] between the synthesized motion and
+motions from motion capture data. Features are extracted
+from motion sequences and the Fr´echet distance is com-
+puted with the extracted features. We measure the natural-
+ness of character root movements FDroot, including root ori-
+entation and velocity, and character joint rotations FDjoint.
+Baselines. We compare our LAMA with the state-of-the-art
+approaches as well as variations of ours.
+• Wang et al. [60] is the state-of-the-art long term mo-
+tion synthesis method for human-scene interactions within
+a given 3D scene. We use the author’s code for evaluation.
+As Wang et al. uses optimization to post-process the synthe-
+sized motion to improve foot contact and reduce collisions,
+we both compare Wang et al. with and without optimization.
+• SAMP [15] generates interactions which can be general-
+ized not only for object variations but also random starting
+points within a given 3D scene. SAMP explicitly exploits a
+path planning module to navigate through cluttered 3D en-
+vironments.
+• Ablative baselines We perform ablation studies on the ac-
+tion controller and task-adaptive motion editing module. We
+perform ablation studies on the scene reward rcoli, and ac-
+tion offset aoffset
+t
+to present the contribution of both terms
+on our system’s capability to generate scene-aware motions.
+We also compare our method without the transition reward
+r∆t and r∆v terms (Sec. 3.4) in the action controller. Fi-
+nally, we demonstrate the strength of our task-adaptive mo-
+tion editing module to edit motions naturally (Sec. 3.5) by
+comparing with inverse kinematics (IK).
+4.2. Comparisons with Previous Work
+Quantitative Evaluation. We compare methods in 6
+different scenarios from various 3D scenes in the PROX
+dataset [16]. Foot contact is automatically labeled based on
+310 for legs and 7 for arms
+Figure 7. Comparison with LAMA (left) and LAMA without ac-
+tion offset (right). The character in original LAMA moves forward
+while tilting its arms to avoid collision with walls, while in LAMA
+without action offset does not.
+positional velocity of the foot joint. Foot slip metric is mea-
+sured by foot joint positions. To compute penetration metric
+in a fair way, SMPL-X outputs of Wang et al. and SAMP are
+converted to box-shaped skeletons as in ours and intersec-
+tion point are counted. Table 1 shows the results.
+As shown, our LAMA outperforms Wang et al both in
+naturalness and physical plausibility. It is noted that Wang
+et al performs optimization as post-processing to explic-
+itly minimize foot slip, and yet LAMA still shows on-par
+performance against it (and better in all other metrics).
+Compared with SAMP, our method shows much better re-
+sults in plausibility metrics (both Slip and Penetration),
+and shows slightly better performance in naturalness. Apart
+from SAMP which relies on a separate navigation mod-
+ule, our RL-based action controller handles collisions in the
+same way of scene-interaction and shows much better per-
+formance in in complex and cluttered 3D scenes.
+A Human Study. To further validate our results, we
+compare the quality of our output over other baselines,
+Wang et al. and SAMP, through A/B testing from human
+observers. For the study, we choose 5 scenarios from dif-
+ferent indoor scenes, and render the results of each method
+using the exactly same view and 3D characters, so that they
+cannot be distinguished from the appearance side. We build
+two separate sets, where in each set the result videos of
+our method are shown with each competitor side by side
+in a random order. Human observers are asked to choose a
+motion clip that is more human-like and plausible in the
+given 3D scene. We perform each set of tests with non-
+overlapping 15 participants. See our supp. mat. for more
+details about the study setup. As the result, the outputs of
+our method are preferred by the majority (more than 50%
+voting) in all cases. By considering all votes independently,
+our method are preferred 80.0% over SAMP and 97.3%
+over Wang et al.’s work. In particular, we found that our
+method greatly outperform the competing methods in terms
+of the naturalism of foot stepping, transition between loco-
+motion and action, and collision avoidance with the scenes.
+See our supp. videos for more results.
+7
+
+LAMA
+LAMA w/o collision rewardoffset
+LAMA w/o a
+LAMA
+LAMAFigure 8. (a) Comparison with LAMA (top) and LAMA without
+manifold and replaced with IK (bottom) of a character opening the
+toilet lid. (b) Comparison with LAMA (top) and LAMA without
+motion editing (bottom) in sitting.
+4.3. Ablation Studies
+Ablation Studies on Action Controller. We quantita-
+tively compare the original LAMA and the LAMA without
+collision reward rcoli. We intend to demonstrate the role of
+rcoli that enforces the action controller to search for optimal
+actions for generating motions without collisions. Ablation
+studies are done in 5 PROX scenes. In the original LAMA,
+penetrations occur in only 1.1% of the frames among the
+whole motion sequences, while the ratio is 15.7% in LAMA
+without collision reward. The result supports that the colli-
+sion reward rcoli enforces the action controller to compute
+optimal actions for synthesizing body movement according
+to the spatial constraint of the given 3D scene. Example re-
+sults are shown in Fig. 6.
+We also compare the contribution of other components
+in the action controller module in generating natural inter-
+actions. As seen in Fig. 7, with the action controller without
+aoffset
+t
+the character fails to avoid penetration with objects
+or walls, as the raw motion from the motion database does
+not have any information of the scene. This demonstrates
+that action offset also plays a role in generating detailed
+scene-aware poses even from raw motion capture data.
+Moreover, the results with the action controller without
+smoothness rewards r∆t and r∆v are not smooth enough,
+showing unnatural movements such as jerking. These abla-
+tion studies justify the advantages of our reward terms.
+Ablation Studies on Task-Adaptive Motion Editing.
+We ablate our motion editing module by replacing it with
+an alternative approach via Inverse-Kinematics (IK). An ex-
+ample result is shown in Fig. 8 (left). For manipulation, the
+results with IK show jerky and awkward motions because
+the temporal and inter-joint correlations in natural human
+motions are not reflected in IK, while original LAMA with
+task-adaptive motion editing module shows much natural
+motions. Our motion editing module can also be used to
+Figure 9. Examples of synthesized manipulation motions. The tar-
+get object for manipulation is colored as orange. Top is a motion
+sequence of walking and opening a toilet lid, and the bottom is a
+sequence of walking and opening doors. The character is colored
+purple at start and aqua at the end.
+further adjust the character movements in different object
+geometries, going over the limit of the motion database. As
+seen in Fig 8 (right), the motion editing module enables the
+character to properly sit in chairs with various sizes.
+5. Discussion
+In this paper, we present a method to synthesize locomo-
+tion, scene-interaction, and manipulation in a unified sys-
+tem. Leveraging a RL framework with motion matching,
+our method enables to produce natural and plausible hu-
+mans motions in complex and cluttered 3D environments
+only with a limited amount of motion-only datasets. Our
+method has been thoroughly evaluated in diverse scenar-
+ios, outperforming previous approaches [15, 60]. We also
+demonstrate the robustness and generalization ability of our
+system by covering a wide range of human interactions in
+many different 3D environments.
+While our RL-based method can be generalized to any
+unseen 3D environments, a new control policy has to be
+trained for each motion sequence. Combining RL with a
+supervised learning framework for better efficiency can be
+an interesting future research direction. Furthermore, al-
+though we assume a fixed skeletal information throughout
+the system, interaction motions may change depending on
+the character’s body shape and sizes. We leave synthesizing
+motions on varying body shapes as future work.
+Acknowledgments: This work was supported by SNU-
+Naver Hyperscale AI Center, SNU Creative-Pioneering Re-
+searchers Program, and NRF grant funded by the Korea
+government (MSIT) (No. 2022R1A2C209272411).
+8
+
+LAMA
+LAMA
+LAMA w/o motion editing
+LAMA w/o manifold + IK梦人庆A. Supplementary Video
+The supplementary video shows the results of our
+method, LAMA, on various scenarios. In the video, we
+show the human motion synthesis results on PROX [16],
+Matterport3D [7], and also our own home-brewed 3D scene
+produced by Polycam App [1] in an iPad pro. We use
+SAPIEN [63] object meshes for manipulation examples. As
+shown, our method successfully produces plausible and nat-
+ural human motions in many challenging scenarios. Our
+supplementary video contains several ablation studies of
+our method by showing the importance of collision reward
+rcoli in Eq. (4), transition reward (r∆t , r∆v) in Eq. (8), pos-
+ture offset aoffset
+t
+in Action Controller (Sec. 3.2), and our
+motion editing modules (Sec. 3.5) compared to the tradi-
+tional Inverse Kinematics (IK). We also show the compari-
+son with previous state-of-the arts [15, 59, 60] and demon-
+strate that our results produces better quality of motions
+with better collision avoidance performance in complicated
+3D scenes.
+B. Additional Details on Implementations
+B.1. Action Controller
+Implementation Details.
+For the action controller A and
+motion synthesizer module S, we use the animation library
+DART [27]. We also use a publicly available PPO imple-
+mentation [32, 41], where we remove the variable time-
+stepping functions stepping in [32] by following the origi-
+nal PPO algorithm. The details of the training regarding the
+policy and value network of the action controller are written
+in Table 2.
+Early Termination Conditions.
+As written in the main
+paper, the episode is terminated (1) when the character
+moves out of the scene bounding box; (2) when the colli-
+sion reward rcoli is under a certain threshold; or (3) when
+the root of the human character is located in the blocked
+(occupied) regions of the scenes in 2D grid space during
+the locomotion status.
+Name
+Value
+Learning rate of policy network
+2e-4
+Learning rate of value network
+0.001
+Discount factor (γ)
+0.95
+GAE and TD (λ)
+0.95
+Clip parameter (ϵ)
+0.2
+# of tuples per policy update
+30000
+Batch size for policy/value update
+512
+Table 2. Details on the hyper-parameters for learning the control
+policy of the action controller A.
+B.2. Motion Synthesizer
+Motion Database Information.
+As described in our
+main paper, we pre-process the motion segments by selec-
+tively collecting and clipping from Ubisoft La Forge [14],
+COUCH [70], and SAMP [15]. The length (in frames)
+of motion segments (“Seg. Length” in tables), number of
+motion segment (“Seg. Count” in tables), and the number
+of total frames (“Total Frames” in tables) are summarized
+in Table 3.
+Action-Specific Feature Definition.
+The motion feature,
+as defined in our main paper Sec 3.3, represents both the
+current state of the motion and a short term future move-
+ments: f(m) = {{pj}, { ˙pj}, θup, c, ofuture}. In particu-
+lar the action specific feature ofuture = {{pdt
+0 }, {rdt
+0 }}
+contains future motions so that the motion search process
+can take into account the future motion consistency, where
+pdt
+0 , rdt
+0 ∈ R2 are the position and orientation of root joint at
+dt frames later from the current target frame. For locomo-
+tion, we extract dt = 10, 20, and 30 frames in the future (at
+30Hz) following [9], as addressed in our main paper. For sit-
+ting, we specifically choose dt as the frame where the char-
+acter completes the sit-down motion. The major motivation
+of this design choice is encourage the motion synthesizer to
+search the motion clips with the desired target action.
+Computation Cost for Searching.
+The computation time
+for searching the motion database is done between 1-2 mil-
+liseconds in CPU, where we test on AMD Ryzen 5950X
+CPU. The number of search times varies and is dependent
+to the 3D scenes and desired motions. In one of our sce-
+narios, total 17 searches in locomotion(walk) and 14 in ac-
+tion(sit) were done. For locomotion, the searching time is
+average 1.743 milliseconds (standard deviation 0.46) and
+for action(sit) 1.103 milliseconds (standard deviation 0.63).
+B.3. Motion Editing via Motion Manifold
+Implementation Details.
+For the convolutional autoen-
+coder of task-adaptive motion editing, we use PyTorch [42],
+FairMotion [12], and PyTorch3d [48]. The autoencoder is
+trained with the Adam optimizer with learning rate 0.0001.
+We use 3 layers of 1D temporal-convolutions with kernel
+width of 25 and stride 2, and the channel dimension of each
+output feature is 256. The training datasets are summarized
+in Table 4. Note that we use different pre-processing steps
+between Motion editing module and Motion Synthesizer.
+Reconstruction Loss.
+The encoder Ψ and decoder Ψ−1
+are trained based on reconstruction loss Lrecon = ||X −
+Ψ−1(Ψ (X)) ||2, where:
+Lrecon = wcLcontact + wrLroot + wqLquat + wpLpos.
+(9)
+9
+
+Lcontact, Lroot, and Lquat are the MSE losses of foot con-
+tact labels, root status (height and transform relative to the
+previous frame projected on the XZ plane), and the joint
+rotations in 6D representations [73]. To penalize errors ac-
+cumulating along the kinematic chain, we perform forward
+kinematics (FK) and measure the global position distance of
+joints between original and reconstructed motion. As global
+positions of the joints are highly dependent on the root po-
+sitions, for the early epochs, the distance is measured based
+on root-centric coordinates to ignore the global location of
+roots, which we found empirically more stable.
+Motion Editing Loss
+For motion editing, the positional
+loss and regularization loss are defined as follows.
+L = wpLpos + wfLfoot + wrLroot,
+where
+Lpos =
+�
+j,qj∈φ
+∥pj − qj∥2, if φ exists at t
+Lfoot =
+�
+foot
+∥pe
+foot − pi
+foot∥2,
+Lroot = wr∥re
+xz − ri
+xz∥2 + w∆r∥˙re
+xz − ˙ri
+xz∥2.
+(10)
+pj denotes positions of joint j, and r, ˙r denotes root po-
+sitions and velocities respectively. Superscript e and i in-
+dicates whether it is from edited or initial motion, respec-
+tively. Subscript xz indicates the vector is projected onto
+the XZ plane. The loss term L enforces the edited motion
+to maintain contact and root trajectory (in the XZ plane) of
+the initial motion, while generating natural movements of
+the other joints to meet the sparse positional constraints.
+Generating Interaction Cue for Manipulation
+To syn-
+thesize character’s arm motions naturally interacting with
+the movements of articulated target objects, we produce
+desired interaction cues by producing the 3D trajectories
+of a chosen 3D position of the object at which the hand
+part of the character are expected to touch. Specifically,
+we apply the expected articulated motion of the 3D object
+model to produce the 3D trajectory of a chosen object ver-
+tex, v(Rt, Tt, θt), where Rt, Tt, are the global orientation
+and translation of the object and θt is the parameters for the
+object articulation (e.g., the hinge angle of the cover of a
+laptop) at time t. v(·) represents the 3D location of the cho-
+sen vertex v. To this end, we input the produced trajectory
+as the desired 3D interaction cue for a character’s joint (e.g.,
+a hand joint) assuming the joint is touching this object tra-
+jectory for manipulation φ = [v(Rt, Tt, θt)]t. Note that, in
+our visualization, we apply the desired articulated motions
+for the 3D object at each time, synced to the produced in-
+teraction cues.
+Label
+Seg. Length
+Seg. Count
+Total Frames
+Locomotion
+10
+11063
+11498
+Sit
+50 – 85
+5842
+14942
+Table 3. Details on pre-processed motion datasets per each action
+category for training our motion synthesizer S.
+Name
+Value
+Motion sequence length
+120
+Number of sequence (training)
+11397
+Number of sequence (validation)
+3135
+Number of sequence (test)
+2139
+Table 4. Details on pre-processed motion datasets for training our
+motion editing module M.
+C. More Details on Experiments
+C.1. Frechet Distance Features
+FDroot is computed by root feature vector, which is a con-
+catenated vector of root orientation in angle-axis represen-
+tation, root up vector, and root transform relative to the pre-
+vious frame. We note that all of the motions for comparison
+have the same up axis (y) and floor plane (xz). FDjoint is
+computed by joint feature vector, represented as joint orien-
+tations in angle-axis representation, excluding the root.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf,len=779
+page_content='Locomotion-Action-Manipulation:  Synthesizing Human-Scene Interactions in Complex 3D Environments  Jiye Lee  Hanbyul Joo  Seoul National University  {kay2353,hbjoo}@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='kr  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Our system, LAMA, produces high-quality and realistic 3D human motions that include locomotion, scene interactions, and  manipulations given a 3D environment and designated interaction cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Abstract  Synthesizing interaction-involved human motions has  been challenging due to the high complexity of 3D environ-  ments and the diversity of possible human behaviors within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  We present LAMA, Locomotion-Action-MAnipulation, to  synthesize natural and plausible long term human move-  ments in complex indoor environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The key motivation  of LAMA is to build a unified framework to encompass a  series of motions commonly observable in our daily lives,  including locomotion, interactions with 3D scenes, and ma-  nipulations of 3D objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' LAMA is based on a reinforce-  ment learning framework coupled with a motion matching  algorithm to synthesize locomotion and scene interaction  seamlessly under common constraints and collision avoid-  ance handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' LAMA also exploits a motion editing frame-  work via manifold learning to cover possible variations  in interaction and manipulation motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We quantitatively  and qualitatively demonstrate that LAMA outperforms ex-  isting approaches in various challenging scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Project  page: https://lama-www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Introduction  In our daily lives, we can easily observe that humans do  not live in isolation nor in voids, but continuously interact  with a complex environment surrounded by many objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Notably, humans perform such a diverse set of daily life  actions effortlessly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Imagine that we visit a new indoor en-  vironment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', a hotel room) we have never been before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  It is expected that we can still easily figure out how to move  from rooms to rooms, how to sit on a chair, how to open the  doors of closets, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' However, endowing machines  or virtual humans with such abilities is still a largely unex-  plored area, despite its importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Synthesizing scene interactions within real-life 3D envi-  ronments has been a challenging research problem due to  its complexity and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Human movements in real life  consists of various types of behaviors, including locomotion  with avoiding cluttered areas, diverse interactions with 3D  scenes, and sophisticated object-manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In particu-  lar, the spatial constraint that arises from real-life 3D envi-  ronments where many objects are cluttered makes motion  synthesis highly constrained and complex, and various pos-  sible arrangements of 3D environments make generalization  difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' As human-scene interactions cover a wide range of  technical challenges, previous approaches have focused on  sub-problems, such as (1) modeling static poses [17,24,49,  64,69,71,72] or (2) human object interactions with a single  target object or interaction type [10, 47, 53–55, 66, 67, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Recent methods [15,59,60] extend to synthesizing dynamic  interaction motions in cluttered real-world 3D scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' How-  ever, the performance of these methods are fundamentally  limited due to the lack of 3D ground truth data that contains  both human motions and paired 3D environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='02667v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='CV]  9 Jan 2023  In this paper, we present LAMA, Locomotion-Action-  MAnipulation, to synthesize natural and plausible long term  human motions in complex indoor environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The key  motivation of LAMA is to build a unified framework to  include locomotion, interactions with 3D scenes, and ma-  nipulations of 3D objects, which are the series of motions  commonly observable in our daily lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' LAMA is based  on a reinforcement learning framework coupled with a mo-  tion matching algorithm to synthesize locomotion and scene  interaction seamlessly while adapting to complicated 3D  scenes with collision avoidance handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The reinforce-  ment learning framework interprets the 3D information of  the given scene and optimally traverses among the motion  capture database via motion matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' As an advantage, our  system does not require any “scene-paired” datasets where  human movements are captured with the surrounding 3D  environments simultaneously, which is rarely available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' To  further cover the numerous variations of interaction mo-  tions, we also exploit an autoencoder based motion editing  approach to learn the motion manifold space [20] in which  the editing is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Through extensive quantitative  and qualitative evaluations against existing approaches, we  demonstrate that our method outperforms previous methods  in various challenging scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Our contributions are summarized as follows: (1) we  present the first method to generate realistic long term mo-  tions combined with locomotion, interaction with scene,  and manipulation in complicated cluttered scenes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' (2) we  propose a novel, unified framework that synthesizes loco-  motion and human-scene interactions in a seamless man-  ner, by introducing scene interpretation terms to a reinforce-  ment learning based approach to automatically generate op-  timal transitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' and (3) our outputs show the state-of-the-  art motion synthesis quality with longer duration (more than  10 sec) than previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Related Work  Generating Human-Scene Interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Generating  natural human motion has been a widely researched topic  in the computer vision community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Early methods focus on  synthesizing or predicting human movements by exploiting  neural networks [11,13,35,35,38,46,56,58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' However, these  approaches primarily address the synthesis of human mo-  tion itself, without taking into account the surrounding 3D  environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Recent approaches begin to tackle modeling  and synthesizing human interactions within 3D scenes, or  with objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Most of the researches focus on statically pos-  ing humans within the given 3D environment [16,24,69,71],  by generating human scene interaction poses from vari-  ous types of input including object semantics [17], im-  ages [21,23,64,65,68], and text descriptions [49,72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  More recently, there have been approaches to synthesize  dynamic human object interactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', sitting on chairs,  Encoder  Decoder  Task-Adaptive Motion Editing  Motion Generation  Action  Controller  3D Scene  Interaction  Cue  Action  Posture  Motion  Synthesizer  Optimization  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Overview of LAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  carrying boxes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Starke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' [53] introduce an autoregres-  sive learning framework with object geometry-based envi-  ronmental encodings to synthesize various human-object  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Later work [15, 70] extends this by synthe-  sizing motions conditioned with variations of objects and  contact points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Other approaches [47, 54, 55, 66, 67] focus  on generating natural hand movements for manipulation,  which is extended by including full body motions [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Physics-based character control to synthesize human object  interactions has been also explored in [8,10,39,47,66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Al-  though these approaches cover a wide range of human ob-  ject interactions, most of them solely focus on the relation-  ship between human and the target object without long-term  navigation in cluttered 3D scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  More recent approaches include generating natural hu-  man scene interactions within a complex 3D scene clut-  tered with many objects [6, 59–61], closely related to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  These methods are trained using human motion datasets  paired with 3D scenes, which require both ground truth mo-  tions and simultaneously captured 3D scenes for supervi-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Due to such difficulties, some methods exploit syn-  thetic datasets [6,61] or data fitted from depth videos [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  In previous approaches [15,59], navigation to move through  cluttered environments is often performed by a separate  module via a path planning algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', A∗ algorithm)  by approximating the volume of a human as a cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' This  path planning based methods approximate the spatial infor-  mation of the scene and the human body and therefore have  limitations under highly complex conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Motion Synthesis and Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Synthesizing natural  human motions by leveraging motion capture data has also  been a long-researched topic in computer graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Some  approaches [26,37] construct motion graphs, where plausi-  ble transitions are inserted as edges and motion synthesis is  done by traversing through the constructed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Similar  approaches [31, 51] connect motion patches to synthesize  interactions in a virtual environment or multi-person inter-  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Due to its versatility and simplicity, a number of  variations have been made on the graph based approach,  such as motion grammar [22] which enforces traversing  rules in the motion graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Motion matching [5, 9] can also  be understood as a special case of motion graph traver-  sal, where the plausible transitions are not precomputed but  searched during runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Recent advances in deep learning  allow to leverage motion capture data for motion manifold  2  rendel  reset  prev  play  nextlearning [19, 20, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Autoregressive approaches based on  variational autoencoders (VAE) [36, 46] and recurrent neu-  ral networks [14,29,41] are also used to forecast future mo-  tions based on past frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' These frameworks are general-  ized to synthesizing a diverse set of motions including lo-  comotion on terrains [19] mazes [36], action-specified mo-  tions [46], and interaction-involved sports [29, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Neural  network-based methods are also reported to be successful  in various motion editing tasks such as skeleton retarget-  ing [2], style transfer [3,20], and inbetweening [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Reinforcement learning (RL) has also been successful in  combination with both data-driven and physics-based ap-  proaches for synthesizing human motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Combined with  data-driven approaches, these RL frameworks serve as a  control module that generates corresponding motions to a  given user input by traversing through motion graphs [28],  latent space [34, 36, 57], and precomputed transition ta-  bles [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Deep reinforcement learning (DRL) has been  widely used recently in physics simulation as well to syn-  thesize physically plausible movements with a diverse set  of motor skills [4,32,41,43–45,62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Method  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Overview  Our system, dubbed as LAMA, outputs a sequence of  human poses M = {mt}T  t=1 by taking the 3D surrounding  cues W and desired interaction cues Φ, as inputs:  M = LAMA(W, Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  (1)  The output posture at time t, mt = (p0, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', rJ) ∈  R3J+3, is represented by a concatenated vector of global  root position p0 ∈ R3 and local joint orientations of J  joints where each j-th joint is in angle-axis representations  rj ∈ so(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Throughout our system, the skeleton tree struc-  ture and joint offsets are fixed and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We  represent the 3D environments W = {wi} as a set of 3D  object and environment meshes, including the background  scene mesh and other object meshes targeted for manip-  ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The interaction cues, Φ = [φ1, φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='φn], are an  ordered list of desired interaction inputs φi = {qj}j∈Ji  where qj ∈ R3 indicates desired positions of j-th joint,  and Ji is a set of specified joints for interaction (in prac-  tice, few joints such as root 1 or end-effectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Examples of  the 3D environment W and interaction inputs φi are shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 5 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Intuitively, φi specifies the expected positions  of selected joints of the human character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Note that we do  not specify the exact timing of the interaction, as the timing  is automatically determined by our action controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' More  details are addressed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  To synthesize locomotion, interaction, and manipulation  together, LAMA is designed via a three-level system com-  1For root, orientation in angle-axis representation is also included in φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' (a) Skeleton with joints and box nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' (b) Automatically  detected collision points (colored as red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  posed of the action controller A and the motion synthesizer  S, followed by a manifold-based motion editor E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' By taking  3D scene cues W and desired interaction cues Φ as input,  the action controller A makes the use of a reinforcement  learning (RL) framework by training the control policy π to  sample an action at time t, π(at|st, W, Φ), where at con-  tains the plausible next action cues including predicted ac-  tion types and short-term future forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' st is the state  cues to represent the current status of human characters in-  cluding its body posture, surrounding scene occupancy, and  current target interaction cue, which can be computed via  a function ψ, st = ψ(mt−1, mt, W, Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Intuitively, action  controller A predicts the plausible next action cues at by  considering the current character-scene state st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The gener-  ated action signals at from the action controller A is pro-  vided as the input for the motion synthesizer S, which then  determines the posture at the next time step mt+1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=',  S(mt, at) = mt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Afterwards, the character’s next state  can be computed again via st+1 = ψ(mt, mt+1, W, Φ),  which is input to the action controller recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Followed by the initial motion generation part from A  and S, our system furthermore applies a motion editor  E(M) = ˜M, where ˜M = { ˜Mt}T  t=1 is the edited motions  to further express the motions involving complex human-  object interactions such as manipulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' moving ob-  jects, opening doors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 2 shows the overview of LAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Scene-Aware Action Controller  Based on reinforcement learning, our action controller  A enables the character to perform locomotion and desired  actions with fulfilling the interaction cues Φ and avoiding  collisions in the 3D environment W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' A is a trained control  policy π where π(at|st, W, Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Different from previous  approaches where navigation and scene-object interactions  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', sitting) are performed by separate modules [15, 59],  our RL-based framework performs both in a unified way  with a common objective by automatically determining the  transition from navigation to specific actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' As a key ad-  vantage, LAMA can be robustly generalized to challenging  unseen 3D clutters in long-term human motion synthesis  and also outperforms previous methods by avoiding colli-  sions throughout the whole process, including navigation  3  Joint  Node  Jointand interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  State.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The state st = ψ(mt−1, mt, W, Φ) at time t is  a feature vector representing the current status of the hu-  man character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' st = (sbody  t  , sscene  t  , sinter  t  ) is composed of  body configuration sbody, 2D scene occupancy sscene, and  desired current target interaction sinter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Body configuration  sbody = {r, ˙r, θup, h, pe} includes r, ˙r ∈ RJ′×6 that are the  joint rotations and velocities respectively for the J′ joints  excluding the root in 6D representations [73], θup ∈ R that  is the up vector of the root (represented by the angle w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='t  the Y-axis), h ∈ R that is the root height from the floor,  and pe ∈ Re×3 that is the end-effector positions in person-  centric coordinate (where e is the number of end-effectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  sscene = {gocc, groot} includes scene occupancy informa-  tion in 2D floor plane, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' gocc ∈ Rn2 rep-  resents the 2D occupancy grid on the floor plane of neigh-  boring n cells around the agent and groot ∈ R2 denote the  current 2D global root position of the character in the dis-  cretized grid plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' sinter is an element of Φ and represents  the interaction cue the character is currently targeting, that  is sinter = φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Given the current status of the character st,  the control policy π outputs the feasible action at  =  (atype  t  , afuture  t  , aoffset  t  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' atype  t  provides the probabilities  of next action type among all possible actions, determining  the transition timing between actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', from locomo-  tion to sitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' afuture  t  predict future motion cues such as  plausible root position for the next 10, 20, and 30 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  aoffset  t  is intended to update the raw motion data searched  from the motion database in motion synthesizer module S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Intuitively, our learned control policy generates an optimal  posture offset aoffset  t  which is applied to the closest plau-  sible raw posture in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' This enables the character  to perform more plausible scene-aware human poses, allow-  ing our system to be generalized to any unseen 3D scenes  given a limited amount of motion capture data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' More details  are addressed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Motion Synthesizer  Given the current motion output mt and actions sig-  nals at from the action controller A as inputs, the mo-  tion synthesizer produces the next plausible character pos-  ture: S(mt, at) = mt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' As the first step, motion synthe-  sizer searches for the motion from a motion database that  best matches the closest motion feature, then modifies the  searched raw motion to be more suitable for the scene envi-  ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' To this end, motion synthesizer’s output mt+1 is  in turn fed into the action controller recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We exploit  a modified version of motion matching algorithm [5, 9, 18]  for the first step of motion synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In motion matching,  motion synthesis is performed periodically by searching the  most plausible next shot motion segments from a motion  DB, and compositing them into a long connected sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Visual representation of the 2D occupancy grid near the  root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Grid on the right represents top view of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Blue in-  dicates root position while gray represents the space is occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Occupied cells near the root are colored as black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Motion features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Motion feature represents the charac-  teristic of each frame in the short motion segment and is  computed as f(m) = {{pj}, { ˙pj}, θup, c, ofuture}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' From  a posture m, the positions and velocities pj, ˙pj  ∈ R3  are extracted for the selected joints j ∈ {Head, Hand,  Foot}, which are defined in a person-centric coordinate  of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' θup ∈ R3 is the up-vector of the root joint, and  c ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='5, 1} indicates automatically computed foot con-  tact cues of the left and right foot (0 for non-contact, 1 for  contact, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='5 for non-contact but close to the floor within  a threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' ofuture = {{pdt  0 }, {rdt  0 }} contains the cues  for the short-term future postures, where pdt  0 and rdt  0 are  the position and orientation of root joint at dt frames later  from the current target frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' ofuture are computed in 2D  XZ plane in person-centric coordinate of the current tar-  get motion m, and thus pdt  0 , rdt  0  ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The selected fu-  ture frames are action-type specific, and for locomotion we  extract 10, 20, and 30 frames in the future (at 30Hz) fol-  lowing [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Intuitively, the motion feature extracts the target  frame’s posture and temporal cues by considering neigh-  boring frames 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We pre-compute motion features for every  frame of the motion clips in the motion database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Motion  feature of the current state of the character, or the query  feature, is also computed in the same way based on posture  mt−1, mt and afuture  t  produced by the action controller,  that is xt = f(mt−1, mt, atype  t  , afuture  t  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The component  afuture  t  serves as ofuture in the query feature, which can be  understood as the action controller providing cues for pre-  dicted future postures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Motion searching and updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The query motion fea-  ture xt from the current character is computed as addressed  above, and let the motion features in motion database de-  noted as yk for the k-th clips in the DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Motion searching  finds the best matches in the motion database by computing  the weighted euclidean distances between the query feature  and DB features:  k∗ = arg min  k  ||wT  f (xt − yk)||2,  (2)  where wf is a fixed weight vector to control the impor-  2In practice, the input of feature extractor function f should take into  account the motions of neighboring timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  4  2D  occupancy gridtance of feature elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' After finding the best match ˆmk∗  from motion database, the motion synthesizer further up-  dates it with the predicted motion offset aoffset  t  from at,  that is τ( ˆmk∗+1, aoffset) = mt+1, where ˆmk∗+1 is the  next plausible character posture and τ is an update function  to update selected joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In practice, the motion searching  is performed periodically (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', every N-th frames) to make  the synthesized motion temporally more coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Learning for Scene-Aware Action Controller  In the reinforcement learning framework, the objective is  to learn the optimal policy which maximizes the discounted  cumulative reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In our method, we design rewards to  guide the agent to perform locomotion towards the target  objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', sofa) and also perform desired interaction with  the object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', sitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In particular, our RL-framework  performs both navigation and interaction with common con-  straints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', smooth transitions, collision avoidance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Our reward function consist of the following terms:  Rtotal = wtrRtr + wactRact + wregRreg,  (3)  where wtr, wact, and wreg are the weights to balance among  reward terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The trajectory reward Rtr is obtained when  the character moves towards the desired interaction input φ  while meeting the spatial constraints from the surrounding  3D scene, described below:  Rtr = rcoli · rpos · rroot, where  (4)  rcoli = exp  �  − 1  σ2  coli  �  b∈B  wbρ(b, W)  �  ,  (5)  rpos = exp  �  �−  1  σ2  root  �  j∈J  ∥p0 − qj∥2  �  � ,  (6)  rvel =  �  1  when ˙proot ≥ σth  σvel∥ ˙p0∥2  else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  (7)  The collision-avoidance reward rcoli penalizes collisions  with 3D scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3 (a), body limbs in  the skeletal structure are represented as a set of box-shaped  nodes B with a fixed width, where each element b ∈ B is  a 3D box representation of legs and arms (we exclude torso  and head).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The function ρ(b, W) detects the collision be-  tween edges of a box-shaped node b with 3D scene meshes  W and returns the number of intersection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3  (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' wb is the weights to control importance of each limb b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  The collision-avoidance reward is maximized when no pen-  etration occurs, making the control policy π to find the opti-  mal trajectory and pose offset to avoid physically implausi-  ble collisions and penetrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' rpos are obtained when the  agent moves to reach the targeting interaction cue φ, by en-  couraging agent’s root position p0 to be closer to the target  interaction cue {qj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' rvel encourages the character to move  by penalizing when the root velocity ˙proot is less than a  threshold σth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' σcoli, σroot, and vel are weights to control  the balance between terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Action reward Ract enforces the synthesized motion to  fulfill the given interaction cue φ = {qj}:  Ract = rinter · r∆t · r∆v,  where  rinter = exp  �  �−  1  σ2  inter  �  j∈J  ∥pj − qj∥2  �  � ,  r∆t = exp  �  −σ2  ∆tCtr  �  ,  r∆v = exp  �  −σ2  ∆vCvel  �  ,  (8)  where interaction reward term rinter is maximized when the  performed action meets the positional constraints provided  by interaction cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Smoothness reward terms r∆t and r∆v  minimizes the transition cost, which is based on the subpart  of the feature distances defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 2, where Ctr is the  weighted feature distances of pj, θup, and c, and Cvel is  from ˙p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' These are intended to penalize the case where the  character makes abrupt changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Regularization reward Rreg penalizes the aoffset  t  exces-  sively modifying the original posture brought from the mo-  tion synthesizer, denoted as ˆmt, and maintains temporal  consistency among frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Rreg = exp  �  −  1  σ2reg  �  ∥ ˆmt − mt∥2 + ∥mt − mt−1∥2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  It is reported that [33, 41] multiplying rewards with con-  sistent goals are suitable for learning, as the reward is re-  ceived when the conditions are simultaneously met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Fur-  thermore, to accelerate learning, we use early termination  conditions [43] and limited action transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The episode  is terminated when the character moves out of the scene  bounding box, or when the collision reward rcoli is under a  certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Also, the action controller first checks in  advance whether the action signal is valid when it makes  transitions from locomotion to other actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' When the  nearest feature distance of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 2 in the motion synthesizer  is over a certain threshold, the action controller discards the  transition and continues navigating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The control policy is  learned through Proximal Policy Optimization (PPO) algo-  rithm [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Task-Adaptive Motion Editing  Interaction includes a massively diverse pool of mo-  tions, and these variations cannot be fully handled by lim-  ited amount of motion database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In order to cover such di-  versity, we include a task-adaptive motion editing module  in our motion synthesis framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The goal of our edit-  ing module E is (1) to edit motion M to fit into diverse  5  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Visual representation of system input Φ, W and output  motion sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' On the left, interaction cues are shown as cyan  spheres and arrows (indicating orientation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The right is the syn-  thesized human motion ˜  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  target object geometries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', sitting on a chair with dif-  ferent height), and (2) to generate additional hand move-  ments for manipulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', grasping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In particular, in the  case of manipulation, additional interaction cue φ can be  provided to enforce an end-effector (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', a hand) to fol-  low the desired trajectories to express the manipulation task  on the target object, as shown in Fig 8 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The edited  motion ˜M = E(M) should not only fulfill the sparsely  given positional constraints, but also preserve the temporal  consistency between frames and spatial correlations among  joints in order to maintain its naturalness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We adopt the mo-  tion manifold learning approach with convolutional autoen-  coders [20] to compress motion to a latent vector within a  motion manifold space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Motion editing is done by searching  for an optimal latent vector among the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' For train-  ing the autoencoder, motion sequence, which we denote as  X converted from M, is represented as a time-series of hu-  man postures by concatenating joint rotations in 6D repre-  sentations [73], root height, root transform relative to the  previous frame projected on the XZ plane, and foot contact  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The encoder and decoder module are trained based  on reconstruction loss, ||X − Ψ−1(Ψ (X)) ||2, where Ψ is  the encoder and Ψ−1 is the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  The latent vector from the encoder z = Ψ(X) repre-  sent the motion manifold space by preserving the spatio-  temporal relationship among joints and frames within the  motion sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' As demonstrated in [20], editing motions  in this manifold space ensures the edited motion to be re-  alistic and temporally coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' To this end, we find the  optimal latent vector z∗ by minimizing a loss function L  by constraining the outputs motions to follow the interac-  tion constraint φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We also include additional regularizers in  L so that the output motion to maintain the foot locations  and root trajectories to the original motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' See supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  for more details on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Finally, the edited motion ˜M can be  computed via Ψ−1(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Experiments  We evaluate LAMA’s ability on synthesizing long-term  motions with various human-scene and human-object inter-  Method  Plausibility  Naturalness  Slip  Penetration  FDtotal  FDroot  FDjoint  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' [60]  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='88  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='93  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' [60]*  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='58  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='44  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='00  SAMP [15]  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='5  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='95  LAMA (ours)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='21  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='91  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Baseline comparison Foot slip loss (cm, ↓) averaged over  all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Penetration loss(percentage, ↓) is counted based on in-  tersection points of the 3D environment and the skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Natural-  ness score is based on fr´echet distance (FD ↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' with an  asterisk indicates without post-processing optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  actions involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We exploit an extensive set of quantitative  metrics and perceptual study to evaluate the physical plau-  sibility and naturalness of the synthesized motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' For constructing the database for the motion  synthesizer, motion capture data are selectively collected  and refined from Ubisoft La Forge [14], COUCH [70],  and SAMP [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' All the data used in this system are mo-  tion capture data (in bvh format) with no scene or ob-  ject related information, and are retargeted into a unified  skeletal structure with MotionBuilder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We use PROX [16]  and Matterport3D [7] datasets for 3D environment and  SAPIEN [63] object meshes for manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Our code and  pre-processed data will be publicly released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The policy and the value net-  work of the action controller module consists of 4 and  2 fully connected layers of 256 nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The  encoder and decoder of the task-adaptive motion editing  module consist of three convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Adam opti-  mizer [25] is used for training and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We use  Nvidia RTX 3090 for training the action controller and the  motion editing module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' It takes 10 to 80 minutes to learn  a single control policy, where the training time mainly de-  pends on how difficult the interaction cues are to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  For optimization in the motion editing module, it takes 3 to  4 minutes for 500 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' See supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' for more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Experimental Setup  Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Quantifying motion synthesis quality  is challenging due to the lack of ground-truth data or offi-  cial evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We try to quantify them in terms of  physical plausibility and naturalness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Physical plausibility: We use contact and penetration  metrics to evaluate the physical plausibility of the synthe-  sized motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Contact loss penalizes the foot movement  when the foot is in contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Since foot contact is a critical  element in dynamics, contact-based metric is closely related  in determining the physical plausibility of motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Pene-  tration loss (“Penetration” in Table 1) measures implausible  cases when the body penetrates the objects in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We  compute penetration metric by counting frames where the  6  interaction cue Φ2  interaction cue ΦFigure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Comparison with LAMA (left) and LAMA without col-  lision reward (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' As shown in the right, without collision re-  ward the character fails to avoid collisions with obstacles (marked  as red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  intersection points (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='4) goes over a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3  Naturalness: We measure the naturalness of the synthe-  sized motions by measuring the Fr´echet distance, as re-  ported in [15, 35, 40] between the synthesized motion and  motions from motion capture data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Features are extracted  from motion sequences and the Fr´echet distance is com-  puted with the extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We measure the natural-  ness of character root movements FDroot, including root ori-  entation and velocity, and character joint rotations FDjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We compare our LAMA with the state-of-the-art  approaches as well as variations of ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' [60] is the state-of-the-art long term mo-  tion synthesis method for human-scene interactions within  a given 3D scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We use the author’s code for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  As Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' uses optimization to post-process the synthe-  sized motion to improve foot contact and reduce collisions,  we both compare Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' with and without optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  SAMP [15] generates interactions which can be general-  ized not only for object variations but also random starting  points within a given 3D scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' SAMP explicitly exploits a  path planning module to navigate through cluttered 3D en-  vironments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Ablative baselines We perform ablation studies on the ac-  tion controller and task-adaptive motion editing module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We  perform ablation studies on the scene reward rcoli, and ac-  tion offset aoffset  t  to present the contribution of both terms  on our system’s capability to generate scene-aware motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  We also compare our method without the transition reward  r∆t and r∆v terms (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='4) in the action controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Fi-  nally, we demonstrate the strength of our task-adaptive mo-  tion editing module to edit motions naturally (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='5) by  comparing with inverse kinematics (IK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Comparisons with Previous Work  Quantitative Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We compare methods in 6  different scenarios from various 3D scenes in the PROX  dataset [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Foot contact is automatically labeled based on  310 for legs and 7 for arms  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Comparison with LAMA (left) and LAMA without ac-  tion offset (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The character in original LAMA moves forward  while tilting its arms to avoid collision with walls, while in LAMA  without action offset does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  positional velocity of the foot joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Foot slip metric is mea-  sured by foot joint positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' To compute penetration metric  in a fair way, SMPL-X outputs of Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' and SAMP are  converted to box-shaped skeletons as in ours and intersec-  tion point are counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Table 1 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  As shown, our LAMA outperforms Wang et al both in  naturalness and physical plausibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' It is noted that Wang  et al performs optimization as post-processing to explic-  itly minimize foot slip, and yet LAMA still shows on-par  performance against it (and better in all other metrics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Compared with SAMP, our method shows much better re-  sults in plausibility metrics (both Slip and Penetration),  and shows slightly better performance in naturalness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Apart  from SAMP which relies on a separate navigation mod-  ule, our RL-based action controller handles collisions in the  same way of scene-interaction and shows much better per-  formance in in complex and cluttered 3D scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  A Human Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' To further validate our results, we  compare the quality of our output over other baselines,  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' and SAMP, through A/B testing from human  observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' For the study, we choose 5 scenarios from dif-  ferent indoor scenes, and render the results of each method  using the exactly same view and 3D characters, so that they  cannot be distinguished from the appearance side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We build  two separate sets, where in each set the result videos of  our method are shown with each competitor side by side  in a random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Human observers are asked to choose a  motion clip that is more human-like and plausible in the  given 3D scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We perform each set of tests with non-  overlapping 15 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' See our supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' for more  details about the study setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' As the result, the outputs of  our method are preferred by the majority (more than 50%  voting) in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' By considering all votes independently,  our method are preferred 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='0% over SAMP and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='3%  over Wang et al.’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In particular, we found that our  method greatly outperform the competing methods in terms  of the naturalism of foot stepping, transition between loco-  motion and action, and collision avoidance with the scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  See our supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' videos for more results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  7  LAMA  LAMA w/o collision rewardoffset  LAMA w/o a  LAMA  LAMAFigure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' (a) Comparison with LAMA (top) and LAMA without  manifold and replaced with IK (bottom) of a character opening the  toilet lid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' (b) Comparison with LAMA (top) and LAMA without  motion editing (bottom) in sitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Ablation Studies  Ablation Studies on Action Controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We quantita-  tively compare the original LAMA and the LAMA without  collision reward rcoli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We intend to demonstrate the role of  rcoli that enforces the action controller to search for optimal  actions for generating motions without collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Ablation  studies are done in 5 PROX scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In the original LAMA,  penetrations occur in only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='1% of the frames among the  whole motion sequences, while the ratio is 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='7% in LAMA  without collision reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The result supports that the colli-  sion reward rcoli enforces the action controller to compute  optimal actions for synthesizing body movement according  to the spatial constraint of the given 3D scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Example re-  sults are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  We also compare the contribution of other components  in the action controller module in generating natural inter-  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 7, with the action controller without  aoffset  t  the character fails to avoid penetration with objects  or walls, as the raw motion from the motion database does  not have any information of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' This demonstrates  that action offset also plays a role in generating detailed  scene-aware poses even from raw motion capture data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Moreover, the results with the action controller without  smoothness rewards r∆t and r∆v are not smooth enough,  showing unnatural movements such as jerking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' These abla-  tion studies justify the advantages of our reward terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Ablation Studies on Task-Adaptive Motion Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  We ablate our motion editing module by replacing it with  an alternative approach via Inverse-Kinematics (IK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' An ex-  ample result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 8 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' For manipulation, the  results with IK show jerky and awkward motions because  the temporal and inter-joint correlations in natural human  motions are not reflected in IK, while original LAMA with  task-adaptive motion editing module shows much natural  motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Our motion editing module can also be used to  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Examples of synthesized manipulation motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The tar-  get object for manipulation is colored as orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Top is a motion  sequence of walking and opening a toilet lid, and the bottom is a  sequence of walking and opening doors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The character is colored  purple at start and aqua at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  further adjust the character movements in different object  geometries, going over the limit of the motion database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' As  seen in Fig 8 (right), the motion editing module enables the  character to properly sit in chairs with various sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Discussion  In this paper, we present a method to synthesize locomo-  tion, scene-interaction, and manipulation in a unified sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Leveraging a RL framework with motion matching,  our method enables to produce natural and plausible hu-  mans motions in complex and cluttered 3D environments  only with a limited amount of motion-only datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Our  method has been thoroughly evaluated in diverse scenar-  ios, outperforming previous approaches [15, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We also  demonstrate the robustness and generalization ability of our  system by covering a wide range of human interactions in  many different 3D environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  While our RL-based method can be generalized to any  unseen 3D environments, a new control policy has to be  trained for each motion sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Combining RL with a  supervised learning framework for better efficiency can be  an interesting future research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Furthermore, al-  though we assume a fixed skeletal information throughout  the system, interaction motions may change depending on  the character’s body shape and sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We leave synthesizing  motions on varying body shapes as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Acknowledgments: This work was supported by SNU-  Naver Hyperscale AI Center, SNU Creative-Pioneering Re-  searchers Program, and NRF grant funded by the Korea  government (MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 2022R1A2C209272411).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  8  LAMA  LAMA  LAMA w/o motion editing  LAMA w/o manifold + IK梦人庆A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Supplementary Video  The supplementary video shows the results of our  method, LAMA, on various scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In the video, we  show the human motion synthesis results on PROX [16],  Matterport3D [7], and also our own home-brewed 3D scene  produced by Polycam App [1] in an iPad pro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We use  SAPIEN [63] object meshes for manipulation examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' As  shown, our method successfully produces plausible and nat-  ural human motions in many challenging scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Our  supplementary video contains several ablation studies of  our method by showing the importance of collision reward  rcoli in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' (4), transition reward (r∆t , r∆v) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' (8), pos-  ture offset aoffset  t  in Action Controller (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='2), and our  motion editing modules (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='5) compared to the tradi-  tional Inverse Kinematics (IK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We also show the compari-  son with previous state-of-the arts [15, 59, 60] and demon-  strate that our results produces better quality of motions  with better collision avoidance performance in complicated  3D scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Additional Details on Implementations  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Action Controller  Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  For the action controller A and  motion synthesizer module S, we use the animation library  DART [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We also use a publicly available PPO imple-  mentation [32, 41], where we remove the variable time-  stepping functions stepping in [32] by following the origi-  nal PPO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The details of the training regarding the  policy and value network of the action controller are written  in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Early Termination Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  As written in the main  paper, the episode is terminated (1) when the character  moves out of the scene bounding box;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' (2) when the colli-  sion reward rcoli is under a certain threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' or (3) when  the root of the human character is located in the blocked  (occupied) regions of the scenes in 2D grid space during  the locomotion status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Name  Value  Learning rate of policy network  2e-4  Learning rate of value network  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='001  Discount factor (γ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='95  GAE and TD (λ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='95  Clip parameter (ϵ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='2  # of tuples per policy update  30000  Batch size for policy/value update  512  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Details on the hyper-parameters for learning the control  policy of the action controller A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Motion Synthesizer  Motion Database Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  As described in our  main paper, we pre-process the motion segments by selec-  tively collecting and clipping from Ubisoft La Forge [14],  COUCH [70], and SAMP [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The length (in frames)  of motion segments (“Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Length” in tables), number of  motion segment (“Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Count” in tables), and the number  of total frames (“Total Frames” in tables) are summarized  in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Action-Specific Feature Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  The motion feature,  as defined in our main paper Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='3, represents both the  current state of the motion and a short term future move-  ments: f(m) = {{pj}, { ˙pj}, θup, c, ofuture}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In particu-  lar the action specific feature ofuture = {{pdt  0 }, {rdt  0 }}  contains future motions so that the motion search process  can take into account the future motion consistency, where  pdt  0 , rdt  0 ∈ R2 are the position and orientation of root joint at  dt frames later from the current target frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' For locomo-  tion, we extract dt = 10, 20, and 30 frames in the future (at  30Hz) following [9], as addressed in our main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' For sit-  ting, we specifically choose dt as the frame where the char-  acter completes the sit-down motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The major motivation  of this design choice is encourage the motion synthesizer to  search the motion clips with the desired target action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Computation Cost for Searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  The computation time  for searching the motion database is done between 1-2 mil-  liseconds in CPU, where we test on AMD Ryzen 5950X  CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The number of search times varies and is dependent  to the 3D scenes and desired motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In one of our sce-  narios, total 17 searches in locomotion(walk) and 14 in ac-  tion(sit) were done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' For locomotion, the searching time is  average 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='743 milliseconds (standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='46) and  for action(sit) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='103 milliseconds (standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Motion Editing via Motion Manifold  Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  For the convolutional autoen-  coder of task-adaptive motion editing, we use PyTorch [42],  FairMotion [12], and PyTorch3d [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The autoencoder is  trained with the Adam optimizer with learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  We use 3 layers of 1D temporal-convolutions with kernel  width of 25 and stride 2, and the channel dimension of each  output feature is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The training datasets are summarized  in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Note that we use different pre-processing steps  between Motion editing module and Motion Synthesizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Reconstruction Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  The encoder Ψ and decoder Ψ−1  are trained based on reconstruction loss Lrecon = ||X −  Ψ−1(Ψ (X)) ||2, where:  Lrecon = wcLcontact + wrLroot + wqLquat + wpLpos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  (9)  9  Lcontact, Lroot, and Lquat are the MSE losses of foot con-  tact labels, root status (height and transform relative to the  previous frame projected on the XZ plane), and the joint  rotations in 6D representations [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' To penalize errors ac-  cumulating along the kinematic chain, we perform forward  kinematics (FK) and measure the global position distance of  joints between original and reconstructed motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' As global  positions of the joints are highly dependent on the root po-  sitions, for the early epochs, the distance is measured based  on root-centric coordinates to ignore the global location of  roots, which we found empirically more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Motion Editing Loss  For motion editing, the positional  loss and regularization loss are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  L = wpLpos + wfLfoot + wrLroot,  where  Lpos =  �  j,qj∈φ  ∥pj − qj∥2, if φ exists at t  Lfoot =  �  foot  ∥pe  foot − pi  foot∥2,  Lroot = wr∥re  xz − ri  xz∥2 + w∆r∥˙re  xz − ˙ri  xz∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  (10)  pj denotes positions of joint j, and r, ˙r denotes root po-  sitions and velocities respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Superscript e and i in-  dicates whether it is from edited or initial motion, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Subscript xz indicates the vector is projected onto  the XZ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' The loss term L enforces the edited motion  to maintain contact and root trajectory (in the XZ plane) of  the initial motion, while generating natural movements of  the other joints to meet the sparse positional constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Generating Interaction Cue for Manipulation  To syn-  thesize character’s arm motions naturally interacting with  the movements of articulated target objects, we produce  desired interaction cues by producing the 3D trajectories  of a chosen 3D position of the object at which the hand  part of the character are expected to touch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Specifically,  we apply the expected articulated motion of the 3D object  model to produce the 3D trajectory of a chosen object ver-  tex, v(Rt, Tt, θt), where Rt, Tt, are the global orientation  and translation of the object and θt is the parameters for the  object articulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', the hinge angle of the cover of a  laptop) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' v(·) represents the 3D location of the cho-  sen vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' To this end, we input the produced trajectory  as the desired 3D interaction cue for a character’s joint (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=',  a hand joint) assuming the joint is touching this object tra-  jectory for manipulation φ = [v(Rt, Tt, θt)]t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Note that, in  our visualization, we apply the desired articulated motions  for the 3D object at each time, synced to the produced in-  teraction cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Label  Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Length  Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Count  Total Frames  Locomotion  10  11063  11498  Sit  50 – 85  5842  14942  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Details on pre-processed motion datasets per each action  category for training our motion synthesizer S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Name  Value  Motion sequence length  120  Number of sequence (training)  11397  Number of sequence (validation)  3135  Number of sequence (test)  2139  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Details on pre-processed motion datasets for training our  motion editing module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' More Details on Experiments  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Frechet Distance Features  FDroot is computed by root feature vector, which is a con-  catenated vector of root orientation in angle-axis represen-  tation, root up vector, and root transform relative to the pre-  vious frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' We note that all of the motions for comparison  have the same up axis (y) and floor plane (xz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' FDjoint is  computed by joint feature vector, represented as joint orien-  tations in angle-axis representation, excluding the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  References  [1] Polycam - lidar and 3d scanner for iphone & android.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  https://poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='cam/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 9  [2] Kfir Aberman, Peizhuo Li, Dani Lischinski, Olga Sorkine-  Hornung, Daniel Cohen-Or, and Baoquan Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Skeleton-  aware networks for deep motion retargeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Graph, 39(4), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3  [3] Kfir Aberman, Yijia Weng, Dani Lischinski, Daniel Cohen-  Or, and Baoquan Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Unpaired motion style transfer from  video to animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', 39(4), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3  [4] Kevin  Bergamin,  Simon  Clavet,  Daniel  Holden,  and  James Richard Forbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Drecon: data-driven responsive con-  trol of physics-based characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' Matterport3d: Learning from rgb-d  data in indoor environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In 3DV, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content='  Learning to sit: Synthesizing human-chair interactions via  hierarchical control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In AAAI, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' Motion matching and the road to next-gen  animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' fairmotion - tools to  load, process and visualize motion capture data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Github,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' A recurrent variational autoen-  coder for human motion synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In BMVC, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' Robust motion in-betweening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Graph, 39(4), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3, 6, 9  [15] Mohamed Hassan, Duygu Ceylan, Ruben Villegas, Jun Saito,  Jimei Yang, Yi Zhou, and Michael Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Stochastic scene-  aware motion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' Black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Resolving 3D human pose ambigu-  ities with 3D scene constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content='  Populating 3d scenes by  learning human-scene interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content='  Learned motion matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content='  ACM  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' ACM  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' Amp: Adversarial motion priors for styl-  ized physics-based character control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+page_content=' Learning to generate long-  term future via hierarchical prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In ICML, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 2  [59] Jingbo Wang, Yu Rong, Jingyuan Liu, Sijie Yan, Dahua Lin,  and Bo Dai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Towards diverse and natural scene-aware 3d  human motion synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 1, 2, 3, 9  [60] Jiashun Wang, Huazhe Xu, Jingwei Xu, Sifei Liu, and Xiao-  long Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Synthesizing long-term 3d human motion and  interaction in 3d scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In CVPR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 1, 2, 6, 7, 8, 9  [61] Jingbo Wang, Sijie Yan, Bo Dai, and Dahua Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Scene-  aware generative network for human motion synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In  CVPR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 2  [62] Jungdam Won, Deepak Gopinath, and Jessica Hodgins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' A  scalable approach to control diverse behaviors for physically  simulated characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', 39(4), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 3  [63] Fanbo Xiang, Yuzhe Qin, Kaichun Mo, Yikuan Xia, Hao  Zhu, Fangchen Liu, Minghua Liu, Hanxiao Jiang, Yifu Yuan,  He Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Sapien: A simulated part-based interactive  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In CVPR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 6, 9  [64] Xianghui Xie, Bharat Lal Bhatnagar, and Gerard Pons-Moll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Chore: Contact, human and object reconstruction from a sin-  gle rgb image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In ECCV, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 1, 2  [65] Xiang Xu, Hanbyul Joo, Greg Mori, and Manolis Savva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  D3d-hoi: Dynamic 3d human-object interactions from  videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' arXiv preprint arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='08420, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 2  [66] Zeshi Yang, Kangkang Yin, and Libin Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Learning to use  chopsticks in diverse gripping styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=',  41(4), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 1, 2  [67] He Zhang, Yuting Ye, Takaaki Shiratori, and Taku Komura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Manipnet: Neural manipulation synthesis with a hand-object  spatial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=', 40(4), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 1, 2  [68] Jason Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Zhang, Sam Pepose, Hanbyul Joo, Deva Ramanan,  Jitendra Malik, and Angjoo Kanazawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content='  Perceiving 3d  human-object spatial arrangements from a single image in  the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In ECCV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 2  [69] Siwei Zhang, Yan Zhang, Qianli Ma, Michael J Black, and  Siyu Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Place: Proximity learning of articulation and con-  tact in 3d environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In 3DV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 1, 2  [70] Xiaohan Zhang, Bharat Lal Bhatnagar, Sebastian Starke,  Vladimir Guzov, and Gerard Pons-Moll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Couch: Towards  controllable human-chair interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In ECCV, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 1, 2,  6, 9  [71] Yan Zhang, Mohamed Hassan, Heiko Neumann, Michael J  Black, and Siyu Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Generating 3d people in scenes with-  out people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In CVPR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 1, 2  [72] Kaifeng Zhao, Shaofei Wang, Yan Zhang, Thabo Beeler, ,  and Siyu Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' Compositional human-scene interaction syn-  thesis with semantic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In ECCV, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 1, 2  [73] Yi Zhou, Connelly Barnes, Lu Jingwan, Yang Jimei, and Li  Hao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' On the continuity of rotation representations in neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' In CVPR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
+page_content=' 4, 6, 10  12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfzAKR/content/2301.02667v1.pdf'}
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+arXiv:2301.02920v1  [math.GR]  7 Jan 2023
+BEAUVILLE STRUCTURES FOR QUOTIENTS
+OF GENERALISED GGS-GROUPS
+ELENA DI DOMENICO, S¸ ¨UKRAN G¨UL, AND ANITHA THILLAISUNDARAM
+Abstract. A finite group with a Beauville structure gives rise to a certain compact
+complex surface called a Beauville surface. G¨ul and Uria-Albizuri showed that quotients
+of the periodic Grigorchuk-Gupta-Sidki (GGS-)groups that act on the p-adic tree, for
+p an odd prime, admit Beauville structures.
+We extend their result by showing that
+quotients of infinite periodic GGS-groups acting on pn-adic trees, for p any prime and
+n ≥ 2, also admit Beauville structures.
+1. Introduction
+A Beauville surface is a compact complex surface isomorphic to (C1 × C2)/G, where
+• C1 and C2 are algebraic curves of genus at least 2, and G is a finite group acting
+freely on C1 × C2 by holomorphic transformations,
+• the group G acts faithfully on the curves Ci such that Ci/G ∼= P1(C) and the
+covering map Ci → Ci/G is ramified over three points, for i ∈ {1, 2}.
+The group G is then said to be a Beauville group.
+One can reformulate the condition for a finite group G to be a Beauville group purely
+in group-theoretical terms: for x, y ∈ G, let
+Σ(x, y) =
+�
+g∈G
+�
+⟨x⟩g ∪ ⟨y⟩g ∪ ⟨xy⟩g�
+,
+that is, the union of all conjugates of the cyclic subgroups generated by x, y and xy.
+Then G is a Beauville group if and only if G is 2-generated and there exist generating sets
+{x1, y1} and {x2, y2} of G such that Σ(x1, y1) ∩ Σ(x2, y2) = {1}. The sets {x1, y1} and
+{x2, y2} are then called a Beauville structure for G.
+Beauville groups have been intensely studied in recent times; see surveys [1, 7, 8, 16].
+For example, the abelian Beauville groups were classified by Catanese [4]: a finite abelian
+group G is a Beauville group if and only if G ∼= Cn×Cn for n > 1 with gcd(n, 6) = 1. After
+abelian groups, the most natural class of finite groups to consider are nilpotent groups.
+The determination of nilpotent Beauville groups is easily reduced to the case of p-groups.
+In [1], it was shown that there are non-abelian Beauville p-groups of order pn for ev-
+ery p ≥ 5 and every n ≥ 3. The first explicit infinite family of Beauville 2-groups was
+constructed in [2]. In [19], Stix and Vdovina constructed an infinite family of Beauville
+p-groups, for every prime p, by considering quotients of ordinary triangle groups. In [9],
+Date: January 10, 2023.
+2010 Mathematics Subject Classification. Primary 20E08; Secondary 20D15, 14J29.
+Key words and phrases. Groups acting on rooted trees, finite p-groups, Beauville structures.
+The first and the second authors are supported by the Spanish Government, grant MTM2017-86802-
+P, partly with FEDER funds, and by the Basque Government, grant IT974-16. The first author is also
+supported by the National Group for Algebraic and Geometric Structures, and their Applications (GN-
+SAGA - INdAM). The third author acknowledges support from EPSRC, grant EP/T005068/1, and from
+the Lincoln Institute of Advanced Studies. She also thanks the University of the Basque Country for its
+hospitality. The first and second authors would like to thank the University of Lincoln for its hospitality
+while this research was conducted.
+1
+
+2
+E. DI DOMENICO, S¸. G¨UL, AND A. THILLAISUNDARAM
+Fern´andez-Alcober and G¨ul extended Catanese’s criterion for abelian Beauville groups to
+finite p-groups satisfying certain conditions which are much weaker than commutativity.
+They also give the first explicit infinite family of Beauville 3-groups, and they show that
+there are Beauville 3-groups of order 3n for every n ≥ 5.
+In [15], G¨ul and Uria-Albizuri showed that the quotients of periodic Grigorchuk-Gupta-
+Sidki (GGS-)groups admit Beauville structures, giving another infinite family of Beauville
+p-groups, for p an odd prime. The GGS-groups were some of the early examples of groups
+acting on rooted trees, which first arose as easily describable examples of infinite finitely
+generated periodic groups. Since then, groups acting on rooted trees have provided many
+other interesting and exotic examples, such as infinite finitely generated groups of inter-
+mediate word growth, infinite finitely generated amenable but not elementary amenable
+groups, etc; see for instance [14, 3].
+The GGS-groups are infinite groups acting faithfully on the p-adic tree. The natural
+quotients of a GGS-group G are G/ StG(n), for n ∈ N, where StG(n) is the normal subgroup
+of the elements of G that pointwise fix the vertices at the nth level of the tree. The quotient
+G/ StG(n) acts on the finite tree consisting of the first n layers of the full p-adic tree.
+We extend the result of [15] to a generalisation of the GGS-groups to the pn-adic tree,
+for any prime p and n ≥ 2, by showing that infinitely many quotients of infinite periodic
+GGS-groups acting on the pn-adic tree do admit Beauville structures.
+Briefly, a GGS-group acting on the pn-adic tree is a group G = ⟨a, b⟩, where a =
+(1 2 · · · pn) permutes the first-level vertices, whereas b fixes the first-level vertices and is
+defined recursively by b = (ae1, . . . , aepn−1, b), for e1, . . . , epn−1 ∈ Z/pnZ with not all ei
+being zero. Here (ae1, . . . , aepn−1, b) refers to the respective independent actions at the pn
+maximal subtrees. We write e = (e1, . . . , epn−1) and call it the defining vector of G. We
+refer the reader to Section 2 for background material and precise definitions. These groups
+were constructed by Vovkivsky [20]. A prominent group in this family, that was defined
+earlier in [13], is the second Grigorchuk group, which acts on the 4-adic tree with defining
+vector e = (1, 0, 1).
+For an infinite periodic GGS-group G acting on the pn-adic tree, we associate to G a
+number mG ∈ N that is related to the order of certain group elements; we refer the reader
+to Section 4 for precise details. The main result of this paper is as follows.
+Theorem 1.1. Let G be an infinite periodic GGS-group acting on the pn-adic tree for p
+a prime and n ≥ 2. Then G/ StG(k) admits a Beauville structure for k ≥ mG.
+In particular, these quotients of infinite periodic GGS-groups acting on the 2n-adic tree
+yield infinite families of Beauville 2-groups, and infinite periodic GGS-groups acting on the
+3n-adic tree give many more infinite families of Beauville 3-groups. Note that Beauville 2-
+groups and 3-groups are somewhat rare, precisely because of the small number of maximal
+subgroups. Indeed, for a time it was unclear whether such groups even existed, until the
+first examples were given in [11]. We also want to emphasise that for the GGS-groups
+acting on the p-adic tree, if one were to consider the case p = 2 there is only one group,
+which is the infinite dihedral group, hence its quotients do not admit Beauville structures,
+and for p = 3 only one out of the three isomorphism classes of such groups has quotients
+admitting Beauville structures; see [15] and [18].
+G¨ul and Uria-Albizuri also showed in [15] that for G a GGS-group acting on the p-adic
+tree, there are quotients of G admitting Beauville structures if and only if G is periodic.
+The situation is not so clear cut for GGS-groups acting on the pn-adic tree, for n ≥ 2.
+In particular, there are non-periodic GGS-groups acting on the pn-adic tree which have
+quotients admitting Beauville structures; see Remark 4.2. Futhermore, our main result
+above deals only with infinite periodic GGS-groups. For finite GGS-groups, some quotients
+
+BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS
+3
+admit Beauville structures; see Remark 4.3. However a large class E of finite GGS-groups
+have no level-stabliser quotients admitting Beauville structures. We refer the reader to
+Section 5 for the definition of E. We have the following result.
+Theorem 1.2. Let G ∈ E be a finite GGS-group acting on the pn-adic tree for p a prime
+and n ≥ 2. Then G/ StG(k) is not a Beauville group for all k ∈ N.
+To prove our theorems, we largely make use of the subgroup structure of the respective
+groups.
+Organisation. Section 2 of this paper consists of background material for groups acting on
+rooted trees and here we also recall the generalisation of the GGS-groups. In Section 3 we
+establish key properties of these groups. In Section 4 we prove Theorem 1.1 and lastly in
+Section 5 we prove Theorem 1.2.
+Acknowledgements. We thank G. A. Fern´andez-Alcober for his valuable feedback and
+helpful suggestions.
+2. Preliminaries
+All trees considered here will be rooted, meaning that there is a distinguished vertex
+called the root, with one degree less than all other vertices. For d ∈ N≥2, let T be the
+d-adic tree, meaning all vertices have d children. Using the alphabet A = {1, . . . , d}, the
+vertices uω of T are labelled bijectively by elements ω of the free monoid A∗ as follows:
+the root of T is labelled by the empty word ∅, and for each word ω ∈ A∗ and letter a ∈ A
+there is an edge connecting uω to uωa. We say that uω precedes uλ whenever ω is a prefix
+of λ. When convenient, we do not differentiate between A∗ and vertices of T.
+There is a natural length function on A∗. The words ω of length |ω| = n, representing
+vertices uω that are at distance n from the root, are the nth-level vertices and form the
+nth layer of the tree. The elements of the boundary ∂T correspond naturally to infinite
+simple rooted paths, and they are in one-to-one correspondence with the d-adic integers.
+Denote by Tu the full subtree of T that has its root at a vertex u and includes all vertices
+succeeding u. For any two vertices u = uω and v = uλ, the map uωτ �→ uλτ, induced by
+replacing the prefix ω by λ, yields an isomorphism between the subtrees Tu and Tv.
+Every automorphism of T fixes the root, and the orbits of Aut(T) on the vertices of the
+tree T are precisely its layers. For f ∈ Aut(T), the image of a vertex u under f is denoted
+by uf. The automorphism f induces a faithful action on the monoid A∗ and for a ∈ A we
+have (ωa)f = ωfa′ where a′ ∈ A is uniquely determined by ω and f. From this we have a
+permutation f(ω) of A where
+(ωa)f = ωfaf(ω),
+and hence
+(uωa)f = uωf af(ω),
+and f(ω) is called the label of f at ω. The collection of all labels of f constitutes the
+portrait of f, and there is a one-to-one correspondence between automorphisms of T and
+portraits.
+The automorphism f is rooted if f(ω) = 1 for ω ̸= ∅.
+It is directed, with
+directed path ℓ for some ℓ ∈ ∂T, if the support {ω | f(ω) ̸= 1} of its labelling is infinite
+and marks only vertices at distance 1 from the set of vertices corresponding to the path ℓ.
+Additionally, the section of f at a vertex u is defined to be the unique automorphism fu
+of T ∼= Tu given by the condition (uv)f = ufvfu for v ∈ A∗.
+2.1. Subgroups of Aut(T). For G ≤ Aut(T), the vertex stabiliser stG(u) is the subgroup
+consisting of elements in G that fix the vertex u.
+For n ∈ N, the nth-level stabiliser
+StG(n) = �
+|ω|=n stG(uω) is the subgroup consisting of automorphisms that fix all vertices
+at level n. Let T[n] be the finite subtree of T of vertices up to level n. Then StG(n) is the
+kernel of the induced action of G on T[n], and we will denote by Gn the quotient G/ StG(n).
+
+4
+E. DI DOMENICO, S¸. G¨UL, AND A. THILLAISUNDARAM
+Each g ∈ StAut(T)(n) can be described completely in terms of its restrictions to the
+subtrees rooted at vertices at level n. Indeed, the map
+ψn : StAut(T)(n) −→
+�
+|ω|=n
+Aut(Tuω) ∼= Aut(T) ×
+dn
+· · · × Aut(T)
+is a natural isomorphism mapping g ∈ StAut(T)(n) to its nth-level sections. For ease of
+notation, we write ψ = ψ1.
+For ω ∈ A∗, we further define:
+ϕω : stAut(T)(uω) −→ Aut(Tuω) ∼= Aut(T)
+g
+�−→
+gω
+A group G ≤ Aut(T) is said to be self-similar if for every g ∈ G and every vertex u, the
+section gu, of g at u, is an element of G. For a self-similar group G ≤ Aut(T), for m ≥ 2
+and i ∈ Am−1, we will write
+ψi
+m = ψ ◦ ϕi.
+That is, for g ∈ stG(ui) with ϕi(g) ∈ StG(1), we have ψi
+m(g) gives the components of ϕi(g)
+corresponding to the children of the (m − 1)th-level vertex i. In addition, we write
+φn,m : StGn(m) −→ Gn−1 ×
+dm
+· · · × Gn−1
+for the corresponding ψm when working in the quotient. Likewise, we write φn = φn,1 for
+simplicity.
+2.2. GGS-groups acting on the pn-adic tree. Let T be the pn-adic tree for a prime p
+and n ∈ N.
+Given a non-zero vector e = (e1, . . . , epn−1) ∈ (Z /pn Z)pn−1, the GGS-
+group G = Ge associated to the defining vector e is the group generated by the rooted
+automorphism a corresponding to the cycle (1 2 · · · pn) and by a directed automorphism
+b ∈ StAut(T)(1) defined recursively via
+ψ(b) = (ae1, ae2, . . . , aepn−1, b).
+Unlike the GGS-groups acting on the p-adic tree, the GGS-groups acting on the pn-adic
+tree for n ≥ 2 are not all infinite. A necessary and sufficient condition for these groups to
+be infinite was given by Vovkivsky [20]. He proved that such a group is infinite if and only
+if there exists an i ≥ 0 such that
+R0 ≤ R1 ≤ · · · ≤ Ri = Ri+1 = · · · < n
+where the sequence Rj is defined recursively as follows: R0 is the largest integer such that
+pR0 | eℓ for all ℓ ∈ {1, . . . , pn−1}; and then for j ≥ 0 and while Rj < n, the number Rj+1 is
+defined as the largest integer such that pRj+1 divides eℓ for all ℓ ∈ {pRj, 2pRj, . . . , pn−pRj}.
+Note that the order of a is pn and the order of b is pn−R0. Further, from [6, Thm. 2.1]
+we have G/G′ ∼= Cpn × Cpn−R0.
+Vovkivsky further proved in [20] that a GGS-group acting on the pn-adic tree is a
+periodic group if and only if for each k ∈ {0, . . . , n − 1},
+S[k] ≡ 0
+(mod pk+1)
+where
+(2.1)
+S[k] = epk + e2pk + · · · + epn−pk.
+We recall the prominent example of such a GGS-group, which is for the case p = n = 2:
+
+BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS
+5
+Example 2.1. Let T be the 4-adic tree. The second Grigorchuk group Γ ≤ Aut(T) is
+generated by two automorphisms a and b, where a is the rooted automorphism corre-
+sponding to the cycle (1 2 3 4), and b ∈ StΓ(1) is recursively defined by ψ(b) = (a, 1, a, b).
+The group Γ, which is periodic, was first defined in [13].
+For more information on GGS-groups acting on the pn-adic tree, see [20, 5, 6].
+3. Properties of GGS-groups acting on the pn-adic tree
+For G = Ge, a GGS-group acting on the pn-adic tree, recall that R0 is the largest integer
+such that pR0 | eℓ for all ℓ ∈ {1, . . . , pn − 1}.
+Suppose now that G is periodic. To each element aiprbjps in G, where 0 ≤ r < n,
+0 ≤ s < n − R0 and i, j ̸≡ 0 (mod p), we associate the following numbers:
+(i) If pn−s ∤ S[r], let t0, t1, . . . , tmr,s ∈ N for some 1 ≤ mr,s < n be such that
+• tk < n − s ≤ tmr,s for all k < mr,s,
+• pt0 | S[r] and pt0+1 ∤ S[r],
+• ptk | S[tk−1 + s] and ptk+1 ∤ S[tk−1 + s] for 1 ≤ k < mr,s,
+• ptmr,s | S[tmr,s−1 + s],
+where the S[λ] for λ ∈ {0, 1, . . . , n − 1} are as in (2.1).
+(ii) If pn−s | S[r], then we set mr,s = 0, and for convenience define t0 = n − s.
+If we are in case (i), note that t0 does not depend on s.
+Lemma 3.1. Let G = ⟨a, b⟩ be a periodic GGS-group acting on the pn-adic tree, let
+0 ≤ r < n, 0 ≤ s < n − R0 and i, j ̸≡ 0 (mod p). Let m := mr,s and t0, t1, . . . , tm ∈ N be
+as above. Then the order of aiprbjps in both G and Gm+3 is pt(r,s) where
+t(r, s) = (m + 2)n − (t0 + · · · + tm−1 + s(m + 1) + r + R0),
+and if pn−s | S[r], that is, when m = 0, we take t0 + · · · + tm−1 = 0.
+Proof. Clearly the order of aiprbjps is a multiple of pn−r. We have
+ψ
+�
+(aiprbjps)pn−r�
+= ψ
+�
+(bjps)a(pn−r−1)ipr
+· · · (bjps)a2ipr
+(bjps)aipr
+bjps�
+=
+�
+(ajps+R0)∗, pr−1
+. . . , (ajps+R0)∗, ajpsS[r](bjps)ajps(eipr +e2ipr +···+epr )
+,
+(ajps+R0)∗, pr−1
+. . . , (ajps+R0)∗, ajpsS[r](bjps)ajps(eipr +e2ipr +···+e2pr )
+, . . . , . . . ,
+(ajps+R0)∗, pr−1
+. . . , (ajps+R0)∗, ajpsS[r](bjps)a
+jps(eipr +e2ipr +···+e(pn−r−1)pr )
+,
+(ajps+R0)∗, pr−1
+. . . , (ajps+R0)∗, ajpsS[r]bjps�
+(3.1)
+where the ∗ denotes an unspecified exponent, and the last equality follows from the fact that
+the set {ipr, 2ipr, . . . , (pn−r − 1)ipr, pn} coincides, modulo pn, with {pr, 2pr, . . . , (pn−r −
+1)pr, pn}. So the components of ψ((aiprbjps)pn−r) in positions a multiple of pr are conju-
+gates of ajpsS[r]bjps and the other components are powers of ajps+R0 whose order is at most
+pn−s−R0.
+Suppose that pn−s | S[r]. Then using (3.1), we have
+ψ
+�
+(aiprbjps)pn−r�
+=
+�
+(ajps+R0)∗, pr−1
+. . . , (ajps+R0)∗, (bjps)ajps(eipr +e2ipr +···+epr )
+, . . . ,
+(ajps+R0)∗, pr−1
+. . . , (ajps+R0)∗, bjps�
+.
+
+6
+E. DI DOMENICO, S¸. G¨UL, AND A. THILLAISUNDARAM
+Note that the order of bjps is pn−s−R0. Thus, the order of aiprbjps in G is pt(r,s), for
+t(r, s) = 2n − (r + s + R0).
+Also since (bjps)pn−s−R0−1 ̸∈ StG(2), it follows that (aiprbjps)pt(r,s)−1 ̸∈ StG(3). Thus the
+element aiprbjps is of order pt(r,s) in G3 as well.
+It remains to settle the case pn−s ∤ S[r], which we will now assume till the end of
+the proof. Since by hypothesis pt0+s+1 ∤ jpsS[r], the order of ajpsS[r]bjps is a multiple of
+pn−(t0+s). Applying ψ we get
+ψpn
+2
+�
+(aiprbjps)pn−rpn−(t0+s)�
+= ψ
+�
+(ajpsS[r]bjps)pn−(t0+s)�
+= ψ
+�
+(bjps)a(pn−(t0+s)−1)jpsS[r] · · · (bjps)a2jpsS[r](bjps)ajpsS[r]bjps�
+.
+As before, it follows that {jpsS[r], 2jpsS[r], . . . , (pn−(t0+s) −1)jpsS[r], pn} coincides, mod-
+ulo pn, with the set {pt0+s, 2pt0+s, . . . , pn − pt0+s, pn}. This means that the b’s appear in
+all positions a multiple of pt0+s and only in these positions, and we can write each of the
+elements in these positions as
+ajpsS[t0+s](bjps)a
+jps�
+ejpsS[r]+e2jpsS[r]+···+eℓpt0+s
+�
+for certain ℓ ∈ {1, . . . , pn−(t0+s)−1}. Since we are interested in the order of these elements,
+up to conjugation the order of these elements coincides with the order of the element
+ajpsS[t0+s]bjps. In the other components of ψ((ajpsS[r]bjps)pn−(t0+s)), there is a power of
+ajps+R0 whose order is at most pn−s−R0. By hypothesis pt1+s+1 ∤ psS[t0 + s] thus the order
+of ajpsS[t0+s]bjps is at least pn−(t1+s). Proceeding in this way, after m − 1 further steps we
+find
+ψpmn
+m+1
+�
+(aiprbjps)pn−rpn−(t0+s)pn−(t1+s)···pn−(tm−1+s)�
+= ψ
+�
+(bjps)a(pn−(tm−1+s)−1)jpsS[tm−2+s] · · · (bjps)a2jpsS[tm−2+s](bjps)ajpsS[tm−2+s](bjps)
+�
+=
+�
+(ajps+R0)∗, ptm−1−1
+. . .
+, (ajps+R0)∗, ajpsS[tm−1+s](bjps)a∗, (ajps+R0)∗, ptm−1−1
+. . .
+, (ajps+R0)∗,
+ajpsS[tm−1+s](bjps)a∗, . . . , . . . , (ajps+R0)∗, ptm−1−1
+. . .
+, (ajps+R0)∗, ajpsS[tm−1+s](bjps)
+�
+=
+�
+(ajps+R0)∗, ptm−1−1
+. . .
+, (ajps+R0)∗, (bjps)a∗, . . . , (ajps+R0)∗, ptm−1−1
+. . .
+, (ajps+R0)∗, bjps�
+where the last equality holds since ptm+s | psS[tm−1 + s], thus pn | psS[tm−1 + s] by
+the definition of tm. Since by hypothesis j ̸≡ 0 (mod p) it follows that the order of the
+elements in positions a multiple of ptm−1 is pn−s−R0 and the orders of the other elements
+are at most pn−s−R0. This proves that the order of aiprbjps is pt(r,s), where
+t(r, s) = (m + 2)n − (t0 + · · · + tm−1 + s(m + 1) + r + R0).
+
+BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS
+7
+Finally, let us prove that the element (aiprbjps)pt(r,s)−1 ̸∈ StG(m + 3), which implies that
+aiprbjps is of order pt(r,s) in Gm+3. We observe that
+ψpmn
+m+1
+�
+(aiprbjps)pt(r,s)−1�
+=
+�
+(ajpn−1)∗, ptm−1−1
+. . .
+, (ajpn−1)∗, (bjpn−R0−1)a∗,
+. . . , . . . , (ajpn−1)∗, ptm−1−1
+. . .
+, (ajpn−1)∗, bjpn−R0−1�
+.
+Since the components (bjpn−R0−1)a∗ are non-trivial in G2, the result follows.
+□
+In the next three results, we identify a sort of branching structure within certain sub-
+groups of GGS-groups. For convenience, we set R−1 = 0.
+Lemma 3.2. Let G = ⟨a, b⟩ be a GGS-group acting on the pn-adic tree, for p a prime
+and n ∈ N.
+Then for j ∈ {0, 1, . . . , n − 1}, we have ϕv(stG(v)) ≥ ⟨apRj−1, b⟩ for any
+vertex v = u1 · · · uj ∈ Aj of length j, where ui ≡ 0 (mod pRi−2) for i ∈ {1, . . . , j}.
+Proof. For a vertex v of length one, the result is clear from considering the sections of
+b, ba, . . . , bapn−1, which are the generators of stG(v). As the subgroup generated by the
+sections of b is ⟨apR0, b⟩, it follows that for a vertex w of length 2,
+b, bapR0 , . . . , bapn−pR0 ∈ ϕv(stG(w)),
+where v is the prefix of w of length 1. It then follows that for vertices w = u1u2, with
+u1, u2 ∈ A such that u2 ≡ 0 (mod pR0), we have ϕw(stG(w)) ≥ ⟨apR1, b⟩. The result now
+follows recursively.
+□
+Lemma 3.3. Let G = ⟨a, b⟩ be a GGS-group acting on the pn-adic tree, for p a prime and
+n ∈ N. Let α ∈ {0, . . . , n − 1} and let pR be the highest power of p dividing eipα for all
+i ∈ {1, . . . , pn−α − 1}. Suppose that
+(i) the highest power of p dividing ejpα+1 for all j ∈ {1, . . . , pn−α−1 − 1} is strictly
+greater than pR, and
+(ii) there exists ℓ ∈ {1, . . . , pn−α − 1} such that pR+1 | eℓpα.
+Then, writing Nα = ⟨apα, b⟩ and NR = ⟨apR, b⟩, we have
+1 ×
+pα−1
+· · · × 1 × γ3(NR) × · · · · · · × 1 ×
+pα−1
+· · · × 1 × γ3(NR) = ψ
+�
+γ3(StNα(1))
+�
+.
+Note that if α = n − 1, condition (i) is vacuous.
+Proof. By assumption (i), there exists k ∈ {1, . . . , pn−α −1} with k ̸≡ 0 (mod p) such that
+ekpα = κpR for some κ ̸≡ 0 (mod p). Up to replacing b with a suitable power of itself,
+we may assume without loss of generality that κ = 1. It suffices to show, since we can
+conjugate by apα, that
+1 ×
+kpα−1
+· · · × 1 × γ3(NR) × 1 × · · · × 1 ≤ ψ
+�
+γ3(StNα(1))
+�
+.
+Case 1: Suppose e(pn−α−k)pα = 0. Then, as γ3(NR) = ⟨[apR, b, apR], [apR, b, b]⟩NR, the
+result follows from
+ψ
+�
+[b, bakpα
+, b]
+�
+= (1, kpα−1
+. . . , 1, [apR, b, apR], 1, . . . , 1)
+ψ
+�
+[b, bakpα
+, bakpα
+]
+�
+= (1, kpα−1
+. . . , 1, [apR, b, b], 1, . . . , 1)
+and using Lemma 3.2.
+
+8
+E. DI DOMENICO, S¸. G¨UL, AND A. THILLAISUNDARAM
+Case 2: Suppose e(pn−α−k)pα ̸= 0 and let χ ∈ {1, . . . , pn−R−1} be such that e(pn−α−k)pα =
+χpR.
+If p | χ, consider
+ψ
+�
+[b, bakpα
+, b]
+�
+= (1, kpα−1
+. . . , 1, [apR, b, apR], 1, . . . , 1, [b, aχpR, b]),
+ψ
+�
+[b, bakpα
+, bakpα
+]
+�
+= (1, kpα−1
+. . . , 1, [apR, b, b], 1, . . . , 1, [b, aχpR, aχpR]).
+Let us refer to those commutators in the final component as the error terms. We proceed
+to cancel those error terms by respectively multiplying with the following:
+ψ
+��
+[bakpα
+, bχ, bakpα
+]−1�a−kpα�
+= (1, . . . , 1, [aχpR, bχ, aχpR]−1, 1, kpα−1
+. . . , 1, [b, aχpR, b]−1),
+ψ
+��
+[bakpα
+, bχ, bχ]−1�a−kpα�
+= (1, . . . , 1, [aχpR, bχ, bχ]−1, 1, kpα−1
+. . . , 1, [b, aχpR, aχpR]−1).
+This introduces a new set of error terms, this time in the (pn − kpα)th component. We
+notice that the new set of error terms involve a higher p-power of a or b. Hence, after
+repeating this process a finite number of times, we will eventually have a set of error
+terms consisting of commutators with at least one component being the trivial element
+apn = bpn−R0. In other words, we have completely cancelled any error terms, showing that
+(1, kpα−1
+. . . , 1, [apR, b, apR], 1, . . . , 1) ∈ ψ
+�
+γ3(StNα(1))
+�
+,
+(1, kpα−1
+. . . , 1, [apR, b, b], 1, . . . , 1) ∈ ψ
+�
+γ3(StNα(1))
+�
+.
+Now suppose that p ∤ χ. By hypothesis, there exists ℓ ∈ {1, . . . , pn−α − 1} such that
+eℓpα = λpR with p | λ. We claim that we may choose ℓ such that e(ℓ−k)pα = ξpR with
+p ∤ ξ. Indeed, suppose on the contrary that for all choices of ℓ satisfying condition (ii), we
+have pR+1 | e(ℓ−k)pα. Since k ̸≡ 0 (mod p), repeatedly replacing ℓ with ℓ − k, this means
+that ejpα ≡ 0 (mod pR+1) for all j ∈ {1, . . . , pn−α − 1}, a contradiction. Hence the claim.
+For convenience, we write e(k−ℓ)pα = ζpR, and let ν ∈ {1, . . . , pn−R} be such that νξ ≡ χ
+(mod pn−R). Then we have
+ψ
+�
+[b, (bν)a(k−ℓ)pα
+, b]−1�
+= (1, (k−ℓ)pα−1
+. . .
+, 1, [aζpR, bν, aζpR]−1, 1, . . . , 1, [b, aχpR, b]−1),
+ψ
+�
+[b, bakpα
+, (bν)a(k−ℓ)pα
+]−1�
+= (1, kpα−1
+. . . , 1, [apR, b, aνλpR]−1, 1, . . . , 1, [b, aχpR, aχpR]−1).
+If p | ζ, we proceed to cancel the error terms as in the above argument.
+If p ∤ ζ, let
+θ ∈ {1, . . . , pn−R} be such that θχ ≡ ζ (mod pn−R). Then, since p | λ, we can use the
+following element to proceed:
+ψ
+�
+[(bθ)akpα
+, bν, (bθν)a(k−ℓ)pα
+]
+�
+= (1, kpα−1
+. . . , 1, [bθ, aνpR, aθνλpR], 1, . . . , 1, [aζpR, bν, aζpR]). □
+Lemma 3.4. Let G = ⟨a, b⟩ be a GGS-group acting on the pn-adic tree, for p an odd prime
+and n ∈ N. Let R ≤ n − 1 be such that ekpR ≡ 0 (mod pR) for all k ∈ {1, . . . , pn−R − 1}
+and suppose that ekpR ̸≡ 0 (mod pR+1) for all k ∈ {1, . . . , pn−R − 1}. Suppose further that
+S[R] ≡ 0 (mod pR+1). Then, writing N = ⟨apR, b⟩, we have
+1 ×
+pR−1
+· · · × 1 × γ3(N) × · · · · · · × 1 ×
+pR−1
+· · · × 1 × γ3(N) = ψ
+�
+γ3(StN(1))
+�
+.
+Proof. Let ekpR = ikpR for k, ik ∈ {1, . . . , pn−R − 1} with ik ̸≡ 0 (mod p). Further, by
+replacing b with an appropriate power, we may assume that i1 = 1. As reasoned in the
+previous proof, it suffices to show that
+1 ×
+pR−1
+· · · × 1 × γ3(N) × 1 × · · · × 1 ≤ ψ
+�
+γ3(StN(1))
+�
+.
+
+BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS
+9
+We proceed as in [10, Lem. 3.2] by considering the following two cases:
+(a) There exists an ℓ ∈ {2, . . . , pn−R − 2} such that i2
+ℓ − iℓ−1iℓ+1 ̸≡ 0 (mod p).
+(b) For all ℓ ∈ {2, . . . , pn−R − 2}, we have i2
+ℓ − iℓ−1iℓ+1 ≡ 0 (mod p).
+Note that if p = 3 and R = n − 1, then Case (b) trivially holds.
+Suppose we are in Case (a). Let us define gℓ as follows:
+gℓ =
+�
+(biℓ)apR
+b−iℓ−1�a−ℓpR
+Writing µ = i2
+ℓ − iℓ−1iℓ+1, we have
+ψ(gℓ) = (∗, pR−1
+. . . , ∗, aµpR, ∗, . . . , ∗, 1)
+where ∗ represents unspecified elements. By assumption, we have µ ̸≡ 0 (mod p). Hence
+there is a power g of gℓ satisfying
+ψ(g) = (∗, pR−1
+. . . , ∗, apR, ∗, . . . , ∗, 1).
+Furthermore, for simplicity let λ := ipn−R−1 ̸≡ 0 (mod p). Then
+ψ
+�
+bapR
+(ba−pR
+)−λ�
+= (∗, pR−1
+. . . , ∗, ba−λe2pR , ∗, . . . , ∗, 1),
+and multiplying by an appropriate power of g we obtain an element h ∈ StN(1) with
+ψ(h) = (∗, pR−1
+. . . , ∗, b, ∗, . . . , ∗, 1).
+Then the result follows from:
+ψ
+�
+[b, bapR
+, g]
+�
+= (1, pR−1
+. . . , 1, [apR, b, apR], 1, . . . , 1)
+ψ
+�
+[b, bapR
+, h]
+�
+= (1, pR−1
+. . . , 1, [apR, b, b], 1, . . . , 1)
+We now suppose that we are in Case (b). It then follows, for j ∈ {2, . . . , pn−R − 1},
+that ij = λj−1 for some λ̸≡ 0 (mod p). However, as S[R] ≡ 0 (mod pR+1), we have λ ̸≡ 1
+(mod p). Further we have λpn−R−1pR ≡ pR (mod pR+1).
+Note that
+ψ
+�
+b(bapR
+)−λ�
+=
+(∗, pR−1
+. . . , ∗, apRb−λ, ∗, pR−1
+. . . , ∗, 1, . . . . . . , ∗, pR−1
+. . . , ∗, 1, ∗, pR−1
+. . . , ∗, ba−pR+νpR+1)
+for some ν, and ∗ represents some power of a. Let χ be such that λχ ≡ 1 (mod pn−R0);
+recall that pn−R0 is the order of b. Then
+ψ
+�
+bχ�
+b−χλ�apR�
+= ψ
+�
+bχ�
+b−1�apR�
+=
+(∗, pR−1
+. . . , ∗, aχpRb−1, ∗, pR−1
+. . . , ∗, 1, . . . . . . , ∗, pR−1
+. . . , ∗, 1, ∗, pR−1
+. . . , ∗, bχ∗).
+Thus, since χ ̸≡ 1 (mod p) and hence
+N = ⟨aχpRb−1, ba−pR+νpR+1⟩ = ⟨a(χ−1)pR+νpR+1, ba−pR+νpR+1⟩,
+the result follows from the observations below:
+ψ
+��
+b(1−νp), bapR
+, bapR
+(ba2pR
+)−λ��
+= (1, pR−1
+. . . , 1, [apR−νpR+1, b, ba−pR+νpR+1], 1, . . . , 1),
+ψ
+��
+(ba2pR
+)λ, bapR
+, bχ�
+b−χλ�apR��
+= (1, pR−1
+. . . , 1, [apR−νpR+1, b, aχpRb−1], 1, . . . , 1).
+□
+Next, we record some properties of certain finite GGS-groups acting on the 2n-adic tree,
+which will be needed in Section 5.
+
+10
+E. DI DOMENICO, S¸. G¨UL, AND A. THILLAISUNDARAM
+Lemma 3.5. Let G be a periodic GGS-group acting on the 2n-adic tree for n ≥ 2, and
+assume that R0 = n − 1. Then StG(4) is trivial. Furthermore StG(3) is non-trivial if and
+only if ei = e2n−1+i for all i ∈ {1, . . . , 2n−1 − 1}.
+Proof. Note that as R0 = n − 1, we have that 2n−1 divides each ei. Thus the ei’s are 0
+or 2n−1. Since G is periodic, it follows that e2n−1 = 0 and hence
+�
+b, ba2n−1� ∼= C2 × C2.
+Let g be an element in StG(4). Certainly ϕω(g) ∈ StG(1) for all ω ∈ A. Since
+ψ(StG(1)) ∩ StG(1) × · · · × StG(1) ⊆
+�
+b, ba2n−1�
+×
+2n
+· · · ×
+�
+b, ba2n−1 �
+,
+it follows that
+ϕω(g) ∈
+�
+1, b, ba2n−1
+, bba2n−1�
+.
+As
+�
+b, ba2n−1�
+∩ StG(3) = 1, the first claim follows.
+For the second statement, let g be a non-trivial element in StG(3), in particular ϕω(g) ∈
+StG(2) for all ω ∈ A. From the previous paragraph, it follows that ϕω(g) ∈
+�
+1, bba2n−1 �
+.
+Thus there exists ω ∈ A such that ϕω(g) = bba2n−1
+. We observe that
+ψ
+�
+bba2n−1�
+= (ae1+e2n−1+1, . . . , ae2n−1−1+e2n−1, b, ae1+e2n−1+1, . . . , ae2n−1−1+e2n−1, b),
+so ϕω(g) ∈ StG(2) if and only if ei = e2n−1+i for all i ∈ {1, . . . , 2n−1 − 1}, as required.
+□
+Lemma 3.6. Let G be a periodic GGS-group acting on the 2n-adic tree for n ≥ 2.
+Suppose R0 = n − 1 and S[0] ≡ 0 (mod pn). Then γ2n+1(G) ≤ StG(1)′. Furthermore
+[b, a, 2n−1
+. . . , a]2 = 1 and [b, a, 2n−1+i
+. . . , a]2 ∈ γ2n+i+2(G) for all i ∈ N.
+Proof. As StG(1)/ StG(1)′ is elementary abelian, we have (G′)2 ≤ StG(1)′.
+Therefore,
+from [17, Prop. 1.1.32(ii)], we have
+1 = [a2n, b] ≡ [b, a, 2n
+. . ., a]
+(mod StG(1)′),
+which gives γ2n+1(G) ≤ StG(1)′, as required.
+Also from [17, Prop. 1.1.32(ii)], we obtain
+[a2n−1, b] ≡ [a, b, a, 2n−1−1
+. . . , a]
+(mod StG(1)′)
+and so [b, a, 2n−1
+. . . , a]2 = 1, as StG(1)′ is elementary abelian and [StG(1), StG(1)′] = 1.
+For the last statement, we proceed by induction on i. For i = 1, the statement follows
+from
+1 =
+�
+[b, a, 2n−1
+. . . , a]2, a
+�
+= [b, a, 2n−1+1
+. . . , a]2�
+b, a, 2n−1+1
+. . . , a, [b, a, 2n−1
+. . . , a]
+�
+,
+and similarly for the inductive step, as
+�
+[b, a, 2n−1+i
+. . . , a]2, a
+�
+= [b, a, 2n−1+i+1
+. . .
+, a]2�
+b, a, 2n−1+i+1
+. . .
+, a, [b, a, 2n−1+i
+. . . , a]
+�
+.
+□
+Lemma 3.7. Let G be a GGS-group satisfying the conditions of Lemma 3.6.
+Then
+γi(G)/γi+1(G) is isomorphic to a subgroup of C2 × C2, for i ≥ 2.
+Furthermore |γi(G) : γi+1(G)| ≤ 2 for i > 2n.
+Proof. We notice that
+ψ(StG(1)′) ⊆ S :=
+��
+[b, a2n−1]ǫ1, . . . , [b, a2n−1]ǫ2n�
+| ǫ1, . . . , ǫ2n ∈ {0, 1}
+�
+.
+Since |S| = 22n, it follows that log2 | StG(1)′| ≤ 2n.
+First we consider the case when the defining vector e is non-symmetric, that is, there
+exists j ∈ {1, . . . , 2n−1 − 1} with ej ̸= e2n−j. Then, there exists some k ∈ {1, . . . , 2n − 1}
+such that
+ψ([b, bak]) =
+�
+1, . . . , 1, [b, a2n−1]
+�
+∈ ψ(StG(1)′).
+
+BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS
+11
+Hence for e non-symmetric, we have log2 | StG(1)′| = 2n, and
+StG(1)′ = ⟨[b, bak], [b, bak]a, [b, bak]a2, . . . , [b, bak]a2n−1⟩
+= ⟨[b, bak], [b, bak, a], [b, bak, a2], . . . , [b, bak, a2n−1]⟩
+= ⟨[b, bak], [b, bak, a], [b, bak, a, a], . . . , [b, bak, a, 2n−1
+. . . , a]⟩.
+Indeed, the last equality follows from the more general fact that
+⟨h, [h, a], [h, a2], . . . , [h, aℓ]⟩ = ⟨h, [h, a], [h, a, a], . . . , [h, a,
+ℓ. . ., a]⟩
+for h ∈ StG(1)′ and 0 ≤ ℓ ≤ 2n − 1. This is proved by induction on ℓ, using the identity
+[h, aℓ+1] = [h, a][h, aℓ][h, aℓ, a].
+From this, we deduce that for each i ≥ 2, either StG(1)′ ∩ γi(G) = StG(1)′ ∩ γi+1(G) or
+(StG(1)′ ∩ γi(G))/(StG(1)′ ∩ γi+1(G)) ∼= C2. As
+γi(G) = ⟨[b, a, i−1
+. . ., a]⟩(StG(1)′ ∩ γi(G))γi+1(G),
+the result follows, with the final statement coming from the fact that [b, a, 2n
+. . ., a] ∈ StG(1)′.
+Now we suppose that e is symmetric, so ej = e2n−j for all j ∈ {1, . . . , 2n−1 − 1}. Then
+there exists a minimal k ∈ {1, . . . , 2n−1 − 2} such that
+�
+1, . . . , 1, [b, a2n−1], 1, k−1
+. . ., 1, [b, a2n−1], 1, . . . , 1
+�
+∈ ψ(StG(1)′).
+Let g ∈ StG(1)′ be such an element, chosen such that g ∈ γℓ(G) with ℓ minimal. We
+observe that k = 2λ for some λ ∈ {0, 1, . . . , n − 2}, else k will not be minimal. Also,
+�
+1, . . . , 1, [b, a2n−1]
+�
+/∈ ψ(StG(1)′).
+Looking at the 2n × 2n circulant matrix C defined by (1, 0, 2λ−1
+. . . , 0, 1, 0, . . . , 0) ∈ (F2)2n,
+a direct generalisation of [10, Lem. 2.7(i)] yields that the rank of C is 2n − 2λ, as the
+polynomial associated to the circulant matrix is 1 + x2λ = (1 + x)2λ. Therefore we obtain
+that log2 | StG(1)′| = 2n − 2λ, and
+StG(1)′ = ⟨g, ga, ga2, . . . , ga2n−2λ−1⟩
+= ⟨g, [g, a], [g, a, a], . . . , [g, a, 2n−2λ−1
+. . .
+, a]⟩.
+The result now follows as in the non-symmetric case.
+□
+Lemma 3.8. Let G be a GGS-group satisfying the conditions of Lemma 3.6.
+Then
+⟨[b, a, 2n−1
+. . . , a, b]⟩G = {[(ab)2n, g] | g ∈ G}.
+Proof. Setting h = (ab)2n, we know from [17, Prop. 1.1.32(i)] that
+h = [b, a, 2n−1−1
+. . . , a]2[b, a, 2n−1
+. . . , a]c
+for some c ∈ γ2n(G) ∩ StG(1)′. Using Lemma 3.6, we observe that for all 1 ≤ i ≤ 2n − 1,
+we have [h, ai] ∈ γ2n+1(G) ≤ StG(1)′. Furthermore, one can check that
+[h, a] = [h, b] = [b, a, 2n−1
+. . . , a, b].
+Let j > 2n be maximal such that [b, a, 2n−1
+. . . , a, b] ∈ γj(G). Thus, since log2 | StG(1)′| ≤ 2n
+as seen in the proof of Lemma 3.7, we have
+γj(G) = ⟨[h, b], [h, b, a], . . . , [h, b, a, 2n−1
+. . . , a]⟩.
+Our goal is now to show that
+{[h, g] | g ∈ G} = ⟨[h, b], [h, b, a], . . . , [h, b, a, 2n−1
+. . . , a]⟩.
+
+12
+E. DI DOMENICO, S¸. G¨UL, AND A. THILLAISUNDARAM
+To this purpose, as seen in the previous proof it suffices to prove
+(3.2)
+{[h, g] | g ∈ G} = ⟨[h, b], [h, b]a, . . . , [h, b]a2n−1⟩.
+For any 1 ≤ i ≤ 2n − 1, we have that
+(3.3)
+[h, b]ai = [hai, bai] = [h[h, ai], bai] = [h, bai],
+where the last equality follows from the fact that [h, ai] ∈ γ2n+1(G) ≤ StG(1)′. Next we
+note that
+(3.4)
+[h, ak] = [h, b][h, b]a · · · [h, b]ak−1
+for all 1 ≤ k ≤ 2n − 1. Now, every element g ∈ G is of the form g = aλbaℓ1 · · · baℓd for some
+d ≥ 0 and λ, ℓ1, . . . , ℓd ∈ {0, 1, . . . , 2n − 1}. Then taking into account (3.3) and (3.4), we
+get
+[h, g] = [h, b]aℓd · · · [h, b]aℓ1[h, aλ],
+from which we deduce that (3.2) holds.
+□
+4. Beauville structures for quotients of infinite GGS-groups
+acting on the pn-adic tree
+We are now ready to prove that certain quotients of G are Beauville groups, for G
+an infinite periodic GGS-group acting on the pn-adic tree where p is a prime and n ≥ 2.
+Recall that for a 2-generator finite group H, the sets {x1, x2} and {y1, y2} form a Beauville
+structure for H, if and only if
+(4.1)
+⟨x⟩ ∩ ⟨yg⟩ = 1
+for all x ∈ X = {x1, x2, x1x2}, y ∈ Y = {y1, y2, y1y2} and g ∈ H. We also remark that we
+will only consider G/StG(k) for k large enough such that the order of each element in X
+(or in Y ) is the same in G and in G/StG(k). The reason for this will become apparent
+when we prove Theorem 1.1 later in this section.
+Recall mr,s from the beginning of Section 3. For G an infinite GGS-group, recall that
+there exists some 1 ≤ q ≤ n − 1 such that Rq = Rq+1 < n. We write pd for the or-
+der of [apRq, b, apRq ]b in G, and we denote by λ′ the minimal level where all sections of
+([apRq , b, apRq ]b)pd−1 that involve b are of the form (b∗)a∗ for some unspecified exponents ∗.
+For the rest of this section, we set m = max{m0,0 + λ′, m1,0}.
+Theorem 4.1. Let G be an infinite periodic GGS-group acting on the pn-adic tree, for p
+a prime and n ≥ 2. Then the quotient G/StG(m + 3) is a Beauville group.
+Proof. Case 1: Suppose that ekpn−1 ̸= 0 for some k ∈ {1, . . . , p − 1}. Observe that by
+assumption the prime p must be odd. Let 1 ≤ q ≤ n−1 be such that R := Rq = Rq+1 < n.
+Suppose first that eℓpn−1 = 0 for some ℓ ∈ {1, . . . , p−1}. Then writing µ = m0,0 and using
+Lemmata 3.3 and 3.2 repeatedly we obtain
+c = ψ−1
+µ+1
+�
+(1, p(µ+1)n−1
+. . .
+, 1, [apR, b, apR])
+�
+∈ γ3(G).
+By abuse of notation, we still write a and b for their images in Gm+3. We set
+X = {ap−1, ab, apb}
+with
+Y = {ac, b, acb}.
+Assume first that x = ap−1, and consider y = ac. We write pτ for the order of ac
+in Gm+3, which is strictly greater than pn. In order to prove our claim for this choice of x
+and y, it suffices to show that
+⟨apn−1⟩ ̸= ⟨(ac)pτ−1⟩g,
+
+BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS
+13
+for all g ∈ Gm+3. Note that the element (ac)pτ−1 is in StGm+3(1) and so is any conjugate
+of (ac)pτ−1. However apn−1 ̸∈ StGm+3(1), hence (4.1) holds here. Similarly for y ∈ {b, acb}.
+Suppose next that x = ab, and consider y = b. By Lemma 3.1, the element ab has
+order pt(0,0). It suffices to consider the intersection of ⟨(ab)pt(0,0)−1⟩ and ⟨bpn−R0−1⟩g. We
+observe that ψ((bpn−R0−1)g) has exactly one component consisting of purely a non-rooted
+automorphism in StG(1), whereas ψ((ab)pt(0,0)−1) has pn such components. Therefore (4.1)
+holds. For y = ac, we observe that ψµ+1((ac)pτ−1) has pn components consisting of non-
+rooted automorphisms in γ3(G), whereas for ψµ+1((ab)pt(0,0)−1) all components consisting
+of non-rooted automorphisms are in StG(1)\γ3(G). Hence the result holds for this pair.
+Let x = apb. Since (apb)pn−1 has pn−1 non-rooted automorphisms at the first level,
+whereas both (ac)pn and (acb)pn have pn such automorphisms and bpn−R0−1 has one such
+automorphism, we have that (4.1) holds here.
+Hence, it remains to establish (4.1) for x = ab with y = acb. Recall from the proof of
+Lemma 3.1 that
+ψpµn
+µ+1
+�
+(ab)pt(0,0)−n+R0
+�
+=
+�
+(apR0)∗, ptµ−1−1
+. . .
+, (apR0)∗, ba∗,
+. . . , . . . , (apR0)∗, ptµ−1−1
+. . .
+, (apR0)∗, b
+�
+.
+However,
+ψpµn
+µ+1
+�
+(acb)pt(0,0)−n+R0
+�
+=
+�
+(apR0)∗, ptµ−1−1
+. . .
+, (apR0)∗, ([apR, b, apR]b)a∗,
+. . . , . . . , (apR0)∗, ptµ−1−1
+. . .
+, (apR0)∗, [apR, b, apR]b
+�
+.
+As ψ([apR, b, apR]b) has more than one section with a non-rooted automorphism, whereas
+ψ(b) has only one, the result follows. Indeed,
+ψ([apR, b, apR]b) = (a∗, pR−1
+. . . , a∗, (ba−epn−pRb)a
+epR , a∗, pR−1
+. . . , a∗, (aepRb−1aepR)a
+e2pR ,
+a∗, . . . , a∗, b−1a2epn−pR−epn−2pRb),
+and so if epR = epn−pR = 0 then
+ψ([apR, b, apR]b) = (a∗, pR−1
+. . . , a∗, b2, a∗, pR−1
+. . . , a∗, (b−1)
+e2pR , a∗, . . . , a∗, (a−epn−2pR)b)
+and the result is clear.
+If either epR or epn−pR is non-zero, we see from the proof of
+Lemma 3.1 that, writing pχ for the order of acb in G, the minimal level λ of the tree, for
+which all λ-level sections of (acb)pχ−1 that involve b are of the form (b∗)a∗ for some unspec-
+ified exponents ∗, is strictly greater than µ+1. In particular, all these cases imply that the
+number of sections of (acb)pχ−1 at level λ consisting of purely non-rooted automorphisms is
+strictly greater than pn−tµ−1, which is the corresponding number for (ab)pt(0,0)−1 at level λ.
+Suppose now that eℓpn−1 ̸= 0 for all ℓ ∈ {1, . . . , p−1}. The result follows similarly using
+Lemma 3.4.
+Case 2: Suppose that ekpn−1 = 0 for all k ∈ {1, . . . , p − 1}. Since G is infinite, for some
+1 ≤ q ≤ n − 1 we have R := Rq = Rq+1 < n − 1. Then we proceed as in the previous
+proof, however noting that for p = 2, we replace c with
+ψ−1
+µ+1
+�
+(1, 2(µ+1)n−1
+. . .
+, 1, [a2R, b, a2R+1])
+�
+,
+
+14
+E. DI DOMENICO, S¸. G¨UL, AND A. THILLAISUNDARAM
+since then, recalling that R < n − 1,
+ψ([a2R, b, a2R+1]b) =
+�
+a∗, 2R−1
+. . . , a∗, (ba−e2n−2R+1+e2n−2R)ae2R , a∗, 2R−1
+. . . , a∗, (ae2R−e2n−2Rb)ae2R+1 ,
+a∗, 2R−1
+. . . , a∗, (ae2R+1b−1ae2R)ae3·2R , a∗, . . . , . . . , a∗, (ae2n−2R−e2n−3·2R+e2n−2R+1)b�
+.
+□
+Proof of Theorem 1.1. Let G be an infinite periodic GGS-group acting on the pn-adic tree
+with p a prime and n ≥ 2. Let t0 be defined according to r = s = 0, and set mG = m+3. By
+Theorem 4.1, the quotient G/StG(mG) is a Beauville group with triples X = {ap−1, ab, apb}
+and Y = {ac, b, acb}, for c defined as in Theorem 4.1.
+As the orders of each of the elements in X are the same in G/ StG(k) and G/ StG(mG)
+for all k ≥ mG, it follows from [12, Lem. 4.2] that G/ StG(k) is a Beauville group for every
+k ≥ mG.
+□
+Remark 4.2. In the above theorems, we only require that S[i] ≡ 0 (mod pi+1) for i ∈ I,
+for some subset I ⊆ {0, 1, . . . , n − 1}. Therefore there are non-periodic GGS-groups acting
+on the pn-adic tree, for n ≥ 2, with quotients being Beauville groups. For instance, if
+• t0 > 2 for the pairs r = s = 0 and r = 1, s = 0, and
+• R > 2, and
+• S[2] ̸≡ 0 (mod p3), but S[i] ≡ 0 (mod pi+1) for all other i,
+the proofs above still hold.
+Remark 4.3. In some cases, the proofs above can be extended to finite GGS-groups G.
+Since G is finite, there exists d ∈ {0, 1, . . . , n − 1} such that Rd+1 = n. Writing µ = m0,0,
+we observe that d ≥ µ − 1. If d > µ, then by repeated application of Lemmata 3.3 and
+3.2, we have an element
+c := ψ−1
+µ+1
+�
+(1, p(µ+1)n−1
+. . .
+, 1, [apRµ , b, apRµ])
+�
+∈ γ3(G).
+Setting X = {ap−1, ab, apb} and Y = {ac, b, acb}, we are done as in Case 1 of the proof of
+Theorem 4.1.
+If d = µ, then using the same c and the sets X and Y as before, since eℓpRµ = 0 for
+any ℓ, we observe that
+ψ([apRµ, b, apRµ]b) = (a∗, pRµ−1
+. . . , a∗, b2, a∗, pRµ−1
+. . . , a∗, b−1, a∗, . . . , a∗, 1),
+and hence if p is odd, the same argument follows. For p = 2, the same argument can
+clearly be used upon replacing c with
+ψ−1
+µ+1
+�
+(1, 2(µ+1)n−1
+. . .
+, 1, [a2Rµ , b, a2Rµ+1])
+�
+,
+provided Rµ < n − 1, since then
+ψ([a2Rµ , b, a2Rµ+1]b) = (a∗, 2Rµ−1
+. . . , a∗, b, a∗, 2Rµ−1
+. . . , a∗, b, a∗, 2Rµ−1
+. . . , a∗, b−1, a∗, . . . , a∗, 1).
+However, for each of these finite groups G, one would only obtain finitely many (Beauville)
+quotients G/StG(k), for k ≥ mG.
+In the next section, we identify a class of finite GGS-groups where no quotient by a level
+stabiliser is Beauville.
+5. Finite GGS-groups acting on the pn-adic tree
+We define the subclass E of finite GGS-groups to consist of:
+(i) the GGS-groups acting on the pn-adic tree, for p any prime and n ≥ 2, with
+t0 = R0, . . . , tm0,0−1 = Rm0,0−1 and Rm0,0 = n; and
+
+BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS
+15
+(ii) the GGS-groups acting on the 2n-adic tree with R0 = n − 1, S[0] ≡ 0 (mod pn),
+and e2n−1 = 0.
+We show here that quotients of the GGS-groups G ∈ E by level stabilisers do not yield
+Beauville groups. We first consider the subfamily (i) of groups in E.
+Theorem 5.1. Let G ∈ E be a GGS-group acting on the pn-adic tree for n ≥ 2, and
+assume that t0 = R0, . . . , tm0,0−1 = Rm0,0−1 and Rm0,0 = n, where t0, t1, . . . , tm0,0−1 are
+defined according to r = s = 0. Then the quotient G/StG(k) is not a Beauville group for
+all k ∈ N.
+Proof. The result is clear for k = 1 since G/ StG(1) is cyclic, so we assume k ≥ 2. Let
+{x1, x2} and {y1, y2} be two systems of generators for G. At least one of x1, x2, x1x2, call
+it z1, must be in the coset aibjG′ for i, j ̸≡ 0 (mod p). Likewise for y1, y2, y1y2, and call
+it z2. By Lemma 3.1 and its proof, the order of ab is pt(0,0) if k ≥ m0,0 + 3, and p2n−R0 if
+k = 2. For 3 ≤ k ≤ m0,0 + 1, the order of ab is at least
+pnpn−R0pn−R1 · · · pn−Rk−2,
+and for k = m0,0 + 2, the order of ab is at least
+pnpn−R0pn−R1 · · · pn−Rm0,0−1.
+Suppose first that k = 2. For i, j ̸≡ 0 (mod p) and w ∈ G′
+2, as
+φ2((aibjw)pn) = (ajS[0], . . . , ajS[0])
+it follows that ⟨zp2n−R0−1
+1
+⟩ = ⟨zp2n−R0−1
+2
+⟩, which gives the result.
+Now let k ≥ m0,0 + 3. Here, writing t = t(0, 0), we claim that
+⟨(ab)pt−1⟩ = ⟨(aibjw)pt−1⟩ ∼= Cp
+for all i, j ̸≡ 0 (mod p) and w ∈ G′.
+To prove the claim, we first note, writing ψ(w) = (w1, . . . , wpn), that the product of all
+the components w1, . . . , wpn, irrespective of the order, has zero total exponent in a, and
+in b, and if written as
+(bβ1)aα1 · · · (bβd)aαd
+for some d ≥ 2 with β1, . . . , βd ∈ Z/pn−R0Z and α1, . . . , αd ∈ Z/pnZ, then we have αi ≡ 0
+(mod pR0) for all i ∈ {1, . . . , d}.
+Hence, each component of ψ
+�
+(aibjw)pn�
+is of the form
+ajS[0]bjv0,
+for some v0 ∈ ⟨apR0, b⟩′. We note that, for ω ∈ A with ω ̸≡ 0 (mod pR0),
+ϕω
+�
+(ajS[0]bjv0)pn−R0
+�
+= ϕω
+�
+(aS[0]b)jpn−R0
+�
+= ajγω
+where
+γω =
+pn−R0−1
+�
+ℓ=0
+eω+ℓpR0,
+and for ω ∈ A with ω ≡ 0 (mod pR0),
+ϕω
+��
+(ajS[0]bjv0)pn−R0�afpR0 �
+= ajS[R0]bjv1
+for v1 ∈ ⟨apR1, b⟩′ and f ∈ {0, 1, . . . , pn−R0 − 1}.
+
+16
+E. DI DOMENICO, S¸. G¨UL, AND A. THILLAISUNDARAM
+Hence, continuing in this manner, we see that it suffices to compare
+(ajS[Rµ−2]bjvµ−1)pn−Rµ−1
+with
+(aS[Rµ−2]b)pn−Rµ−1,
+where µ = m0,0 and vµ−1 ∈ ⟨apRµ−1, b⟩′. Recalling also that Rµ = n, equivalently the
+components of ψ(b) in positions pRµ−1, 2pRµ−1, . . . , pn − pRµ−1 are trivial, we have
+ψ
+�
+(ajS[Rµ−2]bjvµ−1)pn−Rµ−1�
+= (ajδ1, . . . , a
+jδ
+pRµ−1 −1, bj, . . . , ajδ1, . . . , a
+jδ
+pRµ−1 −1, bj)
+where, for q ∈ {1, . . . , pRµ−1 − 1},
+δq =
+pn−Rµ−1−1
+�
+ℓ=0
+eq+ℓpRµ−1.
+As
+ψ
+�
+(aS[Rµ−2]b)pn−Rµ−1�
+= (aδ1, . . . , a
+δ
+pRµ−1 −1, b, . . . , aδ1, . . . , a
+δ
+pRµ−1 −1, b),
+and, from the above computations, we deduce that all other non-trivial labels in the
+portraits of (ab)jpt−1 and (aibjw)pt−1 are equal powers of a. Hence, the claim follows.
+Thus, we have ⟨zpt−1
+1
+⟩ = ⟨zpt−1
+2
+⟩.
+For 3 ≤ k ≤ µ + 2, writing pˆt for the order of ab modulo StG(k), the analogous result
+⟨zpˆt−1
+1
+⟩ = ⟨zpˆt−1
+2
+⟩ follows similarly from the component-wise description of
+(ajS[0]bjv0)pn−R0, (ajS[R0]bjv1)pn−R1, . . . , (ajS[Rk−4]bjvk−3)pn−Rk−3,
+where we recall that R−1 = 0.
+□
+Theorem 5.2. Let G be a periodic GGS-group acting on the 2n-adic tree for n ≥ 2.
+Suppose that R0 = n − 1 and S[0] ≡ 0 (mod pn). Then, the quotient G/StG(k) is not a
+Beauville group for all k ∈ N.
+Proof. From Lemma 3.5, we only need to consider k ≤ 4. For k = 1 the result is clear
+since Beauville groups are not cyclic. For the remaining values of k, let X = {x1, x2, x1x2}
+and Y = {y1, y2, y1y2} be two systems of generators for Gk.
+Assume now k = 2. Then there exists some z1 ∈ X with z1 ≡ ai (mod G′
+2) for some
+odd i, and there exists z2 ∈ Y with z2 ≡ aj (mod G′
+2) for some odd j. Noting that the
+order of both z1 and z2 is 2n, we now prove that (aiw)2n−1 is conjugate to (ajv)2n−1, for
+w, v ∈ G′
+2.
+We observe that StG2(1), and hence G′
+2, is elementary abelian. Furthermore for each
+odd i ∈ {1, . . . , 2n − 1} we have
+(5.1)
+G′
+2 ≤ ⟨[ai, b], [ai, b, ai], . . . , [ai, b, ai, 2n−2
+. . . , ai]⟩;
+compare Lemma 3.6 and the proof of Lemma 3.7. It then follows from [17, Prop. 1.1.32(i)]
+that
+(aiw)2n−1 = ai2n−1[w, ai, 2n−1−1
+. . . , ai].
+From (5.1) there exist j1, . . . , j2n−1 ∈ {0, 1} such that w can be expressed as:
+w = [ai, b]j1[ai, b, ai]j2 · · · [ai, b, ai, 2n−2
+. . . , ai]j2n−1
+Now for h ∈ StG2(1), bearing in mind that StG2(1) is elementary abelian, we obtain
+from [17, Prop. 1.1.32(ii)] that [ai2n−1, h] = [ai, h, ai, 2n−1−1
+. . . , ai] in G2.
+Thus, setting h = bj1[ai, b]j2 · · · [ai, b, ai, 2n−3
+. . . , ai]j2n−1 we deduce that
+[ai2n−1, h] = [w, ai, 2n−1−1
+. . . , ai].
+
+BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS
+17
+Indeed,
+[ai2n−1, h]
+= [ai, bj1, ai, 2n−1−1
+. . . , ai][ai, [ai, b]j2, ai, 2n−1−1
+. . . , ai] · · · [ai, [ai, b, ai, 2n−3
+. . . , ai]j2n−1, ai, 2n−1−1
+. . . , ai]
+= [[ai, b]j1, ai, 2n−1−1
+. . . , ai][[ai, b, ai]j2, ai, 2n−1−1
+. . . , ai] · · · [[ai, b, ai, 2n−2
+. . . , ai]j2n−1, ai, 2n−1−1
+. . . , ai]
+= [w, ai, 2n−1−1
+. . . , ai]
+where the second equality holds since StG2(1)′ is trivial and [a, x] = [x, a] for all x ∈
+StG2(1).
+Thus we have
+(aiw)2n−1 = ai2n−1[ai2n−1, h] = (ai2n−1)h = (a2n−1)h
+where the last equality holds since a has order 2n and i = 1 + 2d for some d ∈ N. This
+proves that for all i ∈ {1, . . . , 2n − 1} and w ∈ G′
+2 the element (aiw)2n−1 is conjugate
+to a2n−1. Hence z1 and z2 are conjugate and G2 is not a Beauville group.
+Now suppose StG(3) ̸= 1. Then there exists some z1 ∈ X with z1 ≡ aib (mod G′
+3) for
+some odd i, and there exists z2 ∈ Y with z2 ≡ ajb (mod G′
+3) for some odd j. Observe that
+for w ∈ G′
+3, we have
+φ3
+�
+(aibw)2n�
+=
+�
+baλ1·2n−1
+, . . . , baλ2n ·2n−1�
+where λ1, . . . , λ2n ∈ {0, 1}. Since by Lemma 3.5 the elements b and ba2n−1
+coincide modulo
+StG(2), we further have
+φ3((aibw)2n) = (b, . . . , b).
+This proves that z 2n
+1
+and z 2n
+2
+are equal, and thus G3 is not a Beauville group when
+StG(3) ̸= 1.
+We now consider the remaining cases, where here Gk = G. As before there exists some
+z1 ∈ X with z1 ≡ aib (mod G′) for some odd i, and there exists z2 ∈ Y with z2 ≡ ajb
+(mod G′) for some odd j. Therefore it suffices to show, for δ ∈ {0, 1, . . . , 2n−1 − 1} and
+w ∈ G′, that (a1+2δbw)2n is conjugate to (ab)2n in G.
+Recall that (G′)2 ≤ StG(1)′ and that StG(1)′ is elementary abelian. Hence, from [17,
+Prop. 1.1.32(i)] we have
+(ab)2n = [b, a, 2n−1−1
+. . . , a]2[b, a, 2n−1
+. . . , a]c
+for some c ∈ γ2n(G) ∩ StG(1)′. Let i1, i2, i3 ≥ 2n be such that
+�
+[b, a, 2n−1−1
+. . . , a]2, a
+�
+∈ γi1(G)\γi1+1(G)
+[b, a, 2n
+. . ., a] ∈ γi2(G)\γi2+1(G)
+c ∈ γi3(G)\γi3+1(G),
+and set
+ℓ := min{i1, i2, i3 + 1}.
+In other words, the integer ℓ is such that [(ab)2n, a] ∈ γℓ(G)\γℓ+1(G).
+Now
+(a1+2δbw)2n = [bw, a1+2δ, 2n−1−1
+. . . , a1+2δ]2[bw, a1+2δ, 2n−1
+. . . , a1+2δ]cd,
+where d ∈ γi3+1(G) is obtained as follows: starting with the commutator c, which is a
+product of commutators in b and a of weight at least 2n, with weight exactly 2 in b;
+compare [17, Prop. 1.1.32(i)], we have that cd = �c, where �c is obtained from c by replacing
+all occurrences of b with bw. Here we have also used the standard identities [xy, z] =
+[x, z]y[y, z] and [x, yz] = [x, z][x, y]z.
+
+18
+E. DI DOMENICO, S¸. G¨UL, AND A. THILLAISUNDARAM
+Next, using these standard identities together with Lemmata 3.6 and 3.7, we have
+(a1+2δbw)2n ≡ [bw, a1+2δ, 2n−1−1
+. . . , a1+2δ]2[bw, a1+2δ, 2n−1
+. . . , a1+2δ]c
+(mod γℓ(G))
+≡ [b, a, 2n−1−1
+. . . , a]2[b, a, 2n−1
+. . . , a]c
+(mod γℓ(G)).
+Therefore
+(a1+2δbw)2n ≡ (ab)2n
+(mod γℓ(G)).
+In other words, we have (a1+2δbw)2n = (ab)2nv for v ∈ γℓ(G). By Lemma 3.8, we have
+that v = [(ab)2n, g] for some g ∈ G, and hence we are done.
+□
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+[19] J. Stix and A. Vdovina, Series of p-groups with Beauville structure, Monatsh. Math. 181 (2016),
+177–186.
+[20] T. Vovkivsky, Infinite torsion groups arising as generalizations of the second Grigorchuk group, in:
+Algebra (Moscow, 1998), de Gruyter, Berlin, 2000.
+[21] J. S. Wilson, Groups with every proper quotient finite, Proc. Cambridge Philos. Soc. 69 (1971), 373–
+391.
+
+BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS
+19
+Elena Di Domenico: Department of Mathematics, University of Trento, 38123 Trento,
+Italy - University of the Basque Country UPV/EHU, 48080 Bilbao Spain
+Email address: elena.didomenico@yahoo.it
+S¸¨ukran G¨ul: Department of Mathematics, TED University, 06420 Ankara, Turkey
+Email address: sukran.gul@tedu.edu.tr
+Anitha Thillaisundaram:
+Centre for Mathematical Sciences, Lund University, 223 62
+Lund, Sweden
+Email address: anitha.t@cantab.net
+
diff --git a/L9E1T4oBgHgl3EQfHAM0/content/tmp_files/load_file.txt b/L9E1T4oBgHgl3EQfHAM0/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf,len=1427
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='02920v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='GR]  7 Jan 2023  BEAUVILLE STRUCTURES FOR QUOTIENTS  OF GENERALISED GGS-GROUPS  ELENA DI DOMENICO, S¸ ¨UKRAN G¨UL, AND ANITHA THILLAISUNDARAM  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' A finite group with a Beauville structure gives rise to a certain compact  complex surface called a Beauville surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' G¨ul and Uria-Albizuri showed that quotients  of the periodic Grigorchuk-Gupta-Sidki (GGS-)groups that act on the p-adic tree, for  p an odd prime, admit Beauville structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  We extend their result by showing that  quotients of infinite periodic GGS-groups acting on pn-adic trees, for p any prime and  n ≥ 2, also admit Beauville structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Introduction  A Beauville surface is a compact complex surface isomorphic to (C1 × C2)/G, where  C1 and C2 are algebraic curves of genus at least 2, and G is a finite group acting  freely on C1 × C2 by holomorphic transformations,  the group G acts faithfully on the curves Ci such that Ci/G ∼= P1(C) and the  covering map Ci → Ci/G is ramified over three points, for i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  The group G is then said to be a Beauville group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  One can reformulate the condition for a finite group G to be a Beauville group purely  in group-theoretical terms: for x, y ∈ G, let  Σ(x, y) =  �  g∈G  �  ⟨x⟩g ∪ ⟨y⟩g ∪ ⟨xy⟩g�  ,  that is, the union of all conjugates of the cyclic subgroups generated by x, y and xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Then G is a Beauville group if and only if G is 2-generated and there exist generating sets  {x1, y1} and {x2, y2} of G such that Σ(x1, y1) ∩ Σ(x2, y2) = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The sets {x1, y1} and  {x2, y2} are then called a Beauville structure for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Beauville groups have been intensely studied in recent times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' see surveys [1, 7, 8, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  For example, the abelian Beauville groups were classified by Catanese [4]: a finite abelian  group G is a Beauville group if and only if G ∼= Cn×Cn for n > 1 with gcd(n, 6) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' After  abelian groups, the most natural class of finite groups to consider are nilpotent groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  The determination of nilpotent Beauville groups is easily reduced to the case of p-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  In [1], it was shown that there are non-abelian Beauville p-groups of order pn for ev-  ery p ≥ 5 and every n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The first explicit infinite family of Beauville 2-groups was  constructed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' In [19], Stix and Vdovina constructed an infinite family of Beauville  p-groups, for every prime p, by considering quotients of ordinary triangle groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' In [9],  Date: January 10, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Primary 20E08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Secondary 20D15, 14J29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Groups acting on rooted trees, finite p-groups, Beauville structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  The first and the second authors are supported by the Spanish Government, grant MTM2017-86802-  P, partly with FEDER funds, and by the Basque Government, grant IT974-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The first author is also  supported by the National Group for Algebraic and Geometric Structures, and their Applications (GN-  SAGA - INdAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The third author acknowledges support from EPSRC, grant EP/T005068/1, and from  the Lincoln Institute of Advanced Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' She also thanks the University of the Basque Country for its  hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The first and second authors would like to thank the University of Lincoln for its hospitality  while this research was conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  1  2  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' DI DOMENICO, S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' G¨UL, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' THILLAISUNDARAM  Fern´andez-Alcober and G¨ul extended Catanese’s criterion for abelian Beauville groups to  finite p-groups satisfying certain conditions which are much weaker than commutativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  They also give the first explicit infinite family of Beauville 3-groups, and they show that  there are Beauville 3-groups of order 3n for every n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  In [15], G¨ul and Uria-Albizuri showed that the quotients of periodic Grigorchuk-Gupta-  Sidki (GGS-)groups admit Beauville structures, giving another infinite family of Beauville  p-groups, for p an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The GGS-groups were some of the early examples of groups  acting on rooted trees, which first arose as easily describable examples of infinite finitely  generated periodic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Since then, groups acting on rooted trees have provided many  other interesting and exotic examples, such as infinite finitely generated groups of inter-  mediate word growth, infinite finitely generated amenable but not elementary amenable  groups, etc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' see for instance [14, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  The GGS-groups are infinite groups acting faithfully on the p-adic tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The natural  quotients of a GGS-group G are G/ StG(n), for n ∈ N, where StG(n) is the normal subgroup  of the elements of G that pointwise fix the vertices at the nth level of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The quotient  G/ StG(n) acts on the finite tree consisting of the first n layers of the full p-adic tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  We extend the result of [15] to a generalisation of the GGS-groups to the pn-adic tree,  for any prime p and n ≥ 2, by showing that infinitely many quotients of infinite periodic  GGS-groups acting on the pn-adic tree do admit Beauville structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Briefly, a GGS-group acting on the pn-adic tree is a group G = ⟨a, b⟩, where a =  (1 2 · · · pn) permutes the first-level vertices, whereas b fixes the first-level vertices and is  defined recursively by b = (ae1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , aepn−1, b), for e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , epn−1 ∈ Z/pnZ with not all ei  being zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Here (ae1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , aepn−1, b) refers to the respective independent actions at the pn  maximal subtrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We write e = (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , epn−1) and call it the defining vector of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We  refer the reader to Section 2 for background material and precise definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' These groups  were constructed by Vovkivsky [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' A prominent group in this family, that was defined  earlier in [13], is the second Grigorchuk group, which acts on the 4-adic tree with defining  vector e = (1, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  For an infinite periodic GGS-group G acting on the pn-adic tree, we associate to G a  number mG ∈ N that is related to the order of certain group elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' we refer the reader  to Section 4 for precise details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The main result of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G be an infinite periodic GGS-group acting on the pn-adic tree for p  a prime and n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then G/ StG(k) admits a Beauville structure for k ≥ mG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  In particular, these quotients of infinite periodic GGS-groups acting on the 2n-adic tree  yield infinite families of Beauville 2-groups, and infinite periodic GGS-groups acting on the  3n-adic tree give many more infinite families of Beauville 3-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Note that Beauville 2-  groups and 3-groups are somewhat rare, precisely because of the small number of maximal  subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Indeed, for a time it was unclear whether such groups even existed, until the  first examples were given in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We also want to emphasise that for the GGS-groups  acting on the p-adic tree, if one were to consider the case p = 2 there is only one group,  which is the infinite dihedral group, hence its quotients do not admit Beauville structures,  and for p = 3 only one out of the three isomorphism classes of such groups has quotients  admitting Beauville structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' see [15] and [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  G¨ul and Uria-Albizuri also showed in [15] that for G a GGS-group acting on the p-adic  tree, there are quotients of G admitting Beauville structures if and only if G is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  The situation is not so clear cut for GGS-groups acting on the pn-adic tree, for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  In particular, there are non-periodic GGS-groups acting on the pn-adic tree which have  quotients admitting Beauville structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Futhermore, our main result  above deals only with infinite periodic GGS-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For finite GGS-groups, some quotients  BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS  3  admit Beauville structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' However a large class E of finite GGS-groups  have no level-stabliser quotients admitting Beauville structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We refer the reader to  Section 5 for the definition of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G ∈ E be a finite GGS-group acting on the pn-adic tree for p a prime  and n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then G/ StG(k) is not a Beauville group for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  To prove our theorems, we largely make use of the subgroup structure of the respective  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Organisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Section 2 of this paper consists of background material for groups acting on  rooted trees and here we also recall the generalisation of the GGS-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' In Section 3 we  establish key properties of these groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' In Section 4 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1 and lastly in  Section 5 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We thank G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Fern´andez-Alcober for his valuable feedback and  helpful suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Preliminaries  All trees considered here will be rooted, meaning that there is a distinguished vertex  called the root, with one degree less than all other vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For d ∈ N≥2, let T be the  d-adic tree, meaning all vertices have d children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Using the alphabet A = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , d}, the  vertices uω of T are labelled bijectively by elements ω of the free monoid A∗ as follows:  the root of T is labelled by the empty word ∅, and for each word ω ∈ A∗ and letter a ∈ A  there is an edge connecting uω to uωa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We say that uω precedes uλ whenever ω is a prefix  of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' When convenient, we do not differentiate between A∗ and vertices of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  There is a natural length function on A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The words ω of length |ω| = n, representing  vertices uω that are at distance n from the root, are the nth-level vertices and form the  nth layer of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The elements of the boundary ∂T correspond naturally to infinite  simple rooted paths, and they are in one-to-one correspondence with the d-adic integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Denote by Tu the full subtree of T that has its root at a vertex u and includes all vertices  succeeding u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For any two vertices u = uω and v = uλ, the map uωτ �→ uλτ, induced by  replacing the prefix ω by λ, yields an isomorphism between the subtrees Tu and Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Every automorphism of T fixes the root, and the orbits of Aut(T) on the vertices of the  tree T are precisely its layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For f ∈ Aut(T), the image of a vertex u under f is denoted  by uf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The automorphism f induces a faithful action on the monoid A∗ and for a ∈ A we  have (ωa)f = ωfa′ where a′ ∈ A is uniquely determined by ω and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' From this we have a  permutation f(ω) of A where  (ωa)f = ωfaf(ω),  and hence  (uωa)f = uωf af(ω),  and f(ω) is called the label of f at ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The collection of all labels of f constitutes the  portrait of f, and there is a one-to-one correspondence between automorphisms of T and  portraits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  The automorphism f is rooted if f(ω) = 1 for ω ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  It is directed, with  directed path ℓ for some ℓ ∈ ∂T, if the support {ω | f(ω) ̸= 1} of its labelling is infinite  and marks only vertices at distance 1 from the set of vertices corresponding to the path ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Additionally, the section of f at a vertex u is defined to be the unique automorphism fu  of T ∼= Tu given by the condition (uv)f = ufvfu for v ∈ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Subgroups of Aut(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For G ≤ Aut(T), the vertex stabiliser stG(u) is the subgroup  consisting of elements in G that fix the vertex u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  For n ∈ N, the nth-level stabiliser  StG(n) = �  |ω|=n stG(uω) is the subgroup consisting of automorphisms that fix all vertices  at level n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let T[n] be the finite subtree of T of vertices up to level n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then StG(n) is the  kernel of the induced action of G on T[n], and we will denote by Gn the quotient G/ StG(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  4  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' DI DOMENICO, S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' G¨UL, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' THILLAISUNDARAM  Each g ∈ StAut(T)(n) can be described completely in terms of its restrictions to the  subtrees rooted at vertices at level n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Indeed, the map  ψn : StAut(T)(n) −→  �  |ω|=n  Aut(Tuω) ∼= Aut(T) ×  dn  · · × Aut(T)  is a natural isomorphism mapping g ∈ StAut(T)(n) to its nth-level sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For ease of  notation, we write ψ = ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  For ω ∈ A∗, we further define:  ϕω : stAut(T)(uω) −→ Aut(Tuω) ∼= Aut(T)  g  �−→  gω  A group G ≤ Aut(T) is said to be self-similar if for every g ∈ G and every vertex u, the  section gu, of g at u, is an element of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For a self-similar group G ≤ Aut(T), for m ≥ 2  and i ∈ Am−1, we will write  ψi  m = ψ ◦ ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  That is, for g ∈ stG(ui) with ϕi(g) ∈ StG(1), we have ψi  m(g) gives the components of ϕi(g)  corresponding to the children of the (m − 1)th-level vertex i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' In addition, we write  φn,m : StGn(m) −→ Gn−1 ×  dm  · · × Gn−1  for the corresponding ψm when working in the quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Likewise, we write φn = φn,1 for  simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' GGS-groups acting on the pn-adic tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let T be the pn-adic tree for a prime p  and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Given a non-zero vector e = (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , epn−1) ∈ (Z /pn Z)pn−1, the GGS-  group G = Ge associated to the defining vector e is the group generated by the rooted  automorphism a corresponding to the cycle (1 2 · · · pn) and by a directed automorphism  b ∈ StAut(T)(1) defined recursively via  ψ(b) = (ae1, ae2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , aepn−1, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Unlike the GGS-groups acting on the p-adic tree, the GGS-groups acting on the pn-adic  tree for n ≥ 2 are not all infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' A necessary and sufficient condition for these groups to  be infinite was given by Vovkivsky [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' He proved that such a group is infinite if and only  if there exists an i ≥ 0 such that  R0 ≤ R1 ≤ · · · ≤ Ri = Ri+1 = · · · < n  where the sequence Rj is defined recursively as follows: R0 is the largest integer such that  pR0 | eℓ for all ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' and then for j ≥ 0 and while Rj < n, the number Rj+1 is  defined as the largest integer such that pRj+1 divides eℓ for all ℓ ∈ {pRj, 2pRj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−pRj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Note that the order of a is pn and the order of b is pn−R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Further, from [6, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1]  we have G/G′ ∼= Cpn × Cpn−R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Vovkivsky further proved in [20] that a GGS-group acting on the pn-adic tree is a  periodic group if and only if for each k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , n − 1},  S[k] ≡ 0  (mod pk+1)  where  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1)  S[k] = epk + e2pk + · · · + epn−pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  We recall the prominent example of such a GGS-group, which is for the case p = n = 2:  BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS  5  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let T be the 4-adic tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The second Grigorchuk group Γ ≤ Aut(T) is  generated by two automorphisms a and b, where a is the rooted automorphism corre-  sponding to the cycle (1 2 3 4), and b ∈ StΓ(1) is recursively defined by ψ(b) = (a, 1, a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  The group Γ, which is periodic, was first defined in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  For more information on GGS-groups acting on the pn-adic tree, see [20, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Properties of GGS-groups acting on the pn-adic tree  For G = Ge, a GGS-group acting on the pn-adic tree, recall that R0 is the largest integer  such that pR0 | eℓ for all ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Suppose now that G is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' To each element aiprbjps in G, where 0 ≤ r < n,  0 ≤ s < n − R0 and i, j ̸≡ 0 (mod p), we associate the following numbers:  (i) If pn−s ∤ S[r], let t0, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , tmr,s ∈ N for some 1 ≤ mr,s < n be such that  tk < n − s ≤ tmr,s for all k < mr,s,  pt0 | S[r] and pt0+1 ∤ S[r],  ptk | S[tk−1 + s] and ptk+1 ∤ S[tk−1 + s] for 1 ≤ k < mr,s,  ptmr,s | S[tmr,s−1 + s],  where the S[λ] for λ ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , n − 1} are as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  (ii) If pn−s | S[r], then we set mr,s = 0, and for convenience define t0 = n − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  If we are in case (i), note that t0 does not depend on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G = ⟨a, b⟩ be a periodic GGS-group acting on the pn-adic tree, let  0 ≤ r < n, 0 ≤ s < n − R0 and i, j ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let m := mr,s and t0, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , tm ∈ N be  as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then the order of aiprbjps in both G and Gm+3 is pt(r,s) where  t(r, s) = (m + 2)n − (t0 + · · · + tm−1 + s(m + 1) + r + R0),  and if pn−s | S[r], that is, when m = 0, we take t0 + · · · + tm−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Clearly the order of aiprbjps is a multiple of pn−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We have  ψ  �  (aiprbjps)pn−r�  = ψ  �  (bjps)a(pn−r−1)ipr  · · (bjps)a2ipr  (bjps)aipr  bjps�  =  �  (ajps+R0)∗, pr−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (ajps+R0)∗, ajpsS[r](bjps)ajps(eipr +e2ipr +···+epr )  ,  (ajps+R0)∗, pr−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (ajps+R0)∗, ajpsS[r](bjps)ajps(eipr +e2ipr +···+e2pr )  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' ,  (ajps+R0)∗, pr−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (ajps+R0)∗, ajpsS[r](bjps)a  jps(eipr +e2ipr +···+e(pn−r−1)pr )  ,  (ajps+R0)∗, pr−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (ajps+R0)∗, ajpsS[r]bjps�  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1)  where the ∗ denotes an unspecified exponent, and the last equality follows from the fact that  the set {ipr, 2ipr, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (pn−r − 1)ipr, pn} coincides, modulo pn, with {pr, 2pr, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (pn−r −  1)pr, pn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' So the components of ψ((aiprbjps)pn−r) in positions a multiple of pr are conju-  gates of ajpsS[r]bjps and the other components are powers of ajps+R0 whose order is at most  pn−s−R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Suppose that pn−s | S[r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1), we have  ψ  �  (aiprbjps)pn−r�  =  �  (ajps+R0)∗, pr−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (ajps+R0)∗, (bjps)ajps(eipr +e2ipr +···+epr )  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' ,  (ajps+R0)∗, pr−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (ajps+R0)∗, bjps�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  6  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' DI DOMENICO, S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' G¨UL, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' THILLAISUNDARAM  Note that the order of bjps is pn−s−R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Thus, the order of aiprbjps in G is pt(r,s), for  t(r, s) = 2n − (r + s + R0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Also since (bjps)pn−s−R0−1 ̸∈ StG(2), it follows that (aiprbjps)pt(r,s)−1 ̸∈ StG(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Thus the  element aiprbjps is of order pt(r,s) in G3 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  It remains to settle the case pn−s ∤ S[r], which we will now assume till the end of  the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Since by hypothesis pt0+s+1 ∤ jpsS[r], the order of ajpsS[r]bjps is a multiple of  pn−(t0+s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Applying ψ we get  ψpn  2  �  (aiprbjps)pn−rpn−(t0+s)�  = ψ  �  (ajpsS[r]bjps)pn−(t0+s)�  = ψ  �  (bjps)a(pn−(t0+s)−1)jpsS[r] · · · (bjps)a2jpsS[r](bjps)ajpsS[r]bjps�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  As before, it follows that {jpsS[r], 2jpsS[r], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (pn−(t0+s) −1)jpsS[r], pn} coincides, mod-  ulo pn, with the set {pt0+s, 2pt0+s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn − pt0+s, pn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' This means that the b’s appear in  all positions a multiple of pt0+s and only in these positions, and we can write each of the  elements in these positions as  ajpsS[t0+s](bjps)a  jps�  ejpsS[r]+e2jpsS[r]+···+eℓpt0+s  �  for certain ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−(t0+s)−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Since we are interested in the order of these elements,  up to conjugation the order of these elements coincides with the order of the element  ajpsS[t0+s]bjps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' In the other components of ψ((ajpsS[r]bjps)pn−(t0+s)), there is a power of  ajps+R0 whose order is at most pn−s−R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' By hypothesis pt1+s+1 ∤ psS[t0 + s] thus the order  of ajpsS[t0+s]bjps is at least pn−(t1+s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Proceeding in this way, after m − 1 further steps we  find  ψpmn  m+1  �  (aiprbjps)pn−rpn−(t0+s)pn−(t1+s)···pn−(tm−1+s)�  = ψ  �  (bjps)a(pn−(tm−1+s)−1)jpsS[tm−2+s] · · · (bjps)a2jpsS[tm−2+s](bjps)ajpsS[tm−2+s](bjps)  �  =  �  (ajps+R0)∗, ptm−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , (ajps+R0)∗, ajpsS[tm−1+s](bjps)a∗, (ajps+R0)∗, ptm−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , (ajps+R0)∗,  ajpsS[tm−1+s](bjps)a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (ajps+R0)∗, ptm−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , (ajps+R0)∗, ajpsS[tm−1+s](bjps)  �  =  �  (ajps+R0)∗, ptm−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , (ajps+R0)∗, (bjps)a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (ajps+R0)∗, ptm−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , (ajps+R0)∗, bjps�  where the last equality holds since ptm+s | psS[tm−1 + s], thus pn | psS[tm−1 + s] by  the definition of tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Since by hypothesis j ̸≡ 0 (mod p) it follows that the order of the  elements in positions a multiple of ptm−1 is pn−s−R0 and the orders of the other elements  are at most pn−s−R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' This proves that the order of aiprbjps is pt(r,s), where  t(r, s) = (m + 2)n − (t0 + · · · + tm−1 + s(m + 1) + r + R0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS  7  Finally, let us prove that the element (aiprbjps)pt(r,s)−1 ̸∈ StG(m + 3), which implies that  aiprbjps is of order pt(r,s) in Gm+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We observe that  ψpmn  m+1  �  (aiprbjps)pt(r,s)−1�  =  �  (ajpn−1)∗, ptm−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , (ajpn−1)∗, (bjpn−R0−1)a∗,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (ajpn−1)∗, ptm−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , (ajpn−1)∗, bjpn−R0−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Since the components (bjpn−R0−1)a∗ are non-trivial in G2, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  □  In the next three results, we identify a sort of branching structure within certain sub-  groups of GGS-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For convenience, we set R−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G = ⟨a, b⟩ be a GGS-group acting on the pn-adic tree, for p a prime  and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Then for j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , n − 1}, we have ϕv(stG(v)) ≥ ⟨apRj−1, b⟩ for any  vertex v = u1 · · · uj ∈ Aj of length j, where ui ≡ 0 (mod pRi−2) for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For a vertex v of length one, the result is clear from considering the sections of  b, ba, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , bapn−1, which are the generators of stG(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' As the subgroup generated by the  sections of b is ⟨apR0, b⟩, it follows that for a vertex w of length 2,  b, bapR0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , bapn−pR0 ∈ ϕv(stG(w)),  where v is the prefix of w of length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' It then follows that for vertices w = u1u2, with  u1, u2 ∈ A such that u2 ≡ 0 (mod pR0), we have ϕw(stG(w)) ≥ ⟨apR1, b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The result now  follows recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G = ⟨a, b⟩ be a GGS-group acting on the pn-adic tree, for p a prime and  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let α ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , n − 1} and let pR be the highest power of p dividing eipα for all  i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−α − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Suppose that  (i) the highest power of p dividing ejpα+1 for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−α−1 − 1} is strictly  greater than pR, and  (ii) there exists ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−α − 1} such that pR+1 | eℓpα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Then, writing Nα = ⟨apα, b⟩ and NR = ⟨apR, b⟩, we have  1 ×  pα−1  · · × 1 × γ3(NR) × · · · · · · × 1 ×  pα−1  · · × 1 × γ3(NR) = ψ  �  γ3(StNα(1))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Note that if α = n − 1, condition (i) is vacuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' By assumption (i), there exists k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−α −1} with k ̸≡ 0 (mod p) such that  ekpα = κpR for some κ ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Up to replacing b with a suitable power of itself,  we may assume without loss of generality that κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' It suffices to show, since we can  conjugate by apα, that  1 ×  kpα−1  · · × 1 × γ3(NR) × 1 × · · · × 1 ≤ ψ  �  γ3(StNα(1))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Case 1: Suppose e(pn−α−k)pα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then, as γ3(NR) = ⟨[apR, b, apR], [apR, b, b]⟩NR, the  result follows from  ψ  �  [b, bakpα  , b]  �  = (1, kpα−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [apR, b, apR], 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1)  ψ  �  [b, bakpα  , bakpα  ]  �  = (1, kpα−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [apR, b, b], 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1)  and using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  8  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' DI DOMENICO, S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' G¨UL, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' THILLAISUNDARAM  Case 2: Suppose e(pn−α−k)pα ̸= 0 and let χ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−R−1} be such that e(pn−α−k)pα =  χpR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  If p | χ, consider  ψ  �  [b, bakpα  , b]  �  = (1, kpα−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [apR, b, apR], 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [b, aχpR, b]),  ψ  �  [b, bakpα  , bakpα  ]  �  = (1, kpα−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [apR, b, b], 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [b, aχpR, aχpR]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Let us refer to those commutators in the final component as the error terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We proceed  to cancel those error terms by respectively multiplying with the following:  ψ  ��  [bakpα  , bχ, bakpα  ]−1�a−kpα�  = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [aχpR, bχ, aχpR]−1, 1, kpα−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [b, aχpR, b]−1),  ψ  ��  [bakpα  , bχ, bχ]−1�a−kpα�  = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [aχpR, bχ, bχ]−1, 1, kpα−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [b, aχpR, aχpR]−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  This introduces a new set of error terms, this time in the (pn − kpα)th component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We  notice that the new set of error terms involve a higher p-power of a or b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Hence, after  repeating this process a finite number of times, we will eventually have a set of error  terms consisting of commutators with at least one component being the trivial element  apn = bpn−R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' In other words, we have completely cancelled any error terms, showing that  (1, kpα−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [apR, b, apR], 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1) ∈ ψ  �  γ3(StNα(1))  �  ,  (1, kpα−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [apR, b, b], 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1) ∈ ψ  �  γ3(StNα(1))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Now suppose that p ∤ χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' By hypothesis, there exists ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−α − 1} such that  eℓpα = λpR with p | λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We claim that we may choose ℓ such that e(ℓ−k)pα = ξpR with  p ∤ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Indeed, suppose on the contrary that for all choices of ℓ satisfying condition (ii), we  have pR+1 | e(ℓ−k)pα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Since k ̸≡ 0 (mod p), repeatedly replacing ℓ with ℓ − k, this means  that ejpα ≡ 0 (mod pR+1) for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−α − 1}, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Hence the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  For convenience, we write e(k−ℓ)pα = ζpR, and let ν ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−R} be such that νξ ≡ χ  (mod pn−R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then we have  ψ  �  [b, (bν)a(k−ℓ)pα  , b]−1�  = (1, (k−ℓ)pα−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , 1, [aζpR, bν, aζpR]−1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [b, aχpR, b]−1),  ψ  �  [b, bakpα  , (bν)a(k−ℓ)pα  ]−1�  = (1, kpα−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [apR, b, aνλpR]−1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [b, aχpR, aχpR]−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  If p | ζ, we proceed to cancel the error terms as in the above argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  If p ∤ ζ, let  θ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−R} be such that θχ ≡ ζ (mod pn−R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then, since p | λ, we can use the  following element to proceed:  ψ  �  [(bθ)akpα  , bν, (bθν)a(k−ℓ)pα  ]  �  = (1, kpα−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [bθ, aνpR, aθνλpR], 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [aζpR, bν, aζpR]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G = ⟨a, b⟩ be a GGS-group acting on the pn-adic tree, for p an odd prime  and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let R ≤ n − 1 be such that ekpR ≡ 0 (mod pR) for all k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−R − 1}  and suppose that ekpR ̸≡ 0 (mod pR+1) for all k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−R − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Suppose further that  S[R] ≡ 0 (mod pR+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then, writing N = ⟨apR, b⟩, we have  1 ×  pR−1  · · × 1 × γ3(N) × · · · · · · × 1 ×  pR−1  · · × 1 × γ3(N) = ψ  �  γ3(StN(1))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let ekpR = ikpR for k, ik ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−R − 1} with ik ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Further, by  replacing b with an appropriate power, we may assume that i1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' As reasoned in the  previous proof, it suffices to show that  1 ×  pR−1  · · × 1 × γ3(N) × 1 × · · · × 1 ≤ ψ  �  γ3(StN(1))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS  9  We proceed as in [10, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2] by considering the following two cases:  (a) There exists an ℓ ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−R − 2} such that i2  ℓ − iℓ−1iℓ+1 ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  (b) For all ℓ ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−R − 2}, we have i2  ℓ − iℓ−1iℓ+1 ≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Note that if p = 3 and R = n − 1, then Case (b) trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Suppose we are in Case (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let us define gℓ as follows:  gℓ =  �  (biℓ)apR  b−iℓ−1�a−ℓpR  Writing µ = i2  ℓ − iℓ−1iℓ+1, we have  ψ(gℓ) = (∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, aµpR, ∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, 1)  where ∗ represents unspecified elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' By assumption, we have µ ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Hence  there is a power g of gℓ satisfying  ψ(g) = (∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, apR, ∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Furthermore, for simplicity let λ := ipn−R−1 ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then  ψ  �  bapR  (ba−pR  )−λ�  = (∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, ba−λe2pR , ∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, 1),  and multiplying by an appropriate power of g we obtain an element h ∈ StN(1) with  ψ(h) = (∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, b, ∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Then the result follows from:  ψ  �  [b, bapR  , g]  �  = (1, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [apR, b, apR], 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1)  ψ  �  [b, bapR  , h]  �  = (1, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [apR, b, b], 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1)  We now suppose that we are in Case (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' It then follows, for j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−R − 1},  that ij = λj−1 for some λ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' However, as S[R] ≡ 0 (mod pR+1), we have λ ̸≡ 1  (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Further we have λpn−R−1pR ≡ pR (mod pR+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Note that  ψ  �  b(bapR  )−λ�  =  (∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, apRb−λ, ∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, 1, ∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, ba−pR+νpR+1)  for some ν, and ∗ represents some power of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let χ be such that λχ ≡ 1 (mod pn−R0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  recall that pn−R0 is the order of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then  ψ  �  bχ�  b−χλ�apR�  = ψ  �  bχ�  b−1�apR�  =  (∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, aχpRb−1, ∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, 1, ∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ∗, bχ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Thus, since χ ̸≡ 1 (mod p) and hence  N = ⟨aχpRb−1, ba−pR+νpR+1⟩ = ⟨a(χ−1)pR+νpR+1, ba−pR+νpR+1⟩,  the result follows from the observations below:  ψ  ��  b(1−νp), bapR  , bapR  (ba2pR  )−λ��  = (1, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [apR−νpR+1, b, ba−pR+νpR+1], 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1),  ψ  ��  (ba2pR  )λ, bapR  , bχ�  b−χλ�apR��  = (1, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [apR−νpR+1, b, aχpRb−1], 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  □  Next, we record some properties of certain finite GGS-groups acting on the 2n-adic tree,  which will be needed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  10  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' DI DOMENICO, S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' G¨UL, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' THILLAISUNDARAM  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G be a periodic GGS-group acting on the 2n-adic tree for n ≥ 2, and  assume that R0 = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then StG(4) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Furthermore StG(3) is non-trivial if and  only if ei = e2n−1+i for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 2n−1 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Note that as R0 = n − 1, we have that 2n−1 divides each ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Thus the ei’s are 0  or 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Since G is periodic, it follows that e2n−1 = 0 and hence  �  b, ba2n−1� ∼= C2 × C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Let g be an element in StG(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Certainly ϕω(g) ∈ StG(1) for all ω ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Since  ψ(StG(1)) ∩ StG(1) × · · · × StG(1) ⊆  �  b, ba2n−1�  ×  2n  · · ×  �  b, ba2n−1 �  ,  it follows that  ϕω(g) ∈  �  1, b, ba2n−1  , bba2n−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  As  �  b, ba2n−1�  ∩ StG(3) = 1, the first claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  For the second statement, let g be a non-trivial element in StG(3), in particular ϕω(g) ∈  StG(2) for all ω ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' From the previous paragraph, it follows that ϕω(g) ∈  �  1, bba2n−1 �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Thus there exists ω ∈ A such that ϕω(g) = bba2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We observe that  ψ  �  bba2n−1�  = (ae1+e2n−1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ae2n−1−1+e2n−1, b, ae1+e2n−1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ae2n−1−1+e2n−1, b),  so ϕω(g) ∈ StG(2) if and only if ei = e2n−1+i for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 2n−1 − 1}, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G be a periodic GGS-group acting on the 2n-adic tree for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Suppose R0 = n − 1 and S[0] ≡ 0 (mod pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then γ2n+1(G) ≤ StG(1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Furthermore  [b, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]2 = 1 and [b, a, 2n−1+i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]2 ∈ γ2n+i+2(G) for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' As StG(1)/ StG(1)′ is elementary abelian, we have (G′)2 ≤ StG(1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Therefore,  from [17, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='32(ii)], we have  1 = [a2n, b] ≡ [b, a, 2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=', a]  (mod StG(1)′),  which gives γ2n+1(G) ≤ StG(1)′, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Also from [17, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='32(ii)], we obtain  [a2n−1, b] ≡ [a, b, a, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]  (mod StG(1)′)  and so [b, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]2 = 1, as StG(1)′ is elementary abelian and [StG(1), StG(1)′] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  For the last statement, we proceed by induction on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For i = 1, the statement follows  from  1 =  �  [b, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]2, a  �  = [b, a, 2n−1+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]2�  b, a, 2n−1+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a, [b, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]  �  ,  and similarly for the inductive step, as  �  [b, a, 2n−1+i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]2, a  �  = [b, a, 2n−1+i+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , a]2�  b, a, 2n−1+i+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , a, [b, a, 2n−1+i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G be a GGS-group satisfying the conditions of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Then  γi(G)/γi+1(G) is isomorphic to a subgroup of C2 × C2, for i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Furthermore |γi(G) : γi+1(G)| ≤ 2 for i > 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We notice that  ψ(StG(1)′) ⊆ S :=  ��  [b, a2n−1]ǫ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , [b, a2n−1]ǫ2n�  | ǫ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ǫ2n ∈ {0, 1}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Since |S| = 22n, it follows that log2 | StG(1)′| ≤ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  First we consider the case when the defining vector e is non-symmetric, that is, there  exists j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 2n−1 − 1} with ej ̸= e2n−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then, there exists some k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 2n − 1}  such that  ψ([b, bak]) =  �  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [b, a2n−1]  �  ∈ ψ(StG(1)′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS  11  Hence for e non-symmetric, we have log2 | StG(1)′| = 2n, and  StG(1)′ = ⟨[b, bak], [b, bak]a, [b, bak]a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , [b, bak]a2n−1⟩  = ⟨[b, bak], [b, bak, a], [b, bak, a2], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , [b, bak, a2n−1]⟩  = ⟨[b, bak], [b, bak, a], [b, bak, a, a], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , [b, bak, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Indeed, the last equality follows from the more general fact that  ⟨h, [h, a], [h, a2], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , [h, aℓ]⟩ = ⟨h, [h, a], [h, a, a], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , [h, a,  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=', a]⟩  for h ∈ StG(1)′ and 0 ≤ ℓ ≤ 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' This is proved by induction on ℓ, using the identity  [h, aℓ+1] = [h, a][h, aℓ][h, aℓ, a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  From this, we deduce that for each i ≥ 2, either StG(1)′ ∩ γi(G) = StG(1)′ ∩ γi+1(G) or  (StG(1)′ ∩ γi(G))/(StG(1)′ ∩ γi+1(G)) ∼= C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' As  γi(G) = ⟨[b, a, i−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=', a]⟩(StG(1)′ ∩ γi(G))γi+1(G),  the result follows, with the final statement coming from the fact that [b, a, 2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=', a] ∈ StG(1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Now we suppose that e is symmetric, so ej = e2n−j for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 2n−1 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then  there exists a minimal k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 2n−1 − 2} such that  �  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [b, a2n−1], 1, k−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=', 1, [b, a2n−1], 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1  �  ∈ ψ(StG(1)′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Let g ∈ StG(1)′ be such an element, chosen such that g ∈ γℓ(G) with ℓ minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We  observe that k = 2λ for some λ ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , n − 2}, else k will not be minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Also,  �  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 1, [b, a2n−1]  �  /∈ ψ(StG(1)′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Looking at the 2n × 2n circulant matrix C defined by (1, 0, 2λ−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 0) ∈ (F2)2n,  a direct generalisation of [10, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='7(i)] yields that the rank of C is 2n − 2λ, as the  polynomial associated to the circulant matrix is 1 + x2λ = (1 + x)2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Therefore we obtain  that log2 | StG(1)′| = 2n − 2λ, and  StG(1)′ = ⟨g, ga, ga2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ga2n−2λ−1⟩  = ⟨g, [g, a], [g, a, a], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , [g, a, 2n−2λ−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , a]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  The result now follows as in the non-symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G be a GGS-group satisfying the conditions of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Then  ⟨[b, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a, b]⟩G = {[(ab)2n, g] | g ∈ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Setting h = (ab)2n, we know from [17, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='32(i)] that  h = [b, a, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]2[b, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]c  for some c ∈ γ2n(G) ∩ StG(1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='6, we observe that for all 1 ≤ i ≤ 2n − 1,  we have [h, ai] ∈ γ2n+1(G) ≤ StG(1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Furthermore, one can check that  [h, a] = [h, b] = [b, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Let j > 2n be maximal such that [b, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a, b] ∈ γj(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Thus, since log2 | StG(1)′| ≤ 2n  as seen in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='7, we have  γj(G) = ⟨[h, b], [h, b, a], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , [h, b, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Our goal is now to show that  {[h, g] | g ∈ G} = ⟨[h, b], [h, b, a], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , [h, b, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  12  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' DI DOMENICO, S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' G¨UL, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' THILLAISUNDARAM  To this purpose, as seen in the previous proof it suffices to prove  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2)  {[h, g] | g ∈ G} = ⟨[h, b], [h, b]a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , [h, b]a2n−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  For any 1 ≤ i ≤ 2n − 1, we have that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='3)  [h, b]ai = [hai, bai] = [h[h, ai], bai] = [h, bai],  where the last equality follows from the fact that [h, ai] ∈ γ2n+1(G) ≤ StG(1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Next we  note that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='4)  [h, ak] = [h, b][h, b]a · · · [h, b]ak−1  for all 1 ≤ k ≤ 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Now, every element g ∈ G is of the form g = aλbaℓ1 · · · baℓd for some  d ≥ 0 and λ, ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ℓd ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 2n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then taking into account (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='4), we  get  [h, g] = [h, b]aℓd · · · [h, b]aℓ1[h, aλ],  from which we deduce that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Beauville structures for quotients of infinite GGS-groups  acting on the pn-adic tree  We are now ready to prove that certain quotients of G are Beauville groups, for G  an infinite periodic GGS-group acting on the pn-adic tree where p is a prime and n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Recall that for a 2-generator finite group H, the sets {x1, x2} and {y1, y2} form a Beauville  structure for H, if and only if  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1)  ⟨x⟩ ∩ ⟨yg⟩ = 1  for all x ∈ X = {x1, x2, x1x2}, y ∈ Y = {y1, y2, y1y2} and g ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We also remark that we  will only consider G/StG(k) for k large enough such that the order of each element in X  (or in Y ) is the same in G and in G/StG(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The reason for this will become apparent  when we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1 later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Recall mr,s from the beginning of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For G an infinite GGS-group, recall that  there exists some 1 ≤ q ≤ n − 1 such that Rq = Rq+1 < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We write pd for the or-  der of [apRq, b, apRq ]b in G, and we denote by λ′ the minimal level where all sections of  ([apRq , b, apRq ]b)pd−1 that involve b are of the form (b∗)a∗ for some unspecified exponents ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  For the rest of this section, we set m = max{m0,0 + λ′, m1,0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G be an infinite periodic GGS-group acting on the pn-adic tree, for p  a prime and n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then the quotient G/StG(m + 3) is a Beauville group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Case 1: Suppose that ekpn−1 ̸= 0 for some k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , p − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Observe that by  assumption the prime p must be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let 1 ≤ q ≤ n−1 be such that R := Rq = Rq+1 < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Suppose first that eℓpn−1 = 0 for some ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , p−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then writing µ = m0,0 and using  Lemmata 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2 repeatedly we obtain  c = ψ−1  µ+1  �  (1, p(µ+1)n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , 1, [apR, b, apR])  �  ∈ γ3(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  By abuse of notation, we still write a and b for their images in Gm+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We set  X = {ap−1, ab, apb}  with  Y = {ac, b, acb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Assume first that x = ap−1, and consider y = ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We write pτ for the order of ac  in Gm+3, which is strictly greater than pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' In order to prove our claim for this choice of x  and y, it suffices to show that  ⟨apn−1⟩ ̸= ⟨(ac)pτ−1⟩g,  BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS  13  for all g ∈ Gm+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Note that the element (ac)pτ−1 is in StGm+3(1) and so is any conjugate  of (ac)pτ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' However apn−1 ̸∈ StGm+3(1), hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1) holds here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Similarly for y ∈ {b, acb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Suppose next that x = ab, and consider y = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1, the element ab has  order pt(0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' It suffices to consider the intersection of ⟨(ab)pt(0,0)−1⟩ and ⟨bpn−R0−1⟩g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We  observe that ψ((bpn−R0−1)g) has exactly one component consisting of purely a non-rooted  automorphism in StG(1), whereas ψ((ab)pt(0,0)−1) has pn such components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Therefore (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For y = ac, we observe that ψµ+1((ac)pτ−1) has pn components consisting of non-  rooted automorphisms in γ3(G), whereas for ψµ+1((ab)pt(0,0)−1) all components consisting  of non-rooted automorphisms are in StG(1)\\γ3(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Hence the result holds for this pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Let x = apb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Since (apb)pn−1 has pn−1 non-rooted automorphisms at the first level,  whereas both (ac)pn and (acb)pn have pn such automorphisms and bpn−R0−1 has one such  automorphism, we have that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1) holds here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Hence, it remains to establish (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1) for x = ab with y = acb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Recall from the proof of  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1 that  ψpµn  µ+1  �  (ab)pt(0,0)−n+R0  �  =  �  (apR0)∗, ptµ−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , (apR0)∗, ba∗,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (apR0)∗, ptµ−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , (apR0)∗, b  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  However,  ψpµn  µ+1  �  (acb)pt(0,0)−n+R0  �  =  �  (apR0)∗, ptµ−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , (apR0)∗, ([apR, b, apR]b)a∗,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (apR0)∗, ptµ−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , (apR0)∗, [apR, b, apR]b  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  As ψ([apR, b, apR]b) has more than one section with a non-rooted automorphism, whereas  ψ(b) has only one, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Indeed,  ψ([apR, b, apR]b) = (a∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, (ba−epn−pRb)a  epR , a∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, (aepRb−1aepR)a  e2pR ,  a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, b−1a2epn−pR−epn−2pRb),  and so if epR = epn−pR = 0 then  ψ([apR, b, apR]b) = (a∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, b2, a∗, pR−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, (b−1)  e2pR , a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, (a−epn−2pR)b)  and the result is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  If either epR or epn−pR is non-zero, we see from the proof of  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1 that, writing pχ for the order of acb in G, the minimal level λ of the tree, for  which all λ-level sections of (acb)pχ−1 that involve b are of the form (b∗)a∗ for some unspec-  ified exponents ∗, is strictly greater than µ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' In particular, all these cases imply that the  number of sections of (acb)pχ−1 at level λ consisting of purely non-rooted automorphisms is  strictly greater than pn−tµ−1, which is the corresponding number for (ab)pt(0,0)−1 at level λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Suppose now that eℓpn−1 ̸= 0 for all ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , p−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The result follows similarly using  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Case 2: Suppose that ekpn−1 = 0 for all k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , p − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Since G is infinite, for some  1 ≤ q ≤ n − 1 we have R := Rq = Rq+1 < n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then we proceed as in the previous  proof, however noting that for p = 2, we replace c with  ψ−1  µ+1  �  (1, 2(µ+1)n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , 1, [a2R, b, a2R+1])  �  ,  14  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' DI DOMENICO, S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' G¨UL, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' THILLAISUNDARAM  since then, recalling that R < n − 1,  ψ([a2R, b, a2R+1]b) =  �  a∗, 2R−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, (ba−e2n−2R+1+e2n−2R)ae2R , a∗, 2R−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, (ae2R−e2n−2Rb)ae2R+1 ,  a∗, 2R−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, (ae2R+1b−1ae2R)ae3·2R , a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, (ae2n−2R−e2n−3·2R+e2n−2R+1)b�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  □  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G be an infinite periodic GGS-group acting on the pn-adic tree  with p a prime and n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let t0 be defined according to r = s = 0, and set mG = m+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' By  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1, the quotient G/StG(mG) is a Beauville group with triples X = {ap−1, ab, apb}  and Y = {ac, b, acb}, for c defined as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  As the orders of each of the elements in X are the same in G/ StG(k) and G/ StG(mG)  for all k ≥ mG, it follows from [12, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2] that G/ StG(k) is a Beauville group for every  k ≥ mG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' In the above theorems, we only require that S[i] ≡ 0 (mod pi+1) for i ∈ I,  for some subset I ⊆ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Therefore there are non-periodic GGS-groups acting  on the pn-adic tree, for n ≥ 2, with quotients being Beauville groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For instance, if  t0 > 2 for the pairs r = s = 0 and r = 1, s = 0, and  R > 2, and  S[2] ̸≡ 0 (mod p3), but S[i] ≡ 0 (mod pi+1) for all other i,  the proofs above still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' In some cases, the proofs above can be extended to finite GGS-groups G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Since G is finite, there exists d ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , n − 1} such that Rd+1 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Writing µ = m0,0,  we observe that d ≥ µ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' If d > µ, then by repeated application of Lemmata 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='3 and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2, we have an element  c := ψ−1  µ+1  �  (1, p(µ+1)n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , 1, [apRµ , b, apRµ])  �  ∈ γ3(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Setting X = {ap−1, ab, apb} and Y = {ac, b, acb}, we are done as in Case 1 of the proof of  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  If d = µ, then using the same c and the sets X and Y as before, since eℓpRµ = 0 for  any ℓ, we observe that  ψ([apRµ, b, apRµ]b) = (a∗, pRµ−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, b2, a∗, pRµ−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, b−1, a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, 1),  and hence if p is odd, the same argument follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For p = 2, the same argument can  clearly be used upon replacing c with  ψ−1  µ+1  �  (1, 2(µ+1)n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  , 1, [a2Rµ , b, a2Rµ+1])  �  ,  provided Rµ < n − 1, since then  ψ([a2Rµ , b, a2Rµ+1]b) = (a∗, 2Rµ−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, b, a∗, 2Rµ−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, b, a∗, 2Rµ−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, b−1, a∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a∗, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  However, for each of these finite groups G, one would only obtain finitely many (Beauville)  quotients G/StG(k), for k ≥ mG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  In the next section, we identify a class of finite GGS-groups where no quotient by a level  stabiliser is Beauville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Finite GGS-groups acting on the pn-adic tree  We define the subclass E of finite GGS-groups to consist of:  (i) the GGS-groups acting on the pn-adic tree, for p any prime and n ≥ 2, with  t0 = R0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , tm0,0−1 = Rm0,0−1 and Rm0,0 = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' and  BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS  15  (ii) the GGS-groups acting on the 2n-adic tree with R0 = n − 1, S[0] ≡ 0 (mod pn),  and e2n−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  We show here that quotients of the GGS-groups G ∈ E by level stabilisers do not yield  Beauville groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We first consider the subfamily (i) of groups in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G ∈ E be a GGS-group acting on the pn-adic tree for n ≥ 2, and  assume that t0 = R0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , tm0,0−1 = Rm0,0−1 and Rm0,0 = n, where t0, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , tm0,0−1 are  defined according to r = s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then the quotient G/StG(k) is not a Beauville group for  all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' The result is clear for k = 1 since G/ StG(1) is cyclic, so we assume k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let  {x1, x2} and {y1, y2} be two systems of generators for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' At least one of x1, x2, x1x2, call  it z1, must be in the coset aibjG′ for i, j ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Likewise for y1, y2, y1y2, and call  it z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1 and its proof, the order of ab is pt(0,0) if k ≥ m0,0 + 3, and p2n−R0 if  k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For 3 ≤ k ≤ m0,0 + 1, the order of ab is at least  pnpn−R0pn−R1 · · · pn−Rk−2,  and for k = m0,0 + 2, the order of ab is at least  pnpn−R0pn−R1 · · · pn−Rm0,0−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Suppose first that k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For i, j ̸≡ 0 (mod p) and w ∈ G′  2, as  φ2((aibjw)pn) = (ajS[0], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ajS[0])  it follows that ⟨zp2n−R0−1  1  ⟩ = ⟨zp2n−R0−1  2  ⟩, which gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Now let k ≥ m0,0 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Here, writing t = t(0, 0), we claim that  ⟨(ab)pt−1⟩ = ⟨(aibjw)pt−1⟩ ∼= Cp  for all i, j ̸≡ 0 (mod p) and w ∈ G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  To prove the claim, we first note, writing ψ(w) = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , wpn), that the product of all  the components w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , wpn, irrespective of the order, has zero total exponent in a, and  in b, and if written as  (bβ1)aα1 · · · (bβd)aαd  for some d ≥ 2 with β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , βd ∈ Z/pn−R0Z and α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , αd ∈ Z/pnZ, then we have αi ≡ 0  (mod pR0) for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Hence, each component of ψ  �  (aibjw)pn�  is of the form  ajS[0]bjv0,  for some v0 ∈ ⟨apR0, b⟩′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' We note that, for ω ∈ A with ω ̸≡ 0 (mod pR0),  ϕω  �  (ajS[0]bjv0)pn−R0  �  = ϕω  �  (aS[0]b)jpn−R0  �  = ajγω  where  γω =  pn−R0−1  �  ℓ=0  eω+ℓpR0,  and for ω ∈ A with ω ≡ 0 (mod pR0),  ϕω  ��  (ajS[0]bjv0)pn−R0�afpR0 �  = ajS[R0]bjv1  for v1 ∈ ⟨apR1, b⟩′ and f ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn−R0 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  16  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' DI DOMENICO, S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' G¨UL, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' THILLAISUNDARAM  Hence, continuing in this manner, we see that it suffices to compare  (ajS[Rµ−2]bjvµ−1)pn−Rµ−1  with  (aS[Rµ−2]b)pn−Rµ−1,  where µ = m0,0 and vµ−1 ∈ ⟨apRµ−1, b⟩′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Recalling also that Rµ = n, equivalently the  components of ψ(b) in positions pRµ−1, 2pRµ−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pn − pRµ−1 are trivial, we have  ψ  �  (ajS[Rµ−2]bjvµ−1)pn−Rµ−1�  = (ajδ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a  jδ  pRµ−1 −1, bj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ajδ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a  jδ  pRµ−1 −1, bj)  where, for q ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , pRµ−1 − 1},  δq =  pn−Rµ−1−1  �  ℓ=0  eq+ℓpRµ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  As  ψ  �  (aS[Rµ−2]b)pn−Rµ−1�  = (aδ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a  δ  pRµ−1 −1, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , aδ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a  δ  pRµ−1 −1, b),  and, from the above computations, we deduce that all other non-trivial labels in the  portraits of (ab)jpt−1 and (aibjw)pt−1 are equal powers of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Hence, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Thus, we have ⟨zpt−1  1  ⟩ = ⟨zpt−1  2  ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  For 3 ≤ k ≤ µ + 2, writing pˆt for the order of ab modulo StG(k), the analogous result  ⟨zpˆt−1  1  ⟩ = ⟨zpˆt−1  2  ⟩ follows similarly from the component-wise description of  (ajS[0]bjv0)pn−R0, (ajS[R0]bjv1)pn−R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , (ajS[Rk−4]bjvk−3)pn−Rk−3,  where we recall that R−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  □  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let G be a periodic GGS-group acting on the 2n-adic tree for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Suppose that R0 = n − 1 and S[0] ≡ 0 (mod pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then, the quotient G/StG(k) is not a  Beauville group for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='5, we only need to consider k ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For k = 1 the result is clear  since Beauville groups are not cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' For the remaining values of k, let X = {x1, x2, x1x2}  and Y = {y1, y2, y1y2} be two systems of generators for Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Assume now k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then there exists some z1 ∈ X with z1 ≡ ai (mod G′  2) for some  odd i, and there exists z2 ∈ Y with z2 ≡ aj (mod G′  2) for some odd j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Noting that the  order of both z1 and z2 is 2n, we now prove that (aiw)2n−1 is conjugate to (ajv)2n−1, for  w, v ∈ G′  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  We observe that StG2(1), and hence G′  2, is elementary abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Furthermore for each  odd i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 2n − 1} we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1)  G′  2 ≤ ⟨[ai, b], [ai, b, ai], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , [ai, b, ai, 2n−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai]⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  compare Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='6 and the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' It then follows from [17, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='32(i)]  that  (aiw)2n−1 = ai2n−1[w, ai, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1) there exist j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , j2n−1 ∈ {0, 1} such that w can be expressed as:  w = [ai, b]j1[ai, b, ai]j2 · · · [ai, b, ai, 2n−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai]j2n−1  Now for h ∈ StG2(1), bearing in mind that StG2(1) is elementary abelian, we obtain  from [17, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='32(ii)] that [ai2n−1, h] = [ai, h, ai, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai] in G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Thus, setting h = bj1[ai, b]j2 · · · [ai, b, ai, 2n−3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai]j2n−1 we deduce that  [ai2n−1, h] = [w, ai, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  BEAUVILLE STRUCTURES FOR QUOTIENTS OF GENERALISED GGS-GROUPS  17  Indeed,  [ai2n−1, h]  = [ai, bj1, ai, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai][ai, [ai, b]j2, ai, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai] · · · [ai, [ai, b, ai, 2n−3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai]j2n−1, ai, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai]  = [[ai, b]j1, ai, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai][[ai, b, ai]j2, ai, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai] · · · [[ai, b, ai, 2n−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai]j2n−1, ai, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai]  = [w, ai, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , ai]  where the second equality holds since StG2(1)′ is trivial and [a, x] = [x, a] for all x ∈  StG2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Thus we have  (aiw)2n−1 = ai2n−1[ai2n−1, h] = (ai2n−1)h = (a2n−1)h  where the last equality holds since a has order 2n and i = 1 + 2d for some d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' This  proves that for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 2n − 1} and w ∈ G′  2 the element (aiw)2n−1 is conjugate  to a2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Hence z1 and z2 are conjugate and G2 is not a Beauville group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Now suppose StG(3) ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Then there exists some z1 ∈ X with z1 ≡ aib (mod G′  3) for  some odd i, and there exists z2 ∈ Y with z2 ≡ ajb (mod G′  3) for some odd j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Observe that  for w ∈ G′  3, we have  φ3  �  (aibw)2n�  =  �  baλ1·2n−1  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , baλ2n ·2n−1�  where λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , λ2n ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Since by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='5 the elements b and ba2n−1  coincide modulo  StG(2), we further have  φ3((aibw)2n) = (b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  This proves that z 2n  1  and z 2n  2  are equal, and thus G3 is not a Beauville group when  StG(3) ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  We now consider the remaining cases, where here Gk = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' As before there exists some  z1 ∈ X with z1 ≡ aib (mod G′) for some odd i, and there exists z2 ∈ Y with z2 ≡ ajb  (mod G′) for some odd j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Therefore it suffices to show, for δ ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , 2n−1 − 1} and  w ∈ G′, that (a1+2δbw)2n is conjugate to (ab)2n in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Recall that (G′)2 ≤ StG(1)′ and that StG(1)′ is elementary abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Hence, from [17,  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='32(i)] we have  (ab)2n = [b, a, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]2[b, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]c  for some c ∈ γ2n(G) ∩ StG(1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Let i1, i2, i3 ≥ 2n be such that  �  [b, a, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]2, a  �  ∈ γi1(G)\\γi1+1(G)  [b, a, 2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=', a] ∈ γi2(G)\\γi2+1(G)  c ∈ γi3(G)\\γi3+1(G),  and set  ℓ := min{i1, i2, i3 + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  In other words, the integer ℓ is such that [(ab)2n, a] ∈ γℓ(G)\\γℓ+1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Now  (a1+2δbw)2n = [bw, a1+2δ, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a1+2δ]2[bw, a1+2δ, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a1+2δ]cd,  where d ∈ γi3+1(G) is obtained as follows: starting with the commutator c, which is a  product of commutators in b and a of weight at least 2n, with weight exactly 2 in b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  compare [17, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='32(i)], we have that cd = �c, where �c is obtained from c by replacing  all occurrences of b with bw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Here we have also used the standard identities [xy, z] =  [x, z]y[y, z] and [x, yz] = [x, z][x, y]z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  18  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' DI DOMENICO, S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' G¨UL, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' THILLAISUNDARAM  Next, using these standard identities together with Lemmata 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='7, we have  (a1+2δbw)2n ≡ [bw, a1+2δ, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a1+2δ]2[bw, a1+2δ, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a1+2δ]c  (mod γℓ(G))  ≡ [b, a, 2n−1−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]2[b, a, 2n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' , a]c  (mod γℓ(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Therefore  (a1+2δbw)2n ≡ (ab)2n  (mod γℓ(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  In other words, we have (a1+2δbw)2n = (ab)2nv for v ∈ γℓ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='8, we have  that v = [(ab)2n, g] for some g ∈ G, and hence we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  □  References  [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Barker, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Boston, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Fairbairn, A note on Beauville p-groups, Experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 21 (2012),  298–306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Barker, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Boston, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Peyerimhoff, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Vdovina, An infinite family of 2-groups with mixed  Beauville structures, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Notices 11 (2015), 3598–3618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Bartholdi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Grigorchuk and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' ˘Suni´c, Branch groups, in Handbook of algebra 3, North-Holland,  Amsterdam, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Catanese, Fibered surfaces, varieties isogenous to a product and related moduli spaces, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' 122 (2000), 1–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Di Domenico, Some questions regarding groups of automorphisms of primary trees, PhD thesis,  University of Trento and University of the Basque Country, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content='  [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Di Domenico, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
+page_content=' Fern´andez-Alcober, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E1T4oBgHgl3EQfHAM0/content/2301.02920v1.pdf'}
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+MNRAS 000, 1–20 (2022)
+Preprint 27 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Spectroscopy of CASSOWARY gravitationally-lensed galaxies in
+SDSS: characterisation of an extremely bright reionization-era
+analog at 𝑧 = 1.42
+Ramesh Mainali1,2,3★, Daniel P. Stark2, Tucker Jones4†, Richard S. Ellis5, Yashar D. Hezaveh6,7,
+& Jane R. Rigby1
+1 Observational Cosmology Lab, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
+2 Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ, USA
+3Department of Physics, The Catholic University of America, Washington, DC 20064, USA
+4 Department of Physics, University of California Davis, 1 Shields Avenue, Davis, CA 95616, USA
+5 Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, UK
+6Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
+7Département de Physique, Université de Montréal, Montreal, Quebec, Canada H3T 1J4
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We present new observations of sixteen bright (𝑟 = 19 − 21) gravitationally lensed galaxies at
+𝑧 ≃ 1−3 selected from the CASSOWARY survey. Included in our sample is the 𝑧 = 1.42 galaxy
+CSWA-141, one of the brightest known reionization-era analogs at high redshift (g=20.5), with
+a large sSFR (31.2 Gyr−1) and an [OIII]+H𝛽 equivalent width (EW[OIII]+H𝛽=730 Å) that is
+nearly identical to the average value expected at 𝑧 ≃ 7 − 8. In this paper, we investigate the
+rest-frame UV nebular line emission in our sample with the goal of understanding the factors
+that regulate strong CIII] emission. Whereas most of the sources in our sample show weak
+UV line emission, we find elevated CIII] in the spectrum of CSWA-141 (EWCIII]=4.6±1.9 Å)
+together with detections of other prominent emission lines (OIII], Si III], Fe II★, Mg II). We
+compare the rest-optical line properties of high redshift galaxies with strong and weak CIII]
+emission, and find that systems with the strongest UV line emission tend to have young stellar
+populations and nebular gas that is moderately metal-poor and highly ionized, consistent
+with trends seen at low and high redshift. The brightness of CSWA-141 enables detailed
+investigation of the extreme emission line galaxies which become common at 𝑧 > 6. We find
+that gas traced by the CIII] doublet likely probes higher densities than that traced by [OII] and
+[SII]. Characterisation of the spectrally resolved Mg II emission line and several low ionization
+absorption lines suggests neutral gas around the young stars is likely optically thin, potentially
+facilitating the escape of ionizing radiation.
+Key words: galaxies: evolution - galaxies: formation - galaxies: high-redshift
+1
+INTRODUCTION
+Over the last decade, much progress has been made in our under-
+standing of galaxies in the first billion years of cosmic time (for a
+review, see Stark 2016). Deep infrared imaging has uncovered thou-
+sands of photometrically-selected star forming systems thought to
+lie in the redshift range 6 < 𝑧 < 9 (e.g., McLure et al. 2013; Finkel-
+stein et al. 2014; Bouwens et al. 2015, 2021; Livermore et al. 2017;
+Ishigaki et al. 2018), providing a census of UV-selected galaxies
+throughout the reionization era. The spectral energy distributions
+(SEDs) point toward a population undergoing rapid stellar mass
+★ E-mail: ramesh.mainali@nasa.gov
+growth with blue UV continuum spectral slopes (e.g., Rogers et al.
+2013; Bouwens et al. 2014), low stellar masses and large specific star
+formation rates (Stark et al. 2013b; González et al. 2014; Grazian
+et al. 2015; Salmon et al. 2015; Curtis-Lake et al. 2016).
+In the last five years, our first constraints on the nebular emis-
+sion properties of galaxies at these early epochs have emerged.
+Spitzer/IRAC photometry suggests that nearly half of the UV-
+selected galaxies at 𝑧 ≃ 7 have extremely large [OIII]+H𝛽 equiv-
+alent widths (EWs) (Labbé et al. 2013; Smit et al. 2014, 2015;
+Roberts-Borsani et al. 2015; De Barros et al. 2017; Endsley et al.
+2020), indicating that very recent (∼< 50 Myr) activity powers the
+UV and optical luminosity, as expected for galaxies undergoing
+rapidly rising star formation histories. Roughly 20% of the popula-
+© 2022 The Authors
+arXiv:2301.11264v1  [astro-ph.GA]  26 Jan 2023
+
+2
+Mainali et al.
+tion have yet more intense rest-optical nebular emission ([OIII]+H𝛽
+EW > 1000 Å), indicating an extremely young stellar population
+(<10 Myr) is dominating the light, as expected for systems that have
+undergone a recent burst of star formation. Since such extreme emis-
+sion line galaxies (EELGs) are very rare at lower redshifts (Boyett
+et al. 2022), we lack a detailed understanding of the gas and ionizing
+agents in typical reionziation-era systems.
+Ground-based near-infrared spectrographs offer a path toward
+progress at the highest redshifts, providing access to the rest-frame
+ultraviolet where a suite of valuable diagnostic lines are situ-
+ated. The first deep spectra have revealed strong nebular emission
+from high ionization metal species ([CIII],CIII]𝜆𝜆1907,1909 Å,
+OIII]𝜆𝜆1660,1666 Å, CIV𝜆𝜆1548,1550 Å), a significant departure
+from what is commonly seen at lower redshifts (Stark et al. 2015b,a,
+2017; Mainali et al. 2017; Laporte et al. 2017; Mainali et al. 2018;
+Hutchison et al. 2019; Topping et al. 2021). The detection of these
+lines reveals gas under extreme ionization conditions and points to
+a population of intense ionizing agents, potentially AGN in some
+cases (Laporte et al. 2017; Mainali et al. 2018) and low metallic-
+ity massive stars in others (Mainali et al. 2017; Stark et al. 2017;
+Berg et al. 2019, 2021). The detection of CIV𝜆𝜆1548,1550 Å and
+[CIII],CIII]𝜆𝜆1907,1909 Å may further indicate a higher fraction
+of ionizing photons escape from a galaxy(Schaerer et al. 2022).
+While the equivalent width distribution in the total population is
+still subject to limited statistics (Mainali et al. 2018), it appears that
+both the gas and ionizing sources at 𝑧 ∼> 6 are often significantly
+different from that in galaxies at 𝑧 ≃ 2 − 3.
+The presence of strong rest-UV nebular emission in a subset of
+𝑧 ∼> 6 galaxies bodes well for future studies of the reionization era.
+For the next-generation 25-30m ground based optical/infrared ob-
+servatories (which are limited to constraints on the rest-frame UV at
+𝑧 > 6), these lines may provide the only way in which early galaxies
+can be studied spectroscopically. While typically fainter than the
+strong lines in the rest-optical, the suite of emission lines in the
+far-UV provide unique diagnostic power of the ionizing spectrum
+and gas physical conditions (i.e, density, temperature, ionization).
+Meanwhile in the near-UV, the resonant nature of the nebular Mg
+II emission line makes it an ideal probe of the neutral gas opacity
+in early galaxies, potentially providing an indirect indicator of LyC
+leakage at 𝑧 ∼> 6 (e.g., Henry et al. 2018; Chisholm et al. 2020; Xu
+et al. 2022; Izotov et al. 2022; Seive et al. 2022). The advantage of
+Mg II relative to Ly𝛼 is that it is not obscured by the neutral IGM
+at very high redshifts.
+To ensure that the faint rest-UV spectra provide reliable phys-
+ical diagnostics from these future facilities, it is important that we
+understand the gas conditions and stellar populations which support
+prominent UV line emission. With current facilities, this is most eas-
+ily done at lower redshifts where rest-frame optical emission lines
+which constrain the gas-phase metallicity and ionization parameter
+are observable with ground-based facilities. Over the last few years,
+a wide range of observations have been conducted with the goal of
+better understanding the physics regulating rest-UV emission line
+spectra. These studies have demonstrated that prominent CIII] ap-
+pears to be a fairly ubiquitous feature in the rest-UV spectra of dwarf
+star forming galaxies (Erb et al. 2010; Christensen et al. 2012; Stark
+et al. 2014; Rigby et al. 2015; Vanzella et al. 2016, 2017; Senchyna
+et al. 2017; Berg et al. 2018; Mainali et al. 2020; Du et al. 2020;
+Schmidt et al. 2021; Tang et al. 2021; Llerena et al. 2021; Berg et al.
+2022; Mainali et al. 2022), reflecting the large electron temperatures
+associated with metal poor gas and the hard ionizing spectrum of
+young, low metallicity massive stars.
+Here we seek to build on this progress, improving our un-
+derstanding of the connection between UV line emission and the
+physical properties of the massive stars and ionized gas. Over the
+last decade, we have utilized a wide range of ground-based facilities
+(LBT, Keck, Magellan) to obtain deep spectra and imaging of some
+of the brightest known (g=19-21) 𝑧 ≃ 1.5 − 3 gravitationally lensed
+galaxies within Sloan Digital Sky Survey (SDSS). This sample con-
+tains galaxies with a range of physical properties, including two of
+the brightest known systems with the large specific star formation
+rates (and extreme emission lines) that are typical in the reioniza-
+tion era. Central to this observational campaign are newly-acquired
+optical spectra from the Echellette Spectrograph and Imager (ESI;
+Sheinis et al. 2002) on the Keck II telescope (see Jones et al. 2018
+for more details). The ESI spectra provide moderate resolution (R=
+6300) rest-UV spectra, enabling a unique exploration of the na-
+ture of the massive stars and the physical conditions of the ionized
+gas. We supplement the Keck observations (9 galaxies) with optical
+spectra from LBT (1 galaxy) and MMT (6 galaxies), near-infrared
+spectra from Magellan, and new optical and near-infrared imaging.
+The large continuum brightness of the SDSS lensed galaxies
+offers several distinct advantages with respect to the much fainter
+low mass galaxies that are typical of cluster fields imaged by HST
+(e.g., Christensen et al. 2012; Stark et al. 2014; Vanzella et al.
+2017). First, it enables improved constraints on the ionized gas phys-
+ical conditions (metallicity, density, ionization parameter) through
+detection of the full suite of rest-optical strong emission lines as
+well as occasionally allowing detection of multiple temperature and
+density-sensitive lines, providing a more comprehensive view of the
+conditions in the gas and the most important factors regulating the
+rest-UV line spectra. Second, the large continuum S/N in the rest-
+UV spectra makes it possible to detect very weak rest-UV nebular
+lines. This enables the rest-UV lines to be characterized in indi-
+vidual galaxies for a wide range of gas conditions, not only in the
+most extreme and metal-poor systems. This will provide a valuable
+control sample for our analysis, offering insight into what factors
+are most important in regulating rest-UV emission line spectra. This
+insight will be critical in assessing the feasibility of detecting and
+characterizing lines in the rest-frame far and near-UV with future
+facilities. In this paper, we will focus primarily on CIII] emission,
+comparing measured line strengths to other empirical and model-
+based quantities (i.e., gas-phase metallicity, optical line ratios with
+the goal of understanding the factors regulating the CIII] strength.
+The paper is organized as follows. We present the new imaging
+and spectra and discuss population synthesis modeling of broadband
+SEDs in §2. In §3, we provide our results on individual galaxies.
+We discuss implications of our spectra of a reionization-era analog
+in §4 and summarize our findings in §5.
+Throughout the paper, we adopt a Λ-dominated, flat universe
+with ΩΛ = 0.7, Ω𝑀 = 0.3 and H0 = 70 km s−1 Mpc−1. All mag-
+nitudes in this paper are quoted in the AB system and equivalent
+widths (EW) are given in rest-frame, unless stated otherwise.
+2
+OBSERVATIONS AND ANALYSIS
+2.1
+Sample Selection
+The galaxies studied in this paper were originally identified using
+a search algorithm that identifies blue arcs surrounding red early
+type galaxies in Sloan Digital Sky Survey (SDSS) imaging as part
+of the Cambridge And Sloan Survey Of Wide ARcs on the skY
+(CASSOWARY; e.g., Belokurov et al. 2007, 2009). The most recent
+catalog of CASSOWARY lensed galaxies was presented in Stark
+MNRAS 000, 1–20 (2022)
+
+3
+Object
+RA
+DEC
+𝑧𝑠𝑝𝑒𝑐
+mAB
+Dates
+Rest-UV Coverage
+texp
+PA
+MUV
+𝜇
+Instrument
+(Å)
+(ksec)
+(deg)
+CSWA-165
+01:05:19.65
++01:44:56.4
+2.127
+21.1
+12 Dec 2012
+1020-2560
+3.6
+5.0
+-22.0
+5.4𝑎
+MMT/BCS
+CSWA-116
+01:43:50.13
++16:07:39.0
+1.499
+20.8
+29 Sep 2011
+1280-3200
+2.7
+105
+-20.9
+10.7𝑎
+MMT/BCS
+CSWA-103
+01:45:04.18
+-04:55:42.7
+1.959
+22.1
+8-10 Nov 2012
+1350-3425
+25
+115
+-20.9
+4.7𝑎
+Keck/ESI
+CSWA-164
+02:32:49.97
+-03:23:29.3
+2.512
+19.9
+8-10 Nov 2012
+1140-2880
+28
+158
+-22.0
+20.8𝑎
+Keck/ESI
+CSWA-11
+08:00:12.37
++08:12:07.0
+1.409
+21.1
+24 March 2012
+1330-3280
+1.8
+60
+-22.3
+1.9𝑐
+MMT/BCS
+CSWA-139
+08:07:31.51
++44:10:48.5
+2.536
+22.1
+23 Mar 2012
+905-2260
+3.6
+80
+-20.8
+3.8𝑎
+MMT/BCS
+CSWA-141
+08:46:47.53
++04:46:09.3
+1.425
+20.5
+8 Nov 2012
+1655-4180
+5.2
+190
+-22.1
+5.5𝑎
+Keck/ESI
+. . .
+. . .
+. . .
+. . .
+. . .
+11 Nov 2015
+1440-2140
+3.6
+190
+. . .
+. . .
+LBT/MODS
+CSWA-19
+09:00:02.64
++22:34:04.9
+2.032
+20.9
+9-10 Nov 2012
+1320-3330
+20
+86
+-21.9
+6.5𝑎
+Keck/ESI
+CSWA-40
+09:52:40.22
++34:34:46.1
+2.189
+21.4
+5-6 Mar 2013
+1255-3175
+16.2
+70
+-22.3
+3.2𝑎
+Keck/ESI
+CSWA-2
+10:38:43.58
++48:49:17.7
+2.196
+20.9
+5 Mar 2013
+1250-3125
+7.2
+17
+-22.3
+8.4𝑎
+Keck/ESI
+CSWA-16
+11:11:03.68
++53:08:54.9
+1.945
+22.1
+24 Mar 2012
+1085-2715
+2.7
+267
+-20.8
+3.9𝑑
+MMT/BCS
+CSWA-38
+12:26:51.69
++21:52:25.5
+2.925
+21.3
+6 Mar 2013
+1020-2580
+10.8
+130
+-20.2
+40𝑏
+Keck/ESI
+CSWA-13
+12:37:36.20
++55:33:42.9
+1.864
+20.3
+24 Mar 2012
+1120-2790
+1.8
+320
+-23.0
+1.9𝑑
+MMT/BCS
+CSWA-39
+15:27:45.02
++06:52:33.9
+2.762
+20.9
+5-6 Mar 2013
+1065-2695
+18
+105
+-21.5
+15𝑏
+Keck/ESI
+CSWA-128
+19:58:35.65
++59:50:53.6
+2.225
+20.6
+8-10 Nov 2012
+1240-3140
+16.3
+60
+-21.9
+10𝑐
+Keck/ESI
+CSWA-163
+21:58:43.68
++02:57:30.2
+2.081
+22.4
+30 Sep 2011
+1035-2600
+1.8
+20
+-20.4
+6.5𝑎
+MMT/BCS
+𝑎Stark et al. (2013a), 𝑏Koester et al. (2010), 𝑐Leethochawalit et al. (2016), 𝑑This work.
+Table 1. Summary of rest-frame UV observations of our bright gravitationally lensed CASSOWARY galaxy sample. New ultra-deep moderate resolution optical
+spectra have been obtained for nine of these sources using ESI on Keck. We also include an additional seven sources from Stark et al. (2013a) with high quality
+MMT blue channel spectra. From left to right, we present the object ID in the CASSOWARY catalog, the RA and DEC, the spectroscopic redshift, apparent
+magnitude, the date the optical spectra were obtained, the rest-UV wavelength coverage provided by the optical spectra, the exposure time and position angle
+of the optical spectra, the absolute magnitude of the arc in the rest-UV, the magnification factor provided by gravitational lensing, and the optical spectrograph
+utilized. Further details are presented in §2.
+et al. (2013a) following a large spectroscopic campaign aimed at
+obtaining redshifts of the source and lens galaxies. The catalog
+of galaxies released in Stark et al. (2013a) contains more than 50
+gravitationally-lensed galaxies in SDSS. Optical magnitudes of this
+sample are in the range 19.6 < 𝑟 < 22.3.
+Here we present the results from a large observational invest-
+ment targeting sixteen of these bright lensed sources with Keck,
+Magellan, LBT, and MMT (see Table 1). Object identifiers from
+the CASSOWARY survey are abbreviated as “CSWA" in the table
+and subsequent discussion. A mosaic of the galaxies considered in
+this paper is shown in Figure 1. In the following subsections, we
+describe the imaging and spectral datasets that we have obtained
+for this paper. The galaxies considered here were selected from the
+larger sample of Stark et al. (2013a) based on the brightness of
+the continuum. We also preferentially targeted sources at redshifts
+which place rest-optical lines in regions of significant atmospheric
+transmission in the near-IR.
+2.2
+Spectroscopy
+2.2.1
+Optical Spectroscopy
+We have acquired deep Keck/ESI optical spectra of nine galaxies
+from the Stark et al. (2013a) sample (see Table 1 for details), en-
+abling robust constraints to be placed on the strength of nebular UV
+metal line emission. The positioning of the ESI slit on the lensed
+galaxies is shown in Figure 1. The spectra were obtained in two
+observing runs between November 2012 and March 2013. The ESI
+spectra cover observed wavelengths between 4000 to 10100 Å, pro-
+viding rest-frame spectral coverage that typically ranges between
+1300 and 2000 Å (see Table 1). A slit width of 0.′′75 was used, pro-
+viding a resolving power of R=6300 (FWHM=48 km s−1). The data
+were reduced using the ESIRedux code written by J. X. Prochaska.
+Sky subtraction is performed following the bias subtraction and flat
+fielding. The 1D spectra are then extracted using a boxcar aper-
+ture matched to the spatial extent of the arc. When multiple lensed
+images appear on the slit, the traces are extracted separately and
+then combined to maximize the S/N. The continuum is generally
+well detected, with S/N ≃ 10 per resolution element at 6200 Å.
+Example spectra are shown in Figure 2. A full description of the
+ESI observations and spectral reduction is presented in Jones et al.
+(2018).
+One source in the ESI sample, CSWA-141, was also observed
+with the Multi Object Double Spectrograph (MODS, Pogge et al.
+2010) on the LBT. MODS provides bluer wavelength coverage than
+ESI, allowing constraints to be placed on emission from CIV, OIII],
+and He II. We used a long slit of width 0.′′8 with the 400 lines/mm
+grating, providing spectral resolution of ∼ 3 .
+We also include seven other galaxies from Stark et al. (2013a)
+for which the MMT Blue Channel Spectrograph (BCS) discovery
+spectra have sufficient continuum S/N to characterize the rest-UV
+metal emission lines. The MMT BCS data were obtained with the
+300 lines/mm grating, providing total spectral coverage of ∼ 5300
+. For the slit width of 1.′′0 used in the observations, the typical
+spectral resolution is ∼ 6.5 . More details of the MMT observations
+are provided in Stark et al. (2013b).
+In total, the Keck, LBT, and MMT data provides rest-UV spec-
+tra for 16 galaxies. Our aim is to characterize CIII] emission fea-
+ture, typically the strongest rest-UV emission line, along with other
+rest-UV emission features in the whole sample. Using redshift pre-
+sented in Jones et al. (2018) for Keck/ESI data and Stark et al.
+(2013a) for MMT data, we first searched for CIII] emission in indi-
+vidual galaxy spectrum. The line is spectrally resolved in the Keck
+spectra, whereas it remained unresolved in the MMT spectra. For
+the resolved CIII] doublet, we measured emission line fluxes and
+associated errors (for error spectrum) in individual components by
+directly integrating the flux levels within ±1 Å of individual line
+MNRAS 000, 1–20 (2022)
+
+4
+Mainali et al.
+CSWA-141
+CSWA-2
+CSWA-40
+CSWA-19
+CSWA-128
+CSWA-165
+CSWA-13
+CSWA-164
+CSWA-163
+CSWA-11
+CSWA-128
+CSWA-16
+CSWA-139
+CSWA-38
+CSWA-39
+CSWA-165
+CSWA-116
+CSWA-103
+CSWA-141
+CSWA-2
+CSWA-19
+CSWA-13
+CSWA-164
+CSWA-163
+CSWA-11
+CSWA-128
+CSWA-16
+CSWA-139
+CSWA-38
+CSWA-39
+Figure 1. Color images of sixteen gravitationally lensed galaxies. The name of each sources is presented at the top of each image, along with the slit position
+used for the observations. For seven galaxies (CSWA-141, CSWA-13, CSWA-139, CSWA-116, CSWA-16, CSWA-165, CSWA-164) the color images are
+created from g,r and i band images from LBT/MODS. The color images for four galaxies (CSWA-11,CSWA-163,CSWA-128,CSWA-103) are made with g,r
+and i bands images from LBT/LBC. Three color images (CSWA-38, CSWA-39, CSWA-40) are made using Keck/ESI images in V,R and I bands. For other two
+galaxies (CSWA-2, CSWA-19), color images are created using archival HST images. North is up and east is to the left. Each postage stamp is 25” × 25” in size.
+The orientation and centroid of the ESI or MMT slit is overlaid in each image.
+centers (rest-frame). When the line is unresolved, we computed
+total CIII] emission line fluxes and associated line flux errors di-
+rectly from the flux spectra and error spectra, respectively. If the
+line remained undetected at 3𝜎 level, we calculated upper limits by
+integrating error spectrum from 1905 Å to 1912 Å. The emission
+line equivalent width is then computed by dividing the line fluxes
+(or upper limits) by median continuum level measured on either side
+of the CIII] line. We then searched for any other rest-UV emission
+features in the spectra and characterized them when detected. Our
+sample includes 10 galaxies with CIII] detection where the equiva-
+lent widths range from 0.4 Å to 4.6 Å (see Table 2). We will come
+back to interpret the CIII] equivalent widths in §3.2, comparing the
+line strengths to optical line ratios and gas physical properties.
+2.2.2
+Near-Infrared Spectroscopy
+Near-infrared (NIR) spectra supplement the optical spectra de-
+scribed above, providing constraints on the metallicity, ionization
+parameter, and electron density of the ionized gas. Near-infrared
+spectroscopic analysis is limited to the subset of sources from Ta-
+ble 1 which are located at redshifts placing strong rest-optical emis-
+sion lines in spectral windows in which atmospheric transmission
+is near-unity. One of the sources in our sample (CSWA-2) was tar-
+MNRAS 000, 1–20 (2022)
+
+.5
+Object
+EW (Å)
+Ly𝛼
+[CIII]𝜆1907
+CIII]𝜆1909
+CIII]𝜆1908𝑎
+CSWA-141
+. . .
+2.3(1.3)
+2.3(1.4)
+4.6(1.9)
+CSWA-13
+5.4(2.1)
+. . .
+. . .
+4.4(0.9)
+CSWA-139
+. . .
+. . .
+. . .
+3.4(2.6)
+CSWA-2
+. . .
+2.0(1.6)
+1.1(0.9)
+3.1(1.6)
+CSWA-39
+11.8(6.2)
+0.6(0.1)
+0.5(0.1)
+1.1(0.2)
+CSWA-19
+. . .
+0.4(0.1)
+0.3(0.1)
+0.7(0.1)
+CSWA-38
+2.6(1.1)
+. . .
+0.4(0.1)
+CSWA-128
+. . .
+0.3(0.1)
+0.3(0.1)
+0.6(0.1)
+CSWA-103
+. . .
+0.3(0.1)
+0.2(0.1)
+0.5(0.1)
+CSWA-164
+2.0(0.5)
+0.2(0.1)
+0.2(0.1)
+0.4(0.1)
+CSWA-163
+. . .
+. . .
+. . .
+< 3.1
+CSWA-16
+. . .
+. . .
+. . .
+< 2.3
+CSWA-165
+. . .
+. . .
+. . .
+< 1.9
+CSWA-40
+. . .
+< 1.2
+< 1.2
+< 1.7
+CSWA-11
+. . .
+. . .
+. . .
+< 1.4
+CSWA-116
+. . .
+. . .
+. . .
+< 1.1
+𝑎 Total CIII] doublet.
+Table 2. Equivalent width measurements of rest-UV emission lines. The
+numbers within parentheses represent uncertainty. The upper limits are 3𝜎.
+Object
+Dates
+texp
+PA
+Observatory/
+(ksec)
+(deg)
+Instrument
+CSWA-165
+2014 June 22
+7.2
+120
+Magellen/FIRE
+CSWA-164
+2015 Nov 03
+4.8
+58
+Magellen/FIRE
+CSWA-11
+2015 Nov 04
+2.7
+-10
+Magellen/FIRE
+CSWA-141
+2012 Feb 15
+1.2
+280
+Magellen/FIRE
+CSWA-128
+2012 Nov 7
+3.6
+190
+LBT/LUCI
+CSWA-163
+2014 June 22
+9.0
+90
+Magellen/FIRE
+Table 3. Details of near-infrared spectroscopic observations obtained for this
+paper. From left to right, the columns denote the object ID, observation dates,
+exposure time, position angle of slit, and the observatory and instrument used
+to acquire near-IR spectroscopy. Further details are provided in §2.2.
+geted with Keck near-infrared spectroscopy in Jones et al. (2013).
+In observing runs between November 2012 and November 2015,
+we have obtained near-infrared spectroscopic observations of five
+additional sources (CSWA-141, CSWA-164, CSWA-165, CSWA-
+163, CSWA-11) using the Folded-port InfraRed Echellette (FIRE;
+Simcoe et al. 2008) on the Magellan Baade Telescope. We used
+FIRE in its echelle mode, providing continuous spectral coverage
+spanning 0.82-2.51 𝜇m. We adopted a slit width of 0.′′75, delivering
+a spectral resolution of 2.6 in the J band, 3.4 in the H band and 4.7
+in the K band. Spectroscopic data reduction was performed using
+standard routines in the FIREHOSE data reduction pipeline1.
+One additional galaxy (CSWA-128) was observed on 2012
+Nov 7 with the LUCI near-IR spectrograph on the Large Binocular
+Telescope (LBT). We used the N1.8 camera and 200_H+K grating in
+longslit mode. We first observed the lensed source with the HKSpec
+filter centered at 1.93 microns, providing spectral coverage from
+1.50 to 2.30 𝜇m. We also observed CSWA-128 with the zJSpec
+filter centered at 1.1 microns, providing spectral coverage between
+0.95 and 1.40𝜇m. A slit width of 1.′′0 was used, resulting in a spectral
+resolution of 16 . Spectroscopic data reduction was performed using
+standard IDL long slit reduction packages (see Bian et al. 2010 for
+1 wikis.mit.edu/confluence/display/FIRE/FIRE+Data+Reduction
+details). We summarize details of near-infrared spectroscopy of
+CASSOWARY galaxies in Table 3. Example spectra are shown in
+Figure 3.
+Emission line fluxes in the NIR spectra are measured using the
+IDL routine MPFITPEAK which computes line fluxes after fitting
+a gaussian model. In cases where the emission lines are partially
+affected by a skyline, we mask the contaminated region before fit-
+ting the emission line. Additionally, if one of the components of
+[OIII]𝜆𝜆4959,5007 is strongly affected by an emission line, we as-
+sume the theoretical line ratio of [OIII]𝜆5007/[OIII]𝜆4959=2.98
+(Storey & Zeippen 2000) in calculating the total [OIII] line flux.
+We calculate the impact of dust on the nebular lines using the
+Balmer decrement flux ratio of H𝛼/H𝛽. The observed line ratio is
+compared to the line ratio expected in absence of dust (H𝛼/H𝛽=2.86;
+Osterbrock & Ferland 2006) for case B recombination assuming
+T𝑒=10,000 K. In the two cases where the Balmer line ratios are not
+available, we use the stellar reddening inferred from the broadband
+data to estimate the nebular attenuation (see §2.5). We note that
+using stellar reddening for nebular attenuation correction doesn’t
+impact our results. We assume that the nebular gas attenuation
+is similar to the stellar continuum attenuation. We presented the
+observed and dust-corrected emission line measurements in Table 4.
+For one object (CSWA-141) where auroral [OIII]𝜆4363 line
+is detected, we calculated electron temperature using PyNeb (ver-
+sion 1.1.8; Luridiana et al. 2015) using emission line flux ratio of
+OIII]𝜆4363/[OIII]𝜆5007. For our calculations, we adopted the de-
+fault PyNeb atomic data sets. We assumed electron density of 250
+cm−3, which is typical to 𝑧 ∼ 2 galaxies (Sanders et al. 2016a). How-
+ever, we note that our assumed electron density value has negligible
+effect on the derived electron temperature in the low density regime
+(<103 cm−3). Since we don’t have T𝑒([OII]) sensitive emission line
+measurements, we followed the relation given by Izotov et al. (2006)
+for low metallicity to estimate the electron temperature in the O+2
+region. We use our electron temperature measurements to infer the
+direct oxygen abundance. We only use O+/H+ and O+2/H+ to com-
+pute oxygen abundance since the ionization states higher than O+2
+contributes significantly low at less than 1 per cent (Izotov et al.
+2006). O+/H+ and O+2/H+ are calculated from PyNeb using our
+T𝑒([OIII], T𝑒([OII], 𝑛𝑒 and emission line fluxes of [OII], H𝛽 and
+[OIII].
+We use PyNeb to calculate the electron density using the flux
+ratio of the [OII], [SII], and [CIII], CIII] doublets. The typical
+error in measurements is then calculated using the errors in emis-
+sion line fluxes. When an electron temperature measurement is not
+available, we assumed electron temperature of 10,000K following
+Sanders et al. (2016a) to adopt temperature dependent effective col-
+lision strengths. Adopting electron temperature of 7000K (15000K)
+instead would overestimate (underestimate) electron density by 15-
+20% which is effectively lower than density error from our line flux
+measurements.
+2.3
+Imaging
+Each of the CASSOWARY galaxies was discovered in SDSS imag-
+ing. In many cases, the photometric constraints from SDSS are
+unreliable owing to blending with neighbors and low S/N detec-
+tion of diffuse emission associated with the arcs. We have obtained
+deeper optical multi-band imaging for each of the galaxies discussed
+in this paper using cameras on the LBT and Keck. In order to better
+characterize the stellar populations, we have also obtained near-IR
+imaging sampling across the Balmer break for a subset of our tar-
+gets using the LBT, MMT and Keck. In the following subsections,
+MNRAS 000, 1–20 (2022)
+
+6
+Mainali et al.
+Line
+𝜆rest(Å)
+𝜆obs (Å)
+F(𝜆)/F(5007)
+I(𝜆)/I(5007)
+CSWA-141
+[OII]
+3727.13
+9039.2
+0.060 ± 0.002
+0.088 ± 0.003
+[OII]
+3729.92
+9045.9
+0.075 ± 0.004
+0.110 ± 0.006
+[NeIII]
+3869.66
+9384.9
+0.054 ± 0.017
+0.075 ± 0.023
+H𝛿
+4102.90
+9950.5
+0.030 ± 0.004
+0.039 ± 0.005
+H𝛾
+4341.58
+10529.4
+0.063 ± 0.003
+0.076 ± 0.004
+[OIII]
+4365.31
+10586.9
+0.019 ± 0.002
+0.023 ± 0.002
+H𝛽
+4862.55
+11792.9
+0.13 ± 0.003
+0.135 ± 0.003
+[OIII]
+4960.25
+12029.8
+0.338 ± 0.004
+0.342 ± 0.004
+[OIII]
+5008.27
+12146.3
+1.000
+1.000
+[SIII]
+6310.48
+15304.4
+0.007 ± 0.001
+0.005 ± 0.001
+[NII]
+6549.84
+. . .
+< 0.004
+< 0.002
+H𝛼
+6564.61
+15920.7
+0.529 ± 0.003
+0.386 ± 0.002
+[SII]
+6717.96
+16292.7
+0.017 ± 0.001
+0.013 ± 0.001
+[SII]
+6732.34
+16327.5
+0.016 ± 0.001
+0.012 ± 0.001
+CSWA-165
+[OII]
+3727.13
+11657.1
+0.56 ± 0.06
+1.01 ± 0.11
+[OII]
+3729.92
+11665.9
+0.63 ± 0.08
+1.13 ± 0.14
+[NeIII]
+3869.66
+. . .
+< 0.23
+< 0.38
+H𝛽
+4862.55
+15208.9
+0.46 ± 0.08
+0.49 ± 0.08
+[OIII]
+5008.27
+15665.2
+1.00
+1.00
+H𝛼
+6564.61
+20532.4
+2.25 ± 0.15
+1.40 ± 0.09
+[NII]
+6585.28
+20597.5
+0.55 ± 0.09
+0.34 ± 0.06
+CSWA-163
+[OII]
+3727.13
+11482.5
+0.28 ± 0.01
+0.53 ± 0.02
+[OII]
+3729.92
+11491.2
+0.35 ± 0.03
+0.66 ± 0.06
+[NeIII]
+3869.66
+. . .
+< 0.04
+< 0.07
+H𝛾
+4341.58
+13375.6
+0.06 ± 0.03
+0.08 ± 0.04
+H𝛽
+4862.55
+14981.5
+0.22 ± 0.03
+0.23 ± 0.03
+[OIII]
+4960.25
+15282.5
+0.35 ± 0.02
+0.35 ± 0.02
+[OIII]
+5008.27
+15429.9
+1.00
+1.00
+H𝛼
+6564.61
+20225.5
+1.12 ± 0.04
+0.67 ± 0.02
+[NII]
+6585.28
+20289.2
+0.18 ± 0.06
+0.11 ± 0.04
+CSWA-128
+[OII]
+3729.01
+12019.6
+0.28 ± 0.08
+0.40 ± 0.11
+[NeIII]
+3869.66
+12476. 4
+0.10 ± 0.03
+0.14 ± 0.04
+H𝛽
+4862.55
+15681.7
+0.24 ± 0.06
+0.25 ± 0.06
+[OIII]
+4960.25
+15997.7
+0.32 ± 0.04
+0.33 ± 0.04
+[OIII]
+5008.27
+16151.8
+1.00
+1.00
+[NII]
+6549.84
+21121.3
+0.05 ± 0.01
+0.04 ± 0.01
+H𝛼
+6564.61
+21170.3
+0.96 ± 0.05
+0.71 ± 0.04
+[NII]
+6585.28
+21237.2
+0.13 ± 0.02
+0.09 ± 0.01
+[SII]
+6717.96
+21664.9
+0.10 ± 0.02
+0.07 ± 0.01
+[SII]
+6732.34
+21711.1
+0.08 ± 0.04
+0.06 ± 0.02
+CSWA-164
+[OII]
+3727.13
+13090.9
+0.52 ± 0.11
+0.80 ± 0.17
+[OII]
+3729.92
+13100.4
+0.64 ± 0.04
+0.99 ± 0.0.07
+[OIII]
+5008.27
+17590.4
+1.00
+1.00
+H𝛼
+6564.61
+23064.8
+1.91 ± 0.09
+1.34 ± 0.06
+CSWA-11
+[OII]
+3727.13
+8979.9
+0.67 ± 0.13
+0.85 ± 0.16
+[OII]
+3729.92
+8986.6
+0.66 ± 0.09
+0.83 ± 0.11
+[NeIII]
+3869.66
+. . .
+< 0.43
+< 0.53
+[OIII]
+5008.27
+12068.7
+1.00
+1.00
+H𝛼
+6564.61
+15816.5
+1.08 ± 0.06
+0.89 ± 0.05
+Table 4. Rest-optical emission line measurements of six bright lensed
+Cassowary galaxies. Measurements are from Magellan/FIRE (CSWA-141,
+CSWA-165, CSWA-163, CSWA-164, CSWA-11) and LBT/LUCI (CSWA-
+128). Line flux measurements are quoted relative to [OIII]𝜆5007. We adopt
+[OIII]𝜆5007 as a reference line due to its high S/N. Three-sigma upper limits
+are reported for emission lines that are not detected.
+Object
+Observatory /
+Filters
+Dates
+texp
+Instrument
+Observed
+(sec)
+CSWA-165
+LBT/MODS
+g
+2014 Jan 27
+720
+LBT/MODS
+r, z
+2014 Jan 27
+360
+MMT/MMIRS
+J
+2015 Sep 30
+900
+MMT/MMIRS
+K
+2015 Sep 30
+1200
+CSWA-116
+LBT/MODS
+g
+2014 Jan 27
+720
+LBT/MODS
+r, z
+2014 Jan 27
+360
+MMT/MMIRS
+J,K
+2015 Oct 01
+600
+CSWA-103
+LBT/LBC
+g
+2015 Nov 15
+600
+LBT/LBC
+r, i
+2015 Nov 15
+300
+MMT/MMIRS
+J,K
+2015 Oct 01
+600
+CSWA-164
+LBT/MODS
+g
+2014 Jan 27
+720
+LBT/MODS
+r, z
+2014 Jan 27
+360
+CSWA-11
+LBT/LBC
+g
+2015 Nov 15
+600
+LBT/LBC
+r, z
+2015 Nov 15
+300
+Keck/ MOSFIRE
+K𝑠
+2015 Nov 30
+300
+CSWA-139
+LBT/MODS
+g
+2014 Jan 27
+720
+LBT/MODS
+r, z
+2014 Jan 27
+360
+MMT/MMIRS
+H
+2017 Jan 11
+900
+CSWA-141
+LBT/ MODS
+g
+2014 Jan 27
+720
+LBT/ MODS
+r, z
+2014 Jan 27
+360
+Keck/ MOSFIRE
+K𝑠
+2014 Apr 11
+300
+Keck/ MOSFIRE
+Y
+2015 Nov 30
+300
+Keck/ MOSFIRE
+J2
+2015 Nov 30
+300
+CSWA-40
+Keck/ESI
+V, R, I
+2013 Mar 06
+200
+CSWA-16
+LBT/MODS
+g, r
+2014 Jan 27
+360
+CSWA-38
+Keck/ESI
+V, R, I
+2013 Mar 06
+200
+CSWA-13
+LBT/MODS
+g
+2014 Jan 27
+720
+LBT/MODS
+r, z
+2014 Jan 27
+360
+LBT/LUCI
+K𝑠
+2014 Apr 13
+1200
+CSWA-39
+Keck/ESI
+V, R, I
+2013 Mar 06
+200
+CSWA-128
+LBT/LBC
+g
+2015 Nov 15
+600
+LBT/LBC
+r, i
+2015 Nov 15
+300
+Keck/ MOSFIRE
+K𝑠
+2014 Apr 11
+300
+CSWA-163
+LBT/LBC
+g
+2015 Nov 15
+600
+LBT/LBC
+r, i
+2015 Nov 15
+300
+MMT/MMIRS
+J
+2015 Sep 30
+900
+MMT/MMIRS
+K
+2015 Sep 30
+1200
+Table 5. New optical and near-infrared imaging of CASSOWARY lensed
+galaxies obtained with Keck, LBT, and MMT. From left to right, columns
+denote the object ID, observatory and instrument, filters utilized, dates of
+observations, and exposure time.
+we describe the optical and near-infrared imaging observations and
+analysis. Details of the new imaging observations are summarized
+in Table 5.
+2.3.1
+Optical imaging
+We secured images of seven galaxies from Table 1 using the Multi
+Object Double Spectrograph (MODS, Pogge et al. 2010) on LBT in
+January 2014. The dual channel mode of MODS provides images
+in two separate filters simultaneously over a field of view of 6 ×
+6 arc minutes. We used the blue and red channel to obtain g, r,
+and z-band images for CSWA-13, CSWA-16, CSWA-116, CSWA-
+139, CSWA-141, CSWA-164 and CSWA-165. Each arc was first
+observed in the g and r-band for 360 seconds and then in the g
+and z-band for an additional 360 seconds. For one source (CSWA
+16), we obtained imaging only in the g and r-bands. The sky was
+partly covered by clouds throughout the MODS observations and
+the seeing varied between 1.′′0 and 1.′′5. Optical images of three
+MNRAS 000, 1–20 (2022)
+
+7
+other galaxies (CSWA-39, CSWA-38, CSWA-40) were taken with
+ESI on Keck, providing a field of view of 2×3.5 arc minutes. These
+three sources were observed with V, R and I filters. There was some
+cloud during the observations, and the seeing was between 0.′′8 and
+1.′′2. For the two other sources in Table 1, CSWA-2 and CSWA-19,
+we make use of archival HST images (program 11974, PI: Allam).
+We summarize the details of the imaging observations in Table 5.
+The optical images were reduced using standard routines for
+flat fielding and image combination. The 5𝜎 typical limiting magni-
+tude in the g, r and z bands of MODS images are 25.5, 24.7 and 23.2
+respectively. Similarly, for the ESI images the typical 5𝜎 limiting
+magnitude in V, R and I bands are 22.7, 22.6 and 22.6 respectively.
+For the archival HST images, the typical 5𝜎 limiting magnitudes in
+F450W, F606W, and F814W filters are 26.3, 26.1 and 25.5 respec-
+tively. Each image was flux calibrated using stars that are isolated
+and well detected in SDSS imaging. The mosaic in Figure 1 shows
+the multi-color optical imaging obtained for each source.
+Before performing photometry on the lensed galaxies, we mod-
+eled the foreground lens using GALFIT (Peng et al. 2002) and
+subtracted its contribution to the lensed source. To calculate the ab-
+solute magnitude for the galaxies in our sample, we first measured
+the integrated flux in the g-band. After applying the appropriate
+magnification corrections (see §2.4), we arrive at the absolute UV
+magnitudes shown in Table 1. The values span MUV = −23.1 to
+MUV = −20.4, corresponding to 0.8-9 L★
+UV at 𝑧 ≃ 2 − 3 (Reddy &
+Steidel 2009; Parsa et al. 2016).
+2.3.2
+Near-infrared imaging
+We obtained near-infrared images of CSWA-141, CSWA-11 and
+CSWA-128 in the Ks-band using the Multi-Object Spectrometer for
+Infra-Red Exploration (MOSFIRE, McLean et al. 2012) on Keck I.
+Conditions were photometric with seeing of 0.′′5. We obtained 10
+dithered frames having 6.1×6.1 arc minute field of view, each with
+exposure time of 30 seconds. The 5𝜎 AB limiting magnitude in the
+MOSFIRE Ks-band images is 23.1. With the goal of constraining
+the equivalent width of [OIII] and H-beta emission, we observed
+CSWA-141 with the MOSFIRE J2 medium band filter. The filter
+spans 1.11-1.25 𝜇m, covering H-beta and [OIII].
+We secured a Ks-band image of CSWA-13 using the LUCI
+near-IR Spectrograph on the LBT when the seeing was 1.′′2. We
+utilized the N3.75 camera providing 4×4 arc minutes of field of view
+with a plate scale of 0.12 arcseconds per pixel. The near-IR imaging
+reduction was performed using standard IDL routines designed to
+flat field, sky-subtract and stack the dithered frames. Finally, we
+used isolated and unsaturated stars that are in the 2MASS catalog
+to flux calibrate the images. The 5𝜎 AB limiting magnitude in the
+LUCI Ks-band image is 23.4. For another four galaxies (CSWA-
+116, CSWA-165, CSWA-103, CSWA-163), near-IR imaging was
+obtained using the MMIRS instrument on the MMT. Each of the
+four sources was observed in the J and K bands. MMIRS provides
+imaging over a field of view of 6.9 × 6.9 arc minutes with a plate
+scale of 0.20 arcsec per pixel. The seeing was between 0.′′6-0.′′9
+throughout the observations. The typical 5𝜎 AB limiting magnitude
+in MMIRS J and K-band images are 22.9 and 22.8, respectively.
+2.4
+Lensing Magnification
+Derivation of stellar mass and intrinsic luminosity of lensed systems
+requires magnification correction. In Stark et al. (2013b), magni-
+fication factors were presented for eleven out of sixteen sources
+given in Table 1 and typically range between 𝜇=5 and 10. Three
+out of the five remaining sources have a published lens model. We
+adopted magnifications for CSWA-38 and CSWA-39 from Koester
+et al. (2010) and CSWA-11 from Leethochawalit et al. (2016).
+For the two remaining sources (CSWA-13 & CSWA-16), we
+follow a similar approach to Hezaveh et al. (2013). In brief, we
+assumed a symmetric Gaussian light distribution for the lensed
+source. We also tested the impact of adding a second Gaussian
+component to the source. The foreground lens is modeled as a
+Singular Isothermal Ellipsoid (SIE). Multiple lenses are allowed in
+the modeling procedure. To obtain the best fit model, we perform
+a Markov Chain Monte Carlo (MCMC) analysis to minimize 𝜒2
+in the image plane. In the case of CSWA-16, the observed data
+are better fit after introducing a second Gaussian component to the
+background source. The magnification factor is then calculated as
+the ratio of the lensed to unlensed flux. We derive a magnification
+of 𝜇 = 3.9 ± 0.3 for CSWA-16 and 𝜇 = 1.9 ± 0.2 for CSWA-13.
+2.5
+Stellar Population Synthesis Modeling
+Thirteen of the galaxies shown in Figure 1 have the necessary optical
+and near-IR imaging to derive physical properties from broadband
+SED fitting. For these systems, we infer the stellar mass, specific
+star formation rate and dust attenuation using the Bayesian galaxy
+SED modeling and interpreting tool BEAGLE tool (version 0.20.3;
+Chevallard & Charlot 2016). BEAGLE is based on the photoioniza-
+tion models of star-forming galaxies in Gutkin et al. (2016), com-
+bining the latest version of the Bruzual & Charlot (2003) stellar pop-
+ulation synthesis models with the photoionization code CLOUDY
+(Ferland et al. 2013).
+We fit the broadband photometric fluxes as well as CIII] equiv-
+alent widths. When relative fluxing between rest-UV and optical
+emission lines is possible, we include all emission lines in the fit-
+ting procedure. The models assume constant star formation where
+the maximum stellar age is allowed to vary freely between 5 Myr to
+the Universe age at the given redshift. We use Chabrier (2003) initial
+mass function and the Calzetti et al. (2000) extinction curve. The
+metallicity is allowed to vary in the range of -2.2≤log(Z/Z)⊙≤0.25
+assuming equivalent stellar and nebular metallicity (Z★=Z𝐼 𝑆𝑀).
+The redshift of all objects is fixed to their spectroscopic redshift
+given in Table 1. The ionization parameter (US; here defined as
+the ratio of ionizing-photon to gas densities at the edge of the
+Strömgren sphere) is varied in the range of -4.0≤𝑈S≤-1.0, and the
+dust-to-metal mass ratio spanned within the range of 𝜉𝑑=0.1-0.5.
+We adopt models with hydrogen density (nH=100 cm−3) and C/O
+abundance of 0.5 of solar value [(C/O)⊙ ≈0.44]. Finally, the prior
+on the V-band dust attenuation optical depths ( ˆ𝜏𝑉 ) is taken as an
+exponential distribution after fixing the fraction of attenuation op-
+tical depth arising from the ambient ISM (𝜇) to be 0.4. We note
+that our assumption of constant star formation may underestimate
+the stellar mass by as high as 0.5 dex when the galaxy is dominated
+by a young stellar population, but this would not affect the primary
+conclusions of this paper. We discuss the main results of this SED
+fitting in §3.2.
+3
+RESULTS
+All sixteen sources presented in this paper have optical spectra cov-
+ering CIII] to quantify equivalent widths. We report these values
+(or 3𝜎 upper limits) in Table 2. Of the sixteen sources, six galaxies
+have near-IR spectra enabling rest optical line flux ratios (Table 3).
+MNRAS 000, 1–20 (2022)
+
+8
+Mainali et al.
+1800
+2000
+2200
+2400
+2600
+2800
+3000
+−1
+0
+1
+2
+3
+0
+Relative Fλ
+CIII]
+FeII
+Mg II
+CSWA−141
+1900
+1905
+1910
+1915
+CIII]
+1200
+1400
+1600
+1800
+2000
+−1
+0
+1
+2
+3
+4
+0
+Lyα
+CIII]
+CIV
+HeII
+SiIV
+SiII
+CII
+OI
+CSWA−13
+1890
+1900
+1910
+1920
+CIII]
+1200
+1400
+1600
+1800
+2000
+−1
+0
+1
+2
+3
+0
+Relative Fλ
+CIII]
+OICII
+Al II
+FeII
+CSWA−139
+1890
+1900
+1910
+1920
+CIII]
+1400
+1600
+1800
+2000
+−1
+0
+1
+2
+3
+4
+0
+CIII]
+CIV Fe II
+OI CII
+CSWA−2
+1900
+1905
+1910
+1915
+CIII]
+1200
+1400
+1600
+1800
+2000
+−1
+1
+3
+5
+Relative Fλ
+CIII]
+Lyα
+CIV Fe II Al II
+Al III
+CSWA−39
+1900
+1905
+1910
+1915
+CIII]
+1200
+1400
+1600
+1800
+2000
+−1
+0
+1
+2
+3
+4
+0
+Lyα
+CIII]
+CIV
+SiIV
+SiIV
+SiII
+Fe II Al II
+Al III
+OICII
+CSWA−38
+1900
+1905
+1910
+1915
+CIII]
+1400
+1500
+1600
+1700
+1800
+1900
+2000
+−1
+1
+3
+5
+Relative Fλ
+CIII]
+CIV
+Al II
+SiIV
+CSWA−19
+1900
+1905
+1910
+1915
+CIII]
+1400
+1600
+1800
+2000
+−1
+0
+1
+2
+3
+4
+0
+Fe II Al II
+Al III
+CIII]
+CIV
+SiII
+SiIV
+OI CII
+CSWA−128
+1890
+1900
+1910
+1920
+CIII]
+1400
+1500
+1600
+1700
+1800
+1900
+2000
+Rest Wavelength(Å)
+−1
+0
+1
+2
+3
+0
+Relative Fλ
+CIII]
+SiIV
+SiIICIV
+Fe II
+Al II
+Al III
+CSWA−103
+1900
+1905
+1910
+1915
+CIII]
+1200
+1400
+1600
+1800
+2000
+Rest Wavelength(Å)
+−1
+0
+1
+2
+3
+4
+0
+Lyα
+CIII]
+CIV
+SiIV
+OI CII
+CSWA−164
+1890
+1900
+1910
+1920
+CIII]
+Figure 2. Rest-UV spectra of gravitationally lensed galaxies presented in Table 1 with CIII] detections. The upper left side of each image contains the
+CASSOWARY-ID. The dotted dashed line below and above the UV continuum represents absorption and emission features identified in the spectra. The upper
+right of each panel shows zoom in spectral coverage near CIII]𝜆𝜆1907,1909. The vertical dashed lines in the inset show the location of CIII]𝜆𝜆1907,1909
+doublet. The CIII] doublet remain unresolved in the MMT/BCS spectra of CSWA-13 and CSWA-139.
+MNRAS 000, 1–20 (2022)
+
+9
+4850
+4855
+4860
+4865
+4870
+4875
+0
+1
+2
+3
+4
+Flux(10−17erg−1s−1cm−2Å−1)
+Hβ
+CSWA−141
+4980
+4990
+5000
+5010
+5020
+5030
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+[OIII]λ5007
+6550
+6555
+6560
+6565
+6570
+6575
+6580
+0
+2
+4
+6
+8
+10
+Hα
+4855
+4860
+4865
+4870
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Flux(10−18erg−1s−1cm−2Å−1)
+Hβ
+CSWA−163
+5000
+5005
+5010
+5015
+0
+2
+4
+6
+8
+10
+[OIII]λ5007
+6550
+6560
+6570
+6580
+6590
+0
+1
+2
+3
+4
+5
+6
+Hα
+[NII]
+4855
+4860
+4865
+4870
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Flux(10−18erg−1s−1cm−2Å−1)
+Hβ
+CSWA−165
+5000
+5005
+5010
+5015
+0
+2
+4
+6
+8
+[OIII]λ5007
+6550
+6560
+6570
+6580
+6590
+0
+1
+2
+3
+4
+5
+6
+7
+[NII]
+Hα
+3720
+3725
+3730
+3735
+3740
+0
+2
+4
+6
+8
+10
+Flux(10−18erg−1s−1cm−2Å−1)
+[OII]λλ3727,3729
+CSWA−164
+5000
+5005
+5010
+5015
+0
+2
+4
+6
+8
+10
+[OIII]λ5007
+6550
+6560
+6570
+6580
+6590
+0
+2
+4
+6
+8
+10
+Hα
+3720
+3725
+3730
+3735
+3740
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Flux(10−18erg−1s−1cm−2Å−1)
+[OII]λλ3727,3729
+CSWA−11
+4990
+5000
+5010
+5020
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+[OIII]λ5007
+6550
+6560
+6570
+6580
+6590
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Hα
+4840
+4850
+4860
+4870
+4880
+Rest Wavelength(Å)
+3.5
+4.0
+4.5
+5.0
+5.5
+6.0
+6.5
+7.0
+Flux(10−18erg−1s−1cm−2Å−1)
+Hβ
+CSWA−128
+4980
+4990
+5000
+5010
+5020
+5030
+Rest Wavelength(Å)
+0
+5
+10
+15
+20
+[OIII]λ5007
+6520
+6540
+6560
+6580
+6600
+Rest Wavelength(Å)
+0
+2
+4
+6
+8
+10
+12
+Hα
+[NII]
+[NII]
+Figure 3. Optical emission lines in six gravitationally lensed galaxies discussed in this paper. Each row shows optical emission lines from a single source. The
+object ID is given in the leftmost panel of each row. The black line represents observed flux whereas blue line shows gaussian fit to the line profile calculated
+using IDL routine MPFITPEAK. The region affected by skylines is shown as a grey swath.
+MNRAS 000, 1–20 (2022)
+
+10
+Mainali et al.
+fν(µJy)
+ 
+ 
+ 
+ 
+ 
+1
+10
+100
+1000
+CSWA−141
+EWCIII]=4.6 Å
+sSFR=31.2 Gyr−1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+CSWA−13
+EWCIII]=4.4 Å
+sSFR=43.1 Gyr−1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+CSWA−139
+EWCIII]=3.4 Å
+sSFR=2.9 Gyr−1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+CSWA−2
+EWCIII]=3.1 Å
+sSFR=19.9 Gyr−1
+ 
+ 
+ 
+ 
+ 
+1
+10
+100
+1000
+CSWA−39
+EWCIII]=1.1 Å
+sSFR=11.7 Gyr−1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+CSWA−19
+EWCIII]=0.7 Å
+sSFR=0.5 Gyr−1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+CSWA−128
+EWCIII]=0.6 Å
+sSFR=1.0 Gyr−1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+CSWA−103
+EWCIII]=0.5 Å
+sSFR=0.7 Gyr−1
+ 
+ 
+ 
+ 
+ 
+1
+10
+100
+1000
+CSWA−163
+EWCIII]<3.1 Å
+sSFR=1.8 Gyr−1
+0.5
+1.0
+1.5
+2.0
+0.0
+λobs (µm)
+ 
+ 
+ 
+ 
+CSWA−40
+EWCIII]<1.7 Å
+sSFR=2.1 Gyr−1
+0.5
+1.0
+1.5
+2.0
+0.0
+λobs (µm)
+ 
+ 
+ 
+ 
+CSWA−165
+EWCIII]<1.9 Å
+sSFR=3.9 Gyr−1
+0.5
+1.0
+1.5
+2.0
+0.0
+λobs (µm)
+ 
+ 
+ 
+ 
+CSWA−11
+EWCIII]<1.4 Å
+sSFR=7.4 Gyr−1
+0.5
+1.0
+1.5
+2.0
+0.0
+λobs (µm)
+1
+10
+100
+1000
+CSWA−116
+EWCIII]<1.1 Å
+sSFR=1.6 Gyr−1
+Figure 4. SEDs of galaxies in the sample. The black circle represents broad band photometry data, blue line represents best fit population synthesis model to
+the observed data (see §2.5) and green square shows synthetic broad band flux from the best fit model. The lower right of each panel shows object ID, sSFR
+and CIII] equivalent widths.
+Object
+log (M★/M⊙)
+sSFR
+ˆ𝜏𝑉
+(Gyr−1)
+CSWA-141
+8.6+0.2
+−0.3
+31.2+60.8
+−16.3
+0.17+0.18
+−0.12
+CSWA-13
+9.8+0.3
+−0.3
+43.1+56.6
+−28.7
+0.53+0.13
+−0.27
+CSWA-139
+10.3+0.3
+−0.3
+2.9+14.3
+−2.0
+0.77+0.41
+−0.33
+CSWA-2
+9.1+0.3
+−0.3
+19.9+26.1
+−10.4
+0.96+0.31
+−0.27
+CSWA-39
+9.9+0.3
+−0.4
+11.7+13.8
+−8.1
+0.96+0.24
+−0.31
+CSWA-128
+9.9+0.1
+−0.1
+1.0+0.4
+−0.3
+0.13+0.06
+−0.07
+CSWA-19
+10.5+0.1
+−0.1
+0.5+0.4
+−0.1
+1.09+0.21
+−0.16
+CSWA-103
+10.4+0.1
+−0.2
+0.7+1.1
+−0.3
+0.48+0.31
+−0.23
+CSWA-163
+9.9+0.1
+−0.1
+1.8+0.9
+−0.9
+0.67+0.09
+−0.13
+CSWA-40
+10.8+0.2
+−0.2
+2.1+3.3
+−1.3
+0.43+0.29
+−0.25
+CSWA-165
+10.0+0.2
+−0.2
+3.9+3.0
+−1.9
+0.62+0.12
+−0.13
+CSWA-11
+10.0+0.4
+−0.3
+7.4+14.0
+−5.5
+0.28+0.31
+−0.20
+CSWA-116
+9.4+0.2
+−0.2
+1.3+2.4
+−0.7
+0.27+0.34
+−0.18
+Table 6. Physical properties inferred from BEAGLE fitting for subset of
+CASSOWARY galaxies with optical and near-IR photometry. From left to
+right, the columns give the object name, C III] equivalent widths, magnifica-
+tion corrected stellar mass, specific star formation rates and V-band optical
+depth.
+Ten sources have multi-wavelength photometry with SED infor-
+mation enabling stellar mass and sSFR measurements. Below we
+briefly comment on individual galaxy properties where sources are
+ordered by descending CIII] equivalent width.
+3.1
+Notes on Individual Sources
+CSWA-141 is an extreme equivalent width optical line emitting
+galaxy at 𝑧 = 1.425 with an integrated apparent magnitude of
+𝑟 = 20.7. The Magellan/FIRE spectrum reported in Stark et al.
+(2013a) shows a large number of rest-frame optical emission
+lines, including the temperature sensitive [OIII]𝜆4363 auroral line
+and the density sensitive [SII]𝜆𝜆6717,6731 and [OII]𝜆𝜆3727,3730
+lines (see Figure 3). We have since obtained deep Keck/ESI and
+LBT/MODS optical spectra and optical and near-IR imaging, pro-
+viding constraints on the rest-UV metal lines and the SED.
+The optical to near-infrared SED of CSWA-141 (Figure 4) im-
+plies a very large specific star formation rate (sSFR=31+61
+−16 Gyr−1).
+After correcting for magnification due to lensing (𝜇=5.5), the stellar
+mass and star formation rate of the best fitting model are 3.98×108
+M⊙ and 12+24
+−7 M⊙ yr−1, respectively (Table 6). The flux in the
+medium-band J2 MOSFIRE filter (covering 1.117 to 1.246 𝜇m) is
+significantly in excess of that in adjacent filters, as expected given the
+MNRAS 000, 1–20 (2022)
+
+11
+Line
+𝜆rest
+𝜆obs
+Fline
+EW
+(Å)
+(Å)
+FCIII]1909
+(Å)
+Keck/ESI
+[CIII]
+1906.73
+4624.32
+1.02(0.10)
+2.3(1.3)
+CIII]
+1908.68
+4628.84
+1.00
+2.3(1.4)
+Fe II*
+2626.64
+6370.24
+0.22(0.07)
+0.9(0.5)
+Mg II
+2796.36
+6783.42
+1.30(0.07)
+9.7(6.8)
+2803.53
+6800.79
+0.65(0.04)
+4.8(2.6)
+[OII]
+3727.10
+9039.02
+5.10(0.04)
+26.3(12.4)
+3729.86
+9045.68
+6.62(0.09)
+39.2(19.9)
+[NeIII]
+3870.16
+9385.28
+4.79(0.23)
+19.5(13.8)
+LBT/MODS
+Line
+𝜆rest
+𝜆obs
+Fline
+EW
+(Å)
+(Å)
+FCIII]1908
+(Å)
+[CIII]
+1908
+4622.13
+1.00
+3.5(1.5)
+He II
+1640.52
+3974.16
+< 0.08
+< 0.7
+CIV
+1549
+3758.75
+< 0.11
+< 0.8
+OIII]
+1660.81
+4027.93
+0.09(0.02)
+0.3(0.2)
+1666.15
+4040.15
+0.23(0.03)
+0.7(0.2)
+Si III]
+1882.98
+4566.25
+0.24(0.03)
+0.7(0.1)
+Si III]
+1892.03
+4588.16
+0.11(0.02)
+0.3(0.1)
+Table 7. Rest-UV emission line measurements of CSWA-141. Emission line
+fluxes are presented relative to the CIII]𝜆1909 line flux for the Keck/ESI
+data, while the line fluxes are presented relative to unresolved combined
+CIII] flux for LBT/MODS data. The upper limits are 3𝜎.
+contamination by extremely strong [OIII] and H𝛽 lines. We use the
+J2 flux excess to calculate the equivalent width from [OIII] and H𝛽,
+following the methodology adopted at higher redshift (e.g.,Stark
+et al. 2013b; Smit et al. 2015). This approach yields a rest-frame
+equivalent width of W[OIII]+H𝛽 =730 Å, consistent with the very
+young stellar populations (32 Myr for constant star formation) im-
+plied by the population synthesis modeling. While the optical line
+EW is significantly in excess of what is seen in typical star forming
+galaxies at 𝑧 ≃ 2 (e.g.,Boyett et al. 2022), it is nearly identical to the
+average [OIII]+H𝛽 EW at 𝑧 ≃ 7 − 8, as implied by flux excesses in
+Spitzer/IRAC bandpasses (Labbé et al. 2013; Smit et al. 2015; De
+Barros et al. 2017; Endsley et al. 2020). The continuum brightness
+of CSWA-141 enables a unique and detailed view of this population.
+The ESI spectrum covers 4100-10000 Å, revealing promi-
+nent emission from [CIII]𝜆1907, CIII]𝜆1909, Fe II★𝜆2626, Mg
+II𝜆𝜆2797,2804, [OII]𝜆𝜆3726,3729, [Ne IIII]𝜆3869, and H𝛿 (Ta-
+ble 7). The [CIII], CIII] doublet is easily resolved by ESI (Figure 2)
+with a total rest-frame equivalent width of WCIII]=4.6Å, the largest
+of the sixteen galaxies considered in this paper. Nebular Mg II emis-
+sion also exhibits a very large rest-frame equivalent width (WMgII
+= 15.0 Å), as is common in lower mass galaxies (e.g.,Erb et al.
+2012; Guseva et al. 2019). The MODS spectrum provides better
+blue sensitivity than ESI, yielding detection of OIII]𝜆1661,1666
+and Si III]𝜆𝜆1883,1892. The CIII] doublet is also detected, but it
+is unresolved at the resolution of MODS (Figure 5). The summed
+equivalent widths of the OIII] and Si III] doublet (1.0 Å and 1.0 Å,
+respectively) are considerably lower than CIII] (see Table 7). The
+high ionization lines He II𝜆1640 and CIV𝜆𝜆1548,1550 are not de-
+tected, implying rest-frame equivalent widths below 0.3 and 0.5 Å
+(at 3𝜎), respectively. For emission lines detected by both ESI and
+MODS, we will adopt whichever instrument provides the higher
+S/N EW measurement for our subsequent analysis and discussion.
+The rest-frame optical emission lines detected in the FIRE
+spectrum provide an array of constraints on the nebular gas
+physical conditions. The detection of [OIII]𝜆4363 in the FIRE
+spectrum enables a measure of the nebular electron temperature
+(Te = 1.5 ± 0.1 × 104 K ) and the oxygen abundance via the direct
+T𝑒 method. Following the process discussed in §2.2.3, we find that
+12 + log(O/H) = 7.95 ± 0.08, implying a gas-phase metallicity of
+0.18 Z⊙. Our measurement is very similar to Sanders et al. (2016b)
+who reported oxygen abundance (12 + log(O/H)) of CSWA-141
+based on our line flux measurements given in Table 4. The derived
+metallicity is also consistent with the metallicities derived from the
+N2 and O3N2 indices with the calibration presented in Bian et al.
+(2018) (12 + log(O/H) < 8.0).
+The O32 (6.6) and R23 indices (11.4) point to a large ioniza-
+tion parameter and gas excitation. The exact values depend on the
+input ionizing spectrum. Photoionization modeling using BEAGLE
+resulted in a large ionization parameter of log 𝑈𝑠 = -2.1±0.1, which
+is broadly consistent with O32 vs ionization parameter relationships
+in the literature (e.g., Sanders et al. 2016a; Berg et al. 2019). The
+photoionization modeling further suggested a high ionizing photon
+production efficiency of log (𝜉ion/Hz erg−1) = 25.5±0.1. This is
+higher than the canonical value typically assumed for galaxy-driven
+reionization model (Robertson et al. 2010) but consistent with other
+strong CIII] emitters in the literature (e.g., Nakajima et al. 2018).
+While the measured O32 and R23 are rare among more massive star-
+forming galaxies at 𝑧 ≃ 2, they are consistent with the O32-sSFR
+and O32-R23 trends that are observed at high redshift (e.g., Sanders
+et al. 2016a; Strom et al. 2016).
+The electron densities inferred from the flux ratios of the [OII]
+and [SII] doublets using PyNeb (160+76
+−74 cm−3 and 350+294
+−206 cm−3,
+respectively) are consistent with the median density of more massive
+star forming galaxies at 𝑧 ≃ 2 (250 cm−3; e.g., Sanders et al. 2016a).
+In contrast, the resolved [CIII], CIII] doublet suggests gas at very
+high density (16500+12100
+−7800 cm−3). Such an offset between densities
+derived from CIII] and those inferred from [OII], [SII] have been
+seen in other galaxies at high redshift (e.g., James et al. 2014). We
+will come back to discuss this in more detail in §4.
+Finally, we note that the MODS and FIRE spectra reveal emis-
+sion lines at 3827, 15304, 15612, 15763, 20663 Å which appear
+nearly spatially coincident with CSWA-141 but are not associated
+with the 𝑧 = 1.425 galaxy (see Figure 5). We identify these as Ly𝛼,
+H𝛽, [OIII]𝜆𝜆4959,5007, and H𝛼 in a second fainter gravitationally
+lensed source at 𝑧 = 2.148. This higher redshift galaxy is unre-
+solved from the 𝑧 = 1.425 source in existing ground-based optical
+and near infrared images. Given that the H𝛼 flux of the 𝑧 = 1.425
+galaxy is 7.4× larger than the 𝑧 = 2.148 source, we expect that
+the newly-discovered higher redshift source contributes negligibly
+(at the 5% level) to the broadband flux and the rest-UV continuum
+in the MODS and ESI spectra. Higher resolution imaging will be
+required to disentangle the two sources.
+CSWA-13 is a bright galaxy at 𝑧 = 1.87 that was first confirmed
+in Stark et al. (2013a) through the detection of Ly𝛼 emission and nu-
+merous interstellar absorption lines in an MMT blue channel spec-
+trum. CIII] emission is confidently detected (WCIII]=4.4±0.9 Å) in
+the discovery spectrum. He II is also detected (WHeII=4.3 ± 0.9 Å)
+with a broad FWHM (2430 km s−1) that is indicative of a stellar
+wind origin. We do not detect the OIII] doublet, implying individual
+components with equivalent widths less than 0.8 Å. While this is
+among the strongest CIII] emitters in our sample, the redshift places
+the strong rest-optical lines in regions of poor atmospheric transmis-
+sion. We have obtained multi-wavelength imaging in the optical and
+near-IR (Figure 4). The SED reveals a large sSFR (43.1 Gyr−1), lit-
+MNRAS 000, 1–20 (2022)
+
+12
+Mainali et al.
+tle dust attenuation ( ˆ𝜏𝑉 =0.53), and a magnification-corrected stellar
+mass of log (M★/M⊙) = 9.8.
+CSWA-139 was confirmed to have a redshift of 𝑧 = 2.54 in
+Stark et al. (2013a) based on the presence of Ly𝛼 absorption and in-
+terstellar metal absorption lines in an MMT spectrum. The spectrum
+also shows emission from the [CIII],CIII]𝜆𝜆1907,1909 doublet with
+a total equivalent width of WCIII] = 3.4 ± 2.6Å. Here we present
+new optical and near-IR imaging from the LBT and MMT. The SED
+is best fit with an sSFR of 2.9 Gyr−1, ˆ𝜏𝑉 =0.77, and a stellar mass
+of log (M★/M⊙) = 10.3 after magnification correction.
+CSWA-2 (SDSS J1038+4849) was first reported in Belokurov
+et al. (2009), and the source redshift (𝑧 = 2.20) was subsequently
+confirmed in Jones et al. (2013) through detection of rest-optical
+emission lines. The lens reconstruction described in Jones et al.
+(2013) shows that CSWA-2 is a merger of two systems with a stel-
+lar mass ratio (6 ± 3):1. log (M★/M⊙) = 9.1+0.2
+−0.1 and one of the
+largest sSFR (19.9 Gyr−1). The oxygen abundance 12+log(O/H)
+of the system as calculated by using N2 index is 8.25 (∼0.4 Z⊙).
+The Balmer decrement ratio (H𝛼/H𝛽=3.47) suggests little nebu-
+lar extinction whereas O3 of 1.80 imply relatively larger excita-
+tion from ionized gas. Using the metallicity calibration of Bian
+et al. (2018), we estimate the oxygen abundance 12+log(O/H) of
+the system using the O3 index as 8.4 (∼0.5 Z⊙) In this paper, we
+present new optical spectroscopy of this system obtained with ESI.
+The spectrum is dominated by continuum emission from the lower
+mass source (denoted J1038 North in Jones et al. (2013) which
+is considerably brighter in the optical. The spectrum shows emis-
+sion from [CIII],CIII]𝜆𝜆1907,1909 with a total equivalent width of
+WCIII]=3.1 ± 1.8 Å.
+CSWA-39 (SDSS J1527+0652), a bright (𝑟 = 20.5) 𝑧 = 2.759
+galaxy was first identified as part of the SDSS Giant Arcs Sur-
+vey (SGAS; Hennawi et al. 2008) and was spectroscopically con-
+firmed by Koester et al. (2010) through detection of Ly𝛼 emission
+and interstellar absorption lines. We have obtained an ESI opti-
+cal spectrum and multi-color optical imaging. Both components of
+the [CIII], CIII] doublet are detected in the spectrum, with a total
+S/N=16.1 for the summed doublet flux. The rest-frame equivalent
+width (WCIII]=1.1 Å) is typical of similarly luminous galaxies at
+this redshift (e.g., Shapley et al. 2003; Du et al. 2017). We also
+detect Ly𝛼 emission with a moderate rest-frame equivalent width
+(WLy𝛼=11.8±6.2Å). No other nebular UV lines are detected in the
+ESI spectrum. The upper limit on the OIII]𝜆𝜆1661,1666 compo-
+nents imply equivalent widths below 1 Å, consistent with the OIII]
+emission strengths in the Shapley et al. (2003) composite of 𝑧 ≃ 3
+galaxies with similar Ly𝛼 equivalent widths.
+CSWA-38 (SDSS J1226+2152) was confirmed to lie at 𝑧 =
+2.923 by detection of Ly𝛼 and metal absorption lines in (Koester
+et al. 2010). Like CSWA-39, this source was first identified in
+Sloan Giant Arcs survey. Here we present deep ESI spectroscopy
+and imaging. The optical spectrum reveals a 4.1𝜎 detection of
+the CIII]𝜆1909 component, but the [CIII]𝜆1907 component is situ-
+ated on a skyline, precluding a useful limit. The rest-frame equiv-
+alent width of the CIII]𝜆1909 component (WCIII]𝜆1909=0.4 Å) is
+comparable to CSWA-38 and CSWA-19. Ly𝛼 emission is weak
+(WLy𝛼=0.4 Å), and no other nebular UV lines are detected.
+CSWA-19 (SDSS J0900+2234) was first confirmed in (Diehl
+et al. 2009) at 𝑧 = 2.03 via detection of Ly𝛼 emission and several
+metal absorption features in a spectrum from the dual imaging spec-
+trograph (DIS) on the Astrophysical Research Consortium (ARC)
+3.5 meter telescope at the Apache Point Observatory. We have since
+obtained a deep ESI spectrum and MMT near-infrared imaging of
+this source. The combined optical and near-infrared SED is best-
+fit by a model with a (magnification-corrected) stellar mass of log
+(M★/M⊙) = 10.5, a specific star formation rate of 0.5 Gyr−1, and
+V-band attenuation optical depth of ˆ𝜏𝑉 =1.09. The ESI spectrum
+of the source shows weak detections of [CIII], CIII] 𝜆𝜆1907, 1909
+with an integrated S/N=8.6 across both components of the doublet.
+The total rest-frame equivalent width (WCIII]=0.7 Å) is among the
+lowest in our sample. No other nebular rest-UV lines are detected,
+as Ly𝛼 is situated to the blue of the ESI coverage.
+CSWA-128 (SDSS J1958+5950) is a bright (r=20.7) z=2.22
+galaxy that was spectroscopically confirmed in Stark et al. (2013a)
+through detection of rest-optical emission lines in an LBT/LUCI
+near-infrared spectrum and interstellar metal absorption lines in
+an MMT optical spectrum. We have since obtained optical and
+near-infrared imaging and a deep ESI optical spectrum. The
+magnification-corrected SED (Figure 4) implies a stellar mass of log
+(M★/M⊙) = 9.9+0.3
+−0.3 and an sSFR of 1.0 Gyr−1. The ESI spectrum
+reveals a faint 4.8𝜎 detection of [CIII], CIII] emission at the sys-
+temic redshift (𝑧 = 2.225) defined by the rest-optical lines, implying
+a rest-frame equivalent width of WCIII]=0.7 Å for the doublet. No
+other nebular lines are detected in the rest-UV.
+The flux ratios of rest-optical lines (Table 4) constrain the gas
+physical conditions, providing a framework to understand the weak
+UV nebular line emission. We infer the gas-phase oxygen abun-
+dance using the R23, O32 and O3 calibration presented in Bian
+et al. (2018). The three indices suggest metallicities of 12+log O/H
+= 8.2 (O32), 12+log O/H=8.4 (R23) and 12+log O/H=8.5 (O3). As
+discussed in Sanders et al. (2020), O32 metallicity calibration in
+Bian et al. (2018) may provide a better estimate of oxygen abun-
+dance for high redshift galaxy. This suggests that the metallicity of
+the ionized gas of CSWA-128 is ∼0.3-0.4 Z⊙. The values of O32
+(3.3) and R23 (7.0) are a factor of 2.0 and 1.6 smaller than that of
+the strong CIII] emitter CSWA-141, consistent with expectations
+for nebular gas with a lower ionization parameter and excitation.
+In contrast, the electron density implied by the [SII]𝜆𝜆6717,6731
+doublet (200 cm−3) is similar to that found in both strong CIII]
+emitters such as CSWA-141 and typical 𝑧 ≃ 2−3 galaxies (Sanders
+et al. 2016a).
+CSWA-103 (SDSS J0145-0455) is a 𝑧 = 1.96 galaxy that was
+confirmed in Stark et al. (2013a) through detection of metal absorp-
+tion lines and weak Ly𝛼 emission in an MMT blue channel spec-
+trum. We have since obtained a moderate resolution Keck/ESI spec-
+trum and optical and near-infrared imaging with LBT and MMT,
+respectively. The broadband SED (Figure 4) is best fit by a stel-
+lar synthesis model with a stellar mass (magnification corrected)
+of log (M★/M⊙) = 10.4, sSFR=0.7 Gyr−1, and ˆ𝜏𝑉 =0.48. The ESI
+spectrum constrains rest-frame wavelengths 1350-3425 Å. Positive
+emission is detected from both components of the [CIII], CIII] dou-
+blet, implying a total rest-frame equivalent width of 0.5 Å. No other
+nebular emission lines are observed in the ESI spectrum.
+CSWA-164 (SDSS J0232-0323) was spectroscopically con-
+firmed in Stark et al. (2013a) based on the presence of Ly𝛼 emission
+and interstellar absorption lines in an MMT blue channel spectrum.
+Stark et al. (2013a) also presented detection of [OII], [OIII]𝜆5007,
+and H𝛼 in a Magellan/FIRE near-infrared spectrum of CSWA-164,
+revealing a systemic nebular redshift of 𝑧 = 2.512. We have since
+obtained a deep moderate resolution optical spectrum with ESI
+and optical broadband imaging. The ESI spectrum reveals the Ly𝛼
+emission (W𝐿𝑦𝛼 = 2.0 Å) detected previously together with weak
+emission (S/N=4.1) from the [CIII],CIII]𝜆𝜆1907,1909 doublet. This
+corresponds to a total CIII],CIII] equivalent width of just 0.4 Å, the
+smallest measured value in our sample.
+MNRAS 000, 1–20 (2022)
+
+13
+0.38
+0.40
+0.42
+0.44
+0.46
+0.48
+λobs(µm)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Relative Fλ
+Lyα (z=2.148)
+CIII]1908 (z=1.425)
+Si III]1883 (z=1.425)
+OIII]1666 (z=1.425)
+1.54
+1.56
+1.58
+1.60
+λobs(µm)
+0
+2
+4
+6
+8
+Hα (z=1.425)
+[OIII]λ4959 (z=2.147)
+Hβ (z=2.147)
+[OIII]λ5007 (z=2.147)
+Figure 5. Optical and NIR spectra of CSWA-141. A secondary source (CSWA-141b) is identified in the Optical+NIR spectra (see §3) (Left:) Ly𝛼 emission at
+𝑧 = 2.148 from CSWA-141b in the LBT/MODS spectrum alongside [SiIII]𝜆1883 and CIII]𝜆1908 from CSWA-141 at 𝑧 = 1.425. (Right:) Rest optical lines
+([OIII]𝜆𝜆4959,5007, H𝛽) from CSWA-141b at 𝑧 = 2.147 observed in the FIRE spectrum alongside the H𝛼 emission line from CSWA-141.
+The FIRE spectrum provides insight into the ionized gas phys-
+ical conditions of CSWA-164. Unfortunately both H𝛽 and [NII] are
+obscured by skylines, precluding a robust determination of the oxy-
+gen abundance through standard strong line calibrations. We can
+however estimate oxygen abundance using O32 metallicity calibra-
+tions from Bian et al. (2018). We find that O32=0.7 corresponds
+to the oxygen abundance of 12+log O/H=8.6. Thus the ionized gas
+appear to be reasonably metal rich in CSWA-164. The value of
+ionization-sensitive line ratios (O32 = 0.7) is among the lowest in
+our sample. In contrast to the metallicity and ionization parameter,
+the electron density derived from the [OII] doublet (165 cm−3) is
+consistent with the range spanned by other galaxies in our sample.
+CSWA-40 (SDSS J0952+3434) is a 𝑧 = 2.190 galaxy identified
+through the Sloan Bright Arcs Survey and spectroscopically con-
+firmed by Kubo et al. (2010) using the DIS on the ARC 3.5 meter
+and the RC Spectrograph on the Mayall 4 meter telescope at Kitt
+Peak National Observatory. The spectra reveal Ly𝛼 in absorption
+along with several other metal absorption lines. We have obtained a
+moderate resolution optical spectrum and imaging with Keck/ESI.
+The continuum S/N of the CSWA-40 ESI spectrum is lower than
+other systems. This is the only ESI spectrum that does not reveal
+detection of the [CIII], CIII] doublet, implying a rest-frame equiv-
+alent width below 1.6 Å for the sum of both components. No other
+nebular emission lines are observed in the spectrum.
+CSWA-163 (SDSS J2158+0257) was spectroscopically con-
+firmed in Stark et al. (2013a) based on identification of metal ab-
+sorption lines and Ly𝛼 absorption in an MMT blue channel spec-
+trum. A Magellan/FIRE near-infrared spectrum was also obtained
+in that paper, revealing a systemic nebular redshift of 𝑧 = 2.079
+based on detection of [OII], H𝛾, H𝛽, [OIII], H𝛼, and [NII]. We
+have obtained new optical and near-infrared imaging of this source,
+allowing constraints to be placed on the broadband SED (Figure 4).
+The data are best fit by stellar synthesis models with stellar mass
+log (M★/𝑀⊙) = 9.9+0.1
+−0.1 after magnification correction, V-band at-
+tenuation optical depth of ˆ𝜏𝑉 =0.67, and an sSFR of 1.8 Gyr−1,
+suggesting CSWA-163 is a relatively massive galaxy with a fairly
+evolved stellar population. Perhaps not surprisingly given the low
+sSFR and lack of Ly𝛼 emission, the MMT blue channel spectrum
+does not show any CIII] emission at the systemic redshift. The 3𝜎
+flux upper limit suggests that the rest-frame equivalent width of the
+double must be lower than 3.1 Å.
+The FIRE spectrum provides useful constraints on the ionized
+gas of CSWA-163. The oxygen abundance can be derived from O32
+and O3 calibration from Bian et al. (2018). Both suggest moderately
+enriched gas: 12+log O/H=8.5 (O32) and12+log O/H=8.4 (O3) im-
+plying a nebular oxygen abundance of 0.4-0.5 Z⊙. The ionization-
+sensitive line ratios (O32=1.1, O3=4.3) are consistent with a rel-
+atively low ionization parameter and moderate gas-excitation. The
+electron density inferred from [OII] is 144 cm−3, consistent with
+the other systems studied in this paper.
+CSWA-16 (SDSS J1111+5308) was confirmed to have a red-
+shift of 𝑧 = 1.95 based on the presence of numerous metal absorp-
+tion lines in an MMT blue channel spectrum (Stark et al. 2013a).
+The discovery spectrum shows no emission from the blended [CIII],
+CIII] doublet. The 3𝜎 upper limit on the total flux from the doublet
+requires the rest-frame equivalent width to be smaller than 2.3 Å.
+We have obtained multi-band optical imaging with LBT, allowing
+more robust constraints to be placed on the apparent magnitude
+(r=21.8) and the magnification-corrected UV absolute magnitude
+(MUV=−20.8).
+CSWA-165 (SDSS J0105+0144) is a 𝑧 = 2.13 galaxy that
+was first confirmed in Stark et al. (2013b) through detection of
+strong metal absorption lines and weak Ly𝛼 emission in an MMT
+blue channel spectrum. We have since obtained a Magellan/FIRE
+near-infrared spectrum and optical and near-infrared imaging from
+LBT and MMT respectively. The FIRE spectrum reveals detection
+of [OII], H𝛽, [OIII]𝜆5007, H𝛼, and [NII], indicating a systemic
+redshift of 𝑧 = 2.128. The MMT shows no emission from the
+[CIII], CIII] doublet at the systemic redshift. The 3𝜎 upper limit
+on the flux of the doublet indicates that the total CIII] equivalent
+width must be lower than 1.3 Å. The broadband SED is best-fit by
+a synthesis model with stellar mass of log(M★/M⊙) = 10.0, sSFR
+of 3.9 Gyr−1, and ˆ𝜏𝑉 =0.62.
+The gas-phase metallicity can be inferred from O32, R23, and
+O3 strong line calibrations. All three indicate metal rich ionized
+gas: 12+log O/H = 8.4 (O32) 8.6 (R23), 8.6 (O3), consistent with
+a gas-phase metallicity in the range 0.5-0.8 Z⊙. The ionization-
+sensitive ratios (O32=0.6, O3=2.1) suggest an ionization parameter
+that is lower than average among 𝑧 ≃ 2 − 3 galaxies. In contrast,
+MNRAS 000, 1–20 (2022)
+
+14
+Mainali et al.
+the electron density derived from the [OII] doublet flux ratio (220
+cm−3) is similar to that found in other systems at high redshift.
+CSWA-11 (SDSS J0800+0812) was confirmed at 𝑧 = 1.41 via
+detection of metal absorption lines and [OII] emission in MMT
+blue and red channel spectra (Stark et al. 2013a). The CIII] dou-
+blet is not detected in either the blue or red channel MMT spectra,
+implying a rest-frame equivalent width below 0.9 Å at 3𝜎. We
+have obtained multi-wavelength imaging and near-infrared spec-
+troscopy. The magnification-corrected SED suggests a stellar mass
+of log(M★/M⊙) = 10.0+0.4
+−0.3, an sSFR of 7.4 Gyr−1, and V-band
+attenuation optical depth of ˆ𝜏𝑉 =0.28. The FIRE spectrum reveals
+detections of [OII], [OIII]𝜆5007, and H𝛼, but strong skylines ob-
+scure detections of H𝛽, [OIII]𝜆4959 and [NII]𝜆6586. Using the
+theoretically-expected flux ratio of [OIII]𝜆5007/[OIII]𝜆4959, we
+infer that this system has O32 = 0.92, relatively low for our sam-
+ple. Using the O32 calibration, we estimate metallicity of 12+log
+O/H=8.5. The electron density implied by the [OII] doublet flux
+ratio is 469+416
+−238 cm−3, consistent with the range spanned by other
+galaxies in our sample.
+CSWA-116 (SDSS J0143+1607) is a 𝑧 = 1.50 galaxy con-
+firmed in Stark et al. (2013a) through detection of rest-UV metal
+absorption lines and [OII] in MMT blue and red channel spectra.
+We do not detect the CIII] doublet in the MMT spectra, indicating
+that the rest-frame equivalent width is below 0.7 Å at 3𝜎. We have
+obtained optical and near-infrared imaging of CSWA-116, provid-
+ing constraints on the broadband SED. The data are best fit by a
+stellar model with log(M★/M⊙) = 9.4, an sSFR of 1.3 Gyr−1, and
+V-band attenuation optical depth of ˆ𝜏𝑉 =0.27.
+3.2
+Physical Properties of CASSOWARY galaxy sample
+The data obtained for this paper provide new constraints on the
+ionized gas properties and the stellar populations of lensed galaxies
+at 𝑧 ≃ 1 − 3 identified by the CASSOWARY selection in SDSS.
+Here we briefly describe what these data reveal about the aver-
+age properties in this sample, providing a concise summary of the
+source-by-source description presented in §3.1. Following correc-
+tion for magnification, the rest-frame UV absolute magnitudes are
+found to range between MUV=−20.2 and MUV=−23.0, with a me-
+dian value (MUV=−21.9) that is roughly three times the value of
+L★
+UV at 𝑧 ≃ 2 (e.g., Reddy & Steidel 2009). Both optical and near-
+infrared imaging exist for 11 of the 16 galaxies considered in this
+paper, allowing the stellar content to be characterized through SED
+fitting. The median stellar mass and sSFR of this subset is 1.3×1010
+M⊙ and 2.1 Gyr−1, respectively. The latter is similar to the average
+sSFR of 𝑧 ≃ 2 − 3 galaxies (e.g., Reddy & Steidel 2009).
+Rest-optical line measurements exist for seven of the CAS-
+SOWARY galaxies shown in Figure 1. The median gas-phase oxy-
+gen abundance derived from the rest-optical line ratios is 12+log
+O/H = 8.33, i.e., ionized gas metallicity of 0.4 Z⊙. The ionization-
+sensitive ratio O32 ranges between 0.7 and 6.7 with a median value
+of O32=1.1. While this is slightly lower than the median O32 in the
+KBSS and MOSDEF surveys, it is well within the range spanned
+by galaxies in these samples (e.g., Sanders et al. 2016a; Strom et al.
+2016; Steidel et al. 2016). The Balmer decrements range between
+H𝛼/H𝛽=3.47 and 4.97. The median electron density derived from
+the [OII] or [SII] doublet flux ratios (198 cm−3) is close to the aver-
+age electron density of 𝑧 ≃ 2.3 galaxies from the MOSDEF survey
+(Sanders et al. 2016a).
+All sixteen galaxies in our sample have constraints on rest-UV
+line emission. Most sources show very weak [CIII], CIII] emission.
+The median equivalent width for the summed doublet is 1 Å, similar
+to that seen in composite spectra of 𝑧 ≃ 3 LBGs (Shapley et al. 2003;
+Llerena et al. 2021). The two strongest [CIII], CIII] emitters (CSWA
+141, CSWA -13) have rest-frame equivalent widths of 4.6 Å and
+4.4 Å, respectively. The rest-UV spectrum of CSWA-141 also shows
+prominent nebular emission from OIII], Si III] and Mg II. While
+these lines are absent in CSWA-13, its spectrum does reveal broad
+He II emission, indicative of a significant Wolf Rayet population.
+No high ionization nebular lines are detected in our sample.
+Our sample allows to compare and contrast ISM conditions ex-
+pected in strong and weak CIII] emitters. A typical CIII] equivalent
+widths of star-forming galaxies at 𝑧 ∼ 2−3 is ∼1.7 Å (Shapley et al.
+2003; Du et al. 2017). From here on, we refer to those galaxies with
+CIII] equivalent widths > 2× typically seen at 𝑧 ∼ 2−3 (≳ 3.5 Å) as
+strong CIII] emitters, whereas objects with CIII] equivalent widths
+below this threshold are described as weak CIII] emitters. Together
+with galaxies presented in this paper, we compile sources from the
+literature having constraints from both CIII] equivalent widths and
+optical spectra. The majority of the literature objects are lower red-
+shifts galaxies (Giavalisco et al. 1996; Leitherer et al. 2011; Berg
+et al. 2016, 2019; Senchyna et al. 2017, 2019; Ravindranath et al.
+2020), while majority of sample at 𝑧 > 1 are comprised of gravita-
+tionally lensed systems (Mainali et al. 2020; de Barros et al. 2016;
+Stark et al. 2014; Vanzella et al. 2017; Berg et al. 2018; Bayliss
+et al. 2014; Vanzella et al. 2016; James et al. 2014; Erb et al. 2010;
+Christensen et al. 2012; Rigby et al. 2021; Quider et al. 2009; Hain-
+line et al. 2009; Pettini et al. 2000; Teplitz et al. 2000; Jones et al.
+2013). We present a compilation of 𝑧 > 1 sources in Table 8.
+In Figure 6, we plot empirical relationships between CIII]
+equivalent width and oxygen abundance (left panel), and CIII]
+equivalent width and O32 (right panel). The oxygen abundance
+of the strong CIII] emitters are mostly measured using direct metal-
+licity measurements, while strong optical line calibrations are used
+for the weaker CIII] emitters (see Table 8). As can be seen in the
+Figure 6, the gas-phase metallicity of the strong CIII] emitters is
+consistently below 12+log(O/H)=8.0, suggesting metallicities be-
+low 20% Z⊙. In contrast, most of the weaker CIII] emitters imply
+higher metallicities than this value. This result is consistent with pre-
+vious investigations in the literature (e.g., Rigby et al. 2015; Jaskot
+& Ravindranath 2016; Maseda et al. 2017; Nakajima et al. 2018;
+Schaerer et al. 2018; Senchyna et al. 2019; Ravindranath et al. 2020;
+Du et al. 2020; Tang et al. 2021). The right panel of Figure 6 shows
+that the O32 values of the stronger CIII] emitters (≳ 3.5 Å) are on
+an average six times larger than the those of the weaker CIII] emit-
+ters. The median O32 value of the weaker CIII] emitters are similar
+to typical galaxies at 𝑧 ∼ 2 − 3 (O32=1.3,Sanders et al. 2016a). A
+variety of factors can influence the O32 value of a galaxy. The ele-
+vated values seen in the strongest of CIII] emitters tend to be found
+in galaxies dominated by the light of a very young stellar popula-
+tion (Tang et al. 2021; Sanders et al. 2020), as expected in galaxies
+undergoing a burst of star formation. Overall these empirical rela-
+tionships support a physical picture where galaxies with metal poor
+gas and young stellar populations are able to power strong CIII]
+emission (e.g. Stark et al. 2014; Rigby et al. 2015; Senchyna et al.
+2017). The scatter seen in the CIII] EW at a given metallicity or
+O32 value may stem from differences in ISM conditions (relative
+carbon abundances (C/O), ionization parameters) and stellar age
+and metallicity.
+The compilation of strong CIII] emitters further provides in-
+formation on the global spectral properties expected from typical
+reionization era systems. In Figure 7, we show strong CIII] emit-
+ters (denoted by red square) and weak CIII] emitters (denoted by
+blue square) in log([OIII]/H𝛽) vs sSFR (left panel) and log(O32)
+MNRAS 000, 1–20 (2022)
+
+15
+0.1
+1.0
+10.0
+EWCIII] (Å)
+7.0
+7.5
+8.0
+8.5
+9.0
+12+log(O/H)
+This work
+Mainali+2020
+Ravindranath+2020
+Stark+2014
+James+2014
+Erb+2010
+Stark+2015
+deBarros+2014
+Christensen+2012
+Bayliss+2014
+Vanzella+2016/17
+Rigby+2015/21
+Berg+2016/19
+Leitherer+2011
+Giavalisco+1996
+Quider+2009
+Pettini+2000
+Berg+2018
+Senchyna+2017/19
+1
+10
+EWCIII] (Å)
+1
+10
+O32
+Figure 6. (Left:) Empirical relationship between oxygen abundance (12+log(O/H)) and rest frame CIII] equivalent width (EWCIII]). (Right:) Empirical
+relationship between the line ratio of [OIII]𝜆𝜆4959,5007 to [OII]𝜆𝜆3727,3729 (i.e. O32) and rest-frame CIII] EW (EWCIII]). The red symbol represents bright
+lensed galaxies presented in this paper, while other data points are compilation from the literature.
+Figure 7. The strong (blue) and weak (red) CIII] emitters (see §3.2) in O3 vs sSFR (left) and O32 vs R23 (right) plots. Star forming galaxies at 𝑧 ∼ 2.3 from
+KBSS survey (Strom et al. 2016) are shown in cyan circle whereas MOSDEF survey (Sanders et al. 2016a) are shown in violet triangle. Local galaxies from
+SDSS survey are shown in grey. Galaxies with properties similar to those at z>6 tend to have strongest C III] emission.
+vs log(R23) (right panel) plots. We compare them to typical line
+ratios expected at 𝑧 ∼ 2 using dataset from KBSS survey (Steidel
+et al. 2014; Strom et al. 2016) and MOSDEF survey (Sanders et al.
+2016a), as well as at 𝑧 ∼ 0 (local SDSS galaxies). The strong CIII]
+emitting galaxies appears to be distinct from local SDSS galaxies
+as well as typical galaxies at 𝑧 ∼ 2 in both the plots. However, the
+position of weaker CIII] emitters on both the diagrams is similar to
+typical galaxies at 𝑧 ∼ 2. Taken together, this further demonstrates
+that strong CIII] emitters show highly ionized gas conditions from
+a large sSFR systems. Assuming these strong CIII] emitters repre-
+sentative of a typical reionization era systems, we might expect a
+similar ISM conditions in galaxies at 𝑧 > 6.
+4
+DISCUSSION
+In this paper, we present spectra of some of the brightest-known
+gravitationally-lensed galaxies at 𝑧 ≃ 2−3, discovered over the foot-
+print of SDSS. Included in this sample are CSWA-13 and CSWA-
+141, two exceptionally bright systems with sSFRs (> 20 Gyr−1)
+that are similar to those of the reionization-era. Their spectra reveal
+rest-frame CIII] equivalent widths more than twice what is typical
+at 𝑧 ≃ 2 − 3 (e.g., Shapley et al. 2003; Du et al. 2018; Llerena et al.
+2021). In this section, we explore the ionized gas conditions and
+the properties of the outflowing gas, taking advantage of emission
+and absorption lines that are often too faint to be detected in indi-
+vidual high redshift galaxies with similarly intense emission lines.
+MNRAS 000, 1–20 (2022)
+
+1og([OIII]25008 / Hβ)
+0.5
+0.0
+-0.5
+Strong CIIIl emitters
+Weak CIIIl emitters
+ KBSS z~2.3
+SDSS z~0
+-2
+-1
+0
+2
+1
+log(sSFR /Gyr-1)1.5
+Strong CIIIl emitters
+Weak CIIIl emitters
+1.0
+KBSS z~2.3
+MOSDEF z~2.3
+log(O32)
+0.5
+■SDSS z～0
+0.0
+-0.5
+.1.0
+-0.5
+0.0
+0.5
+1.0
+log(R23)16
+Mainali et al.
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+EWFeII λ2374 (Å)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+EWFeII λ2382 (Å)
+CSWA−141
+optically  thin
+optically  thick
+Erb+2012 (Composite)
+Figure 8. A comparison between equivalent widths of Fe II 𝜆2374 and
+Fe II 𝜆2382. The red square represents CSWA-141 data point whereas black
+circles are composite data points presented in Erb et al. 2012. The two
+dashed black lines represent optically thin and optically thick cases. This
+reflects that Fe II is not optically thick in CSWA-141 and therefore has a
+lower column density c.f. black points
+Our analysis will primarily focus on CSWA-141, as the redshift of
+CSWA-13 places the strong rest-optical lines in regions of low at-
+mospheric transmission. For CSWA 141, the rest-optical lines are
+similar in equivalent width to those found in the reionization era
+(e.g., De Barros et al. 2017; Endsley et al. 2020; Boyett et al. 2022),
+classifying this galaxy as an EELG and revealing a stellar population
+dominated by a recent burst or upturn in star formation.
+The gas properties in such intense line emitting galaxies have
+been the subject of a number of spectroscopic investigations in
+recent years. These studies demonstrate that the nebular gas is gen-
+erally under extreme ionization conditions in EELGs (Tang et al.
+2018), with the ionization parameter reaching its largest values in the
+galaxies powered by the youngest stellar populations (or the largest
+equivalent width rest-optical nebular lines). The gas in CSWA-141
+is consistent with this picture, showing a large O32 ratio (6.7) as
+expected given its elevated sSFR (31.2 Gyr−1) and [OIII]+H𝛽 EW
+(730 Å). Its large ionization parameter implied by the photoion-
+isaiton models (log U = -2.1) suggests a large photon density is
+impingent on the ionised gas. This can be driven be a variety of fac-
+tors, including efficient ionizing photon production (e.g., Chevallard
+et al. 2018; Tang et al. 2021) and a compact configuration of ion-
+ized gas around the sources of ionizing radiation. The latter is the
+norm for galaxies dominated by very young stellar clusters (e.g.,
+Whitmore et al. 2011; Hannon et al. 2019; Chen et al. 2023), and
+thus may be expected in EELGs like CSWA-141.
+Further insight into the gas conditions of EELGs is made possi-
+ble by detection of three density-sensitive emission lines in CSWA-
+141. As we discussed in §3.1, the [CIII], CIII] doublet implies
+very high gas densities (16500+12100
+−7800 cm−3), perhaps again reflect-
+ing a very compact or concentrated gas geometry surrounding the
+extremely young star clusters that power EELGs. Although the un-
+certainty in CIII] density is currently large, such high densities are
+routinely seen in 𝑧 ≃ 2 − 4 galaxies with spectrally-resolved CIII]
+measurements (e.g., Christensen et al. 2012; James et al. 2014, 2018;
+CSWA-141
+EWMgII = 14.5Å
+F2797
+F2803 = 2.0 ± 0.2
+Figure 9. Mg II 2797,2803 line profile in CSWA-141. The black curve
+represents continuum subtracted flux level while the green solid line is
+gaussian fit to the observed data. Both the Mg II doublet are well fitted by a
+single component gaussian. Mg II 2797 is clearly stronger than Mg II 2803
+line, suggesting a Mg II doublet intensity ratio (𝐹2797/𝐹2803) of 2.0±0.2.
+The doublet ratio is consistent with the intrinsic recombination value (2.0)
+indicating minimal scattering by low-ionization ISM along the line of sight.
+Based on the observed Mg II doublet ratio, CSWA-141 has implied escape
+fraction of 𝑓esc(LyC) = 27±4%.
+Bayliss et al. 2014; Acharyya et al. 2019) and are also starting to be
+seen in the first handful of 𝑧 ∼> 7 galaxies with CIII] doublet mea-
+surements (Stark et al. 2017; Jiang et al. 2021). As such, the high
+C III] density is less likely due to a statistical fluctuation, although
+a larger sample should confirm this. In most of the cases, the CIII]
+density is also significantly in excess of that inferred from lower-
+ionization species. The same trend is apparent in CSWA-141, with
+the CIII] density being nearly two orders of magnitude higher than
+those derived from [SII] and [OII]. This is consistent with a picture
+(e.g., Kewley et al. 2019; Acharyya et al. 2019; Berg et al. 2021)
+dictated by the ionization structure of the nebulae, with the lower
+ionization rest-optical lines (i.e., [OII] and [SII]) probing primarily
+the outer layers which are preferentially dominated by lower density
+gas. The higher ionization lines (like CIII]) probe the inner regions
+of the nebula, where densities are expected to be higher. Variations
+in the critical density of the ions will additionally contribute to CIII]
+probing higher density gas than [OII] (e.g., James et al. 2018). A key
+question is whether very young EELGs like CSWA-141 (and many
+of the 𝑧 > 7 galaxies) might have a large fraction of their ionized
+gas in very dense clumps, as is expected at the earliest evolution-
+ary phases following the formation of star clusters (e.g., Kim et al.
+2018; Rigby & Rieke 2004). Currently samples with CIII] densities
+tend to be those that have at least moderately large CIII] EW, gener-
+ally indicative of a relatively young stellar population. Larger CIII]
+density samples are required to test whether the presence of very
+large densities is at all sensitive to the luminosity-weighted stellar
+population age.
+CSWA 141 also provides a unique window on the kinematics
+and covering fraction of the outflowing gas in EELGs, a population
+which becomes commonplace at 𝑧 > 6. As it is this outflowing
+gas which regulates the escape of ionising radiation, systems like
+MNRAS 000, 1–20 (2022)
+
+4
+3
+itive
+Rel
+2792
+2794
+2796
+2798
+2800
+2802
+2804
+2806
+Wavelenqth (A)17
+CSWA 141 offer potential for understanding the likely contribu-
+tion of EELGs to reionisation. At low redshift, (Jaskot et al. 2017;
+Chisholm et al. 2017) find that the most extreme [OIII] emitting
+galaxies (the Green Peas) tend to have low outflow velocities and
+suggest that this is consistent with models of suppressed superwinds,
+where catastrophic cooling prevents the development of large scale
+outflows (e.g., Silich et al. 2007; Silich & Tenorio-Tagle 2017; Gray
+et al. 2019). The low ionization absorption lines in the CSWA 141
+spectra are all blueshifted with respect to the systemic redshift, in-
+dicating the presence of outflows. In the ESI spectrum, we detect
+Fe II and Mg II absorption lines, while the MODS spectrum probes
+slightly bluer wavelengths, allowing detection of Al II𝜆1670. The
+average velocity of low ionization outflowing gas in CSWA 141 is
+76 km s−1. This is slightly lower than outflow velocities of 100-300
+km s−1 that are commonly observed in galaxies at 𝑧 ∼ 1 − 3 (e.g.,
+Shapley et al. 2003; Weiner et al. 2009; Steidel et al. 2010; Jones
+et al. 2012). However given the low stellar mass of CSWA-141, the
+measurement is consistent with expectations from known trends be-
+tween galaxy mass and outflow velocity (e.g., Martin 2005; Weiner
+et al. 2009; Erb et al. 2012; Chisholm et al. 2016).
+The strength of the absorption lines in CSWA 141 provide
+insight into the opacity of neutral gas along the line of sight to the
+young star clusters which dominate the light. The Fe II 𝜆2374 and
+Fe II 𝜆2382 absorption lines are very weak with equivalent widths
+of 0.5 Å and 1.5 Å, respectively. Jones et al. (2018) measured a
+column density N(Fe II) = 1.5+0.7
+−0.2 × 1014 cm−2, which is among
+the lowest of the sample. In Figure 8, we compare these equivalent
+widths with those measured from composite spectra of more typical
+star forming galaxies at 1 < 𝑧 < 2 (Erb et al. 2012). As is clear in
+the figure, most star forming galaxies at these redshifts show similar
+Fe II 𝜆2374 and Fe II 𝜆2382 equivalent widths, as expected for
+optically thick neutral gas. In contrast, the Fe II 𝜆2374 absorption in
+CSWA-141 is significantly weaker than Fe II 𝜆2382. With an Fe II
+𝜆2374 EW that is roughly three times weaker than that found in the
+𝑧 ≃ 1 − 2 composites, the lines are much closer to expectations for
+optically thin neutral gas. We additionally note that the Fe II 𝜆2382
+line is susceptible to emission filling (e.g., Erb et al. 2012) which
+tends to weaken the observed Fe II 𝜆2382 equivalent widths. This
+is particularly likely to be the case for EELGs like CSWA-141.
+In such a scenario, the line ratio would move even closer to that
+expected for optically thin conditions, implying a low covering frac-
+tion and potentially low column density of neutral gas in the outflow.
+Further indications that CSWA 141 has a low column density
+of neutral gas comes from the resonant Mg II𝜆𝜆 2797,2803 emis-
+sion line. Both components of the doublet are confidently detected
+(S/N>10) in emission with total equivalent widths of 14.5 Å (Fig-
+ure 9). This is not only one of the highest measured Mg II EWs at
+z>1, but it is also one of the brightest Mg II emission lines, making
+detailed (and resolved) study of the line uniquely possible. Recent
+studies have pointed out that the flux ratio of the doublet is strongly
+sensitive to the neutral gas column density in the galaxy. As such
+it is thought to correlate closely with the ionizing photon escape
+fraction (Henry et al. 2018; Chisholm et al. 2020; Xu et al. 2022;
+Seive et al. 2022; Xu et al. 2023), perhaps providing one of the best
+indirect indicators of photon leakage. At low HI column densities
+(<1017.2 cm−2), galaxies become optically thin to the resonant Mg
+II line photons (Chisholm et al. 2020). This leads to strong nebu-
+lar Mg II emission, and it drives the doublet ratio (R=F2797/F2803)
+to its intrinsic value of 𝑅 = 2. If the line photons are resonantly
+scattered by Mg+ ions in the neutral gas along the line of sight, the
+doublet ratio will decrease, asymptoting to a value of R=1 if the gas
+is optically thick to Mg II photons (see Chisholm et al. 2020 for a
+detailed discussion).
+The measured Mg II doublet ratio in CSWA-141 is R=2.0±0.2,
+consistent with the intrinsic value produced in the HII regions. This
+suggests that optically thin channels along the line of sight to the
+young star clusters are powering the nebular emission. Both compo-
+nents of the doublet are well-fit by single Gaussians, suggesting that
+the line profile is not significantly impacted by resonant scattering.
+The very large EW of the line also points to minimal attenuation of
+line photons. We note that the Mg II profile does show very weak
+blue-shifted absorption (see Figure 9), suggesting the presence of
+some neutral gas along the line of sight, but this gas must be either
+optically thin or clumpy with low density channels to result in the
+observed line profile.
+Given the derived oxygen abundance of CSWA-141 (see §3.1)
+and nominal assumptions on the Mg/O ratio (Chisholm et al. 2020),
+this can be converted to an estimate of the hydrogen column den-
+sity (See equation 14 of Chisholm et al. 2020). Given the measured
+doublet ratio is consistent with the intrinsic value, CSWA-141 is for-
+mally consistent with a negligible hydrogen column density. Within
+the measurement errors of the flux ratio, we find a 1𝜎 upper limit on
+the HI column density of 3.8×1016 cm−2. This value is well below
+the HI column density at which galaxies become optically thin to
+LyC radiation (<1017.2 cm−2), suggesting that CSWA-141 may be
+a likely candidate for LyC leakage. While more realistic geometries
+(i.e., clumpy gas) would alter the derived column densities, the ob-
+served line profile requires there to be low density channels along
+the line of sight where resonant line photons (and potentially LyC
+emission) are transmitted (Gazagnes et al. 2020; Saldana-Lopez
+et al. 2022). Because of the extreme brightness of CSWA-141, it
+presents a unique opportunity to spatially resolve the absorbing gas
+in an extreme line emitter that is very similar in its properties to
+those systems at 𝑧 > 6. An upcoming HST UVIS grism observa-
+tions (GO-16710, PI: Mainali) will provide more direct constraints
+on the escape of ionizing radiation from the galaxy.
+5
+SUMMARY
+We present new spectroscopic and photometric observations of six-
+teen bright gravitationally lensed galaxies originally identified in
+SDSS via the CASSOWARY program. Observations were con-
+ducted using LBT, Keck, MMT and Magellan. Included in this
+sample is the 𝑧 = 1.42 galaxy CSWA-141, one of the brightest
+known EELGs at high redshift, with an [OIII]+H𝛽 EW (730 Å)
+nearly identical to the average value seen at 𝑧 ≃ 7 − 8. In this
+paper, we focus on the rest-UV spectral properties of the sample,
+leveraging high quality Keck/ESI data. Owing to the brightness of
+our targets (𝑔 ≃ 19-21), we are able to detect rest-UV metal line
+emission in the Keck spectra down to very low EW values. While
+most systems have weak line emission (median CIII] EW =1.7 Å),
+CSWA 141 shows relatively strong emission (CIII] EW 4.6 Å) to-
+gether with detections of a variety of UV lines (OIII], Si III], Fe
+II★, Mg II).
+We compare the properties of the strong (∼> 3.5 Å) and weak
+(∼< 3.5 Å) CIII] emitters in our sample and in the literature. We
+find that the stronger CIII] emitters have larger sSFR and lower gas-
+phase oxygen abundances. We find that the strong CIII] emitters
+can be easily separated by their rest-optical line ratios, with larger
+values of O32 at roughly fixed R23. Overall, these results suggest
+that CIII] tends to be strong in galaxies dominated by young stellar
+populations with low metallicity and extreme ionization conditions.
+MNRAS 000, 1–20 (2022)
+
+18
+Mainali et al.
+Object
+EWCIII]
+R23
+O32
+O3
+Electron Density
+12 + log (O/H)
+Reference
+()
+(cm−3)
+CSWA-141
+4.6±1.9
+11.4±1.0
+6.6±1.0
+7.4±0.2
+160+76
+−74
+𝑎, 350+294
+−206
+𝑏, 16500+12100
+−7800
+𝑐
+7.95±0.08𝑑
+This work
+CSWA-2
+3.1±1.6
+. . .
+. . .
+5.2±2.1
+. . .
+8.4±0.2𝑔
+This work, 16
+CSWA-128
+0.7±0.1
+7.0±1.2
+3.3±0.8
+4.0±1.1
+198+85
+−76
+𝑏
+8.2±0.2𝑒, 8.4±0.2 𝑓 , 8.5±0.2𝑔
+This work
+CSWA-164
+0.4±0.1
+. . .
+0.7±0.2
+. . .
+165+65
+−46
+𝑎
+8.6±0.2𝑒
+This work
+CSWA-163
+< 3.1
+10.9±1.1
+1.1±0.1
+4.3±0.6
+144+45
+−33
+𝑎
+8.5±0.2𝑒, 8.4±0.2𝑔
+This work
+CSWA-165
+< 1.9
+7.1±1.3
+0.6±0.1
+2.1±0.4
+221+75
+−64
+𝑎
+8.4±0.2𝑒, 8.6±0.2 𝑓 , 8.6±0.2𝑔
+This work
+CSWA-11
+< 1.4
+. . .
+0.8±0.2
+. . .
+469+120
+−156
+𝑎
+8.5±0.2𝑒
+This work
+RXCJ0232-588
+21.7
+6.0
+9.4
+4.1
+80𝑎
+7.61 𝑑
+1
+Ion2
+18
+. . .
+> 15
+14.7
+. . .
+8.07ℎ
+2
+860_359
+12.4
+< 9.2
+> 6.91
+6.0
+. . .
+< 8.1 𝑓
+3
+ID14
+11.8
+< 11.7
+> 10
+7.6
+. . .
+< 7.8𝑑
+4
+SL2S0217
+11.7
+. . .
+. . .
+3.1
+300𝑐
+7.5𝑑
+5
+SGAS J1050+0017
+11
+11.4
+9.7
+7.9
+2-3×100𝑎
+> 8.1𝑑
+6
+ID11
+11
+< 11.8
+> 10
+8.4
+. . .
+7.7𝑑
+7
+CSWA-20
+9.1
+7.7
+5.6
+4.9
+276𝑎, 17100𝑏
+7.82𝑑
+8
+BX418
+7.1
+< 9.2
+> 11.6
+6.4
+. . .
+7.8𝑑
+9
+A1689 31.1
+7
+7.8
+8.2
+4.7
+330𝑎, 2900𝑐
+7.76𝑑
+10
+MACS 0451-1.1
+6.7
+< 5.9
+> 8.4
+3.9
+. . .
+< 8.0 𝑓
+3
+SDSS J1723+3411
+4.0
+8.6
+5.5
+5.4
+47𝑎, 1950𝑐
+8.4 𝑓
+11
+MACS 0451-3.1
+2.4
+. . .
+> 2.0
+. . .
+. . .
+. . .
+3
+Cosmic Horseshoe
+0.9
+5.5
+1.31
+2.4
+840-6900𝑏
+8.65 𝑓
+12,13
+MS 1512-cB58
+0.8
+7.9
+1.36
+3.6
+. . .
+8.47 𝑓
+14,15
+𝑎Derived from [OII] doublet, 𝑏Derived from [SII] doublet, 𝑐Derived from CIII] doublet.
+𝑑Direct (T𝑒) method, 𝑒O32 method, 𝑓 R23 method, 𝑔O3 method,
+ℎ Using HII-CHI-mistry code (Perez-Montero 2014)
+1Mainali et al. (2020),2de Barros et al. (2016),3 Stark et al. (2014),4Vanzella et al. (2017),5Berg et al. (2018),6Bayliss et al. (2014),7Vanzella et al.
+(2016),8James et al. (2014), 9Erb et al. (2010),10Christensen et al. (2012), 11Rigby et al. (2021), 12Quider et al. (2009), 13Hainline et al. (2009), 14Pettini
+et al. (2000), 15Teplitz et al. (2000), 16Jones et al. (2013)
+Table 8. Rest-optical emission line properties of galaxies at high redshift with CIII] emission constraints. The upper half shows data from this paper whereas
+lower half shows measurements from the literature.
+This is consistent with trends found in observations at low and high
+redshift (e.g., Rigby et al. 2015; Senchyna et al. 2017, 2019; Maseda
+et al. 2017; Nakajima et al. 2018; Schaerer et al. 2018; Du et al. 2020;
+Ravindranath et al. 2020; Tang et al. 2021) and in photoionization
+models (e.g., Jaskot & Ravindranath 2016; Nakajima et al. 2018).
+The brightness of CSWA-141 enables a detailed investigation
+of an EELG with properties similar to that which become common
+at 𝑧 > 6. This galaxy is characterized by low stellar mass (4.0×108
+M⊙), large sSFR (31.2 Gyr−1), low gas-phase metallicity (12+log
+O/H=7.95) and relatively highly ionized gas (O32=6.7) and has
+likely undergone a recent upturn or burst of star formation. We
+find that the electron density traced by the CIII] doublet (1.65×104
+cm−3) is higher than that traced by [OII] and [SII] doublet (160
+and 350 cm−3, respectively), a discrepancy that is also found in
+other systems (e.g., James et al. 2014; Acharyya et al. 2019). While
+this is likely to reflect the ionization structure of the HII regions
+powering the lines (Kewley et al. 2019), it may also indicate that
+CSWA-141 contains a significant fraction of its ionized gas in very
+dense clumps, as is expected in the earliest stages following the
+formation of star clusters (e.g. Kim et al. 2018).
+The spectra of CSWA-141 provide several probes of the neutral
+gas opacity in the galaxy, including both low ionization absorption
+lines and the resolved Mg II doublet. The Fe II𝜆𝜆2374, 2382 ab-
+sorption lines indicate the presence of outflowing gas with average
+velocity of 76 km s−1. The lines are much weaker than in typical star
+forming galaxies at 𝑧 ≃ 1 − 2, implying a low covering fraction and
+potentially low column density of neutral gas in the outflow. The res-
+onant Mg II𝜆𝜆 2797,2803 emission line supports this picture. The
+Mg II doublet ratio in CSWA-141 ( R= F2797/F2803) is 2.0±0.2, con-
+sistent with the intrinsic value produced in the HII regions. When
+combined with the very large EW of Mg II and the near-Gaussian
+profiles of the doublet components, this suggests minimal resonant
+scattering, consistent with a very low column density of neutral
+hydrogen. These indirect indicators suggest CSWA-141 may be a
+likely candidate for LyC leakage.
+ACKNOWLEDGMENTS
+We would like to thank Ryan Endsley for helping with MMT ob-
+servations of some of the sources presented in this paper. We also
+thank Stephane Charlot and Jacopo Charlot for making the BEA-
+GLE population synthesis tool available to us for this paper.
+DPS acknowledges support from the National Science Founda-
+tion through the grant AST-2109066. TJ acknowledges support from
+the National Science Foundation through the grant AST-2108515.
+RSE acknowledges funding from the European Research Council
+(ERC) under the European Union’s Horizon 2020 research and in-
+novation 0programme (grant agreement No 669253). Y.H. acknowl-
+edges support from the National Sciences and Engineering Council
+of Canada grant RGPIN-2020-05102, the Fonds de recherche du
+Québec grant 2022-NC-301305, and the Canada Research Chairs
+Program.
+Observations presented in this paper were obtained from the
+MNRAS 000, 1–20 (2022)
+
+19
+Keck Observatory, which was made possible by the generous fi-
+nancial support of the W. M. Keck Foundation. The material
+is based upon work supported by NASA under award number
+80GSFC21M0002. The authors acknowledge the very significant
+cultural role that the summit of Mauna Kea has always had within
+the indigenous Hawaiian community. We are most fortunate to have
+the opportunity to conduct observations from this mountain. Some
+of the observations reported here were obtained at the MMT Obser-
+vatory, a joint facility of the University of Arizona and the Smith-
+sonian Institution. This paper includes data gathered with the 6.5
+meter Magellan Telescopes located at Las Campanas Observatory,
+Chile. Some of the data presented in this paper were obtained using
+the Large Binocular Telescope (LBT). The LBT is an international
+collaboration among institutions in the United States, Italy and Ger-
+many. The LBT Corporation partners are: The University of Ari-
+zona on behalf of the Arizona university system; Istituto Nazionale
+di Astrofisica, Italy; LBT Beteiligungsgesellschaft, Germany, rep-
+resenting the Max Planck Society, the Astrophysical Institute Pots-
+dam, and Heidelberg University; The Ohio State University; The
+Research Corporation, on behalf of The University of Notre Dame,
+University of Minnesota and University of Virginia.
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable request
+to the corresponding author.
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+
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+page_content=' The Catholic University of America,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content=' University of California Davis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 1 Shields Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content=' USA  5 Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Gower Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content=' UK  6Center for Computational Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Flatiron Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 162 Fifth Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' NY 10010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' USA  7Département de Physique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Université de Montréal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Montreal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Quebec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Canada H3T 1J4  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  We present new observations of sixteen bright (𝑟 = 19 − 21) gravitationally lensed galaxies at  𝑧 ≃ 1−3 selected from the CASSOWARY survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Included in our sample is the 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='42 galaxy  CSWA-141, one of the brightest known reionization-era analogs at high redshift (g=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5), with  a large sSFR (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2 Gyr−1) and an [OIII]+H𝛽 equivalent width (EW[OIII]+H𝛽=730 Å) that is  nearly identical to the average value expected at 𝑧 ≃ 7 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In this paper, we investigate the  rest-frame UV nebular line emission in our sample with the goal of understanding the factors  that regulate strong CIII] emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Whereas most of the sources in our sample show weak  UV line emission, we find elevated CIII] in the spectrum of CSWA-141 (EWCIII]=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 Å)  together with detections of other prominent emission lines (OIII], Si III], Fe II★, Mg II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We  compare the rest-optical line properties of high redshift galaxies with strong and weak CIII]  emission, and find that systems with the strongest UV line emission tend to have young stellar  populations and nebular gas that is moderately metal-poor and highly ionized, consistent  with trends seen at low and high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The brightness of CSWA-141 enables detailed  investigation of the extreme emission line galaxies which become common at 𝑧 > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We find  that gas traced by the CIII] doublet likely probes higher densities than that traced by [OII] and  [SII].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Characterisation of the spectrally resolved Mg II emission line and several low ionization  absorption lines suggests neutral gas around the young stars is likely optically thin, potentially  facilitating the escape of ionizing radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Key words: galaxies: evolution - galaxies: formation - galaxies: high-redshift  1  INTRODUCTION  Over the last decade, much progress has been made in our under-  standing of galaxies in the first billion years of cosmic time (for a  review, see Stark 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Deep infrared imaging has uncovered thou-  sands of photometrically-selected star forming systems thought to  lie in the redshift range 6 < 𝑧 < 9 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', McLure et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Finkel-  stein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Bouwens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2015, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Livermore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Ishigaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018), providing a census of UV-selected galaxies  throughout the reionization era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The spectral energy distributions  (SEDs) point toward a population undergoing rapid stellar mass  ★ E-mail: ramesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='mainali@nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='gov  growth with blue UV continuum spectral slopes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Bouwens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014), low stellar masses and large specific star  formation rates (Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2013b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' González et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Grazian  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Salmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Curtis-Lake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  In the last five years, our first constraints on the nebular emis-  sion properties of galaxies at these early epochs have emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Spitzer/IRAC photometry suggests that nearly half of the UV-  selected galaxies at 𝑧 ≃ 7 have extremely large [OIII]+H𝛽 equiv-  alent widths (EWs) (Labbé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Roberts-Borsani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' De Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2020), indicating that very recent (∼< 50 Myr) activity powers the  UV and optical luminosity, as expected for galaxies undergoing  rapidly rising star formation histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Roughly 20% of the popula-  © 2022 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='11264v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='GA]  26 Jan 2023  2  Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  tion have yet more intense rest-optical nebular emission ([OIII]+H𝛽  EW > 1000 Å), indicating an extremely young stellar population  (<10 Myr) is dominating the light, as expected for systems that have  undergone a recent burst of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Since such extreme emis-  sion line galaxies (EELGs) are very rare at lower redshifts (Boyett  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2022), we lack a detailed understanding of the gas and ionizing  agents in typical reionziation-era systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Ground-based near-infrared spectrographs offer a path toward  progress at the highest redshifts, providing access to the rest-frame  ultraviolet where a suite of valuable diagnostic lines are situ-  ated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The first deep spectra have revealed strong nebular emission  from high ionization metal species ([CIII],CIII]𝜆𝜆1907,1909 Å,  OIII]𝜆𝜆1660,1666 Å, CIV𝜆𝜆1548,1550 Å), a significant departure  from what is commonly seen at lower redshifts (Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2015b,a,  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Hutchison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Topping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The detection of these  lines reveals gas under extreme ionization conditions and points to  a population of intense ionizing agents, potentially AGN in some  cases (Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018) and low metallic-  ity massive stars in others (Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2019, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The detection of CIV𝜆𝜆1548,1550 Å and  [CIII],CIII]𝜆𝜆1907,1909 Å may further indicate a higher fraction  of ionizing photons escape from a galaxy(Schaerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  While the equivalent width distribution in the total population is  still subject to limited statistics (Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018), it appears that  both the gas and ionizing sources at 𝑧 ∼> 6 are often significantly  different from that in galaxies at 𝑧 ≃ 2 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The presence of strong rest-UV nebular emission in a subset of  𝑧 ∼> 6 galaxies bodes well for future studies of the reionization era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  For the next-generation 25-30m ground based optical/infrared ob-  servatories (which are limited to constraints on the rest-frame UV at  𝑧 > 6), these lines may provide the only way in which early galaxies  can be studied spectroscopically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' While typically fainter than the  strong lines in the rest-optical, the suite of emission lines in the  far-UV provide unique diagnostic power of the ionizing spectrum  and gas physical conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='e, density, temperature, ionization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Meanwhile in the near-UV, the resonant nature of the nebular Mg  II emission line makes it an ideal probe of the neutral gas opacity  in early galaxies, potentially providing an indirect indicator of LyC  leakage at 𝑧 ∼> 6 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Xu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Seive et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The advantage of  Mg II relative to Ly𝛼 is that it is not obscured by the neutral IGM  at very high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  To ensure that the faint rest-UV spectra provide reliable phys-  ical diagnostics from these future facilities, it is important that we  understand the gas conditions and stellar populations which support  prominent UV line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' With current facilities, this is most eas-  ily done at lower redshifts where rest-frame optical emission lines  which constrain the gas-phase metallicity and ionization parameter  are observable with ground-based facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Over the last few years,  a wide range of observations have been conducted with the goal of  better understanding the physics regulating rest-UV emission line  spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' These studies have demonstrated that prominent CIII] ap-  pears to be a fairly ubiquitous feature in the rest-UV spectra of dwarf  star forming galaxies (Erb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Stark  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Vanzella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Senchyna  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Llerena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2022), reflecting the large electron temperatures  associated with metal poor gas and the hard ionizing spectrum of  young, low metallicity massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Here we seek to build on this progress, improving our un-  derstanding of the connection between UV line emission and the  physical properties of the massive stars and ionized gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Over the  last decade, we have utilized a wide range of ground-based facilities  (LBT, Keck, Magellan) to obtain deep spectra and imaging of some  of the brightest known (g=19-21) 𝑧 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 − 3 gravitationally lensed  galaxies within Sloan Digital Sky Survey (SDSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This sample con-  tains galaxies with a range of physical properties, including two of  the brightest known systems with the large specific star formation  rates (and extreme emission lines) that are typical in the reioniza-  tion era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Central to this observational campaign are newly-acquired  optical spectra from the Echellette Spectrograph and Imager (ESI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Sheinis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2002) on the Keck II telescope (see Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018  for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The ESI spectra provide moderate resolution (R=  6300) rest-UV spectra, enabling a unique exploration of the na-  ture of the massive stars and the physical conditions of the ionized  gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We supplement the Keck observations (9 galaxies) with optical  spectra from LBT (1 galaxy) and MMT (6 galaxies), near-infrared  spectra from Magellan, and new optical and near-infrared imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The large continuum brightness of the SDSS lensed galaxies  offers several distinct advantages with respect to the much fainter  low mass galaxies that are typical of cluster fields imaged by HST  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Vanzella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' First, it enables improved constraints on the ionized gas phys-  ical conditions (metallicity, density, ionization parameter) through  detection of the full suite of rest-optical strong emission lines as  well as occasionally allowing detection of multiple temperature and  density-sensitive lines, providing a more comprehensive view of the  conditions in the gas and the most important factors regulating the  rest-UV line spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Second, the large continuum S/N in the rest-  UV spectra makes it possible to detect very weak rest-UV nebular  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This enables the rest-UV lines to be characterized in indi-  vidual galaxies for a wide range of gas conditions, not only in the  most extreme and metal-poor systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This will provide a valuable  control sample for our analysis, offering insight into what factors  are most important in regulating rest-UV emission line spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This  insight will be critical in assessing the feasibility of detecting and  characterizing lines in the rest-frame far and near-UV with future  facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In this paper, we will focus primarily on CIII] emission,  comparing measured line strengths to other empirical and model-  based quantities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', gas-phase metallicity, optical line ratios with  the goal of understanding the factors regulating the CIII] strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We present the new imaging  and spectra and discuss population synthesis modeling of broadband  SEDs in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In §3, we provide our results on individual galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  We discuss implications of our spectra of a reionization-era analog  in §4 and summarize our findings in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Throughout the paper, we adopt a Λ-dominated, flat universe  with ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7, Ω𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3 and H0 = 70 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' All mag-  nitudes in this paper are quoted in the AB system and equivalent  widths (EW) are given in rest-frame, unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2  OBSERVATIONS AND ANALYSIS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1  Sample Selection  The galaxies studied in this paper were originally identified using  a search algorithm that identifies blue arcs surrounding red early  type galaxies in Sloan Digital Sky Survey (SDSS) imaging as part  of the Cambridge And Sloan Survey Of Wide ARcs on the skY  (CASSOWARY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2007, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The most recent  catalog of CASSOWARY lensed galaxies was presented in Stark  MNRAS 000, 1–20 (2022)  3  Object  RA  DEC  𝑧𝑠𝑝𝑒𝑐  mAB  Dates  Rest-UV Coverage  texp  PA  MUV  𝜇  Instrument  (Å)  (ksec)  (deg)  CSWA-165  01:05:19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='8  320  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='081  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4  30 Sep 2011  1035-2600  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8  20  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5𝑎  MMT/BCS  𝑎Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a), 𝑏Koester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2010), 𝑐Leethochawalit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2016), 𝑑This work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Summary of rest-frame UV observations of our bright gravitationally lensed CASSOWARY galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' New ultra-deep moderate resolution optical  spectra have been obtained for nine of these sources using ESI on Keck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We also include an additional seven sources from Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a) with high quality  MMT blue channel spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' From left to right, we present the object ID in the CASSOWARY catalog, the RA and DEC, the spectroscopic redshift, apparent  magnitude, the date the optical spectra were obtained, the rest-UV wavelength coverage provided by the optical spectra, the exposure time and position angle  of the optical spectra, the absolute magnitude of the arc in the rest-UV, the magnification factor provided by gravitational lensing, and the optical spectrograph  utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Further details are presented in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a) following a large spectroscopic campaign aimed at  obtaining redshifts of the source and lens galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The catalog  of galaxies released in Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a) contains more than 50  gravitationally-lensed galaxies in SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Optical magnitudes of this  sample are in the range 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 < 𝑟 < 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Here we present the results from a large observational invest-  ment targeting sixteen of these bright lensed sources with Keck,  Magellan, LBT, and MMT (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Object identifiers from  the CASSOWARY survey are abbreviated as “CSWA" in the table  and subsequent discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' A mosaic of the galaxies considered in  this paper is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In the following subsections, we  describe the imaging and spectral datasets that we have obtained  for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The galaxies considered here were selected from the  larger sample of Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a) based on the brightness of  the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We also preferentially targeted sources at redshifts  which place rest-optical lines in regions of significant atmospheric  transmission in the near-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  Spectroscopy  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1  Optical Spectroscopy  We have acquired deep Keck/ESI optical spectra of nine galaxies  from the Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a) sample (see Table 1 for details), en-  abling robust constraints to be placed on the strength of nebular UV  metal line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The positioning of the ESI slit on the lensed  galaxies is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The spectra were obtained in two  observing runs between November 2012 and March 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The ESI  spectra cover observed wavelengths between 4000 to 10100 Å, pro-  viding rest-frame spectral coverage that typically ranges between  1300 and 2000 Å (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' A slit width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′75 was used, pro-  viding a resolving power of R=6300 (FWHM=48 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The data  were reduced using the ESIRedux code written by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Prochaska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Sky subtraction is performed following the bias subtraction and flat  fielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The 1D spectra are then extracted using a boxcar aper-  ture matched to the spatial extent of the arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' When multiple lensed  images appear on the slit, the traces are extracted separately and  then combined to maximize the S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The continuum is generally  well detected, with S/N ≃ 10 per resolution element at 6200 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Example spectra are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' A full description of the  ESI observations and spectral reduction is presented in Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  One source in the ESI sample, CSWA-141, was also observed  with the Multi Object Double Spectrograph (MODS, Pogge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2010) on the LBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' MODS provides bluer wavelength coverage than  ESI, allowing constraints to be placed on emission from CIV, OIII],  and He II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We used a long slit of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′8 with the 400 lines/mm  grating, providing spectral resolution of ∼ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  We also include seven other galaxies from Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a)  for which the MMT Blue Channel Spectrograph (BCS) discovery  spectra have sufficient continuum S/N to characterize the rest-UV  metal emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The MMT BCS data were obtained with the  300 lines/mm grating, providing total spectral coverage of ∼ 5300  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' For the slit width of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′0 used in the observations, the typical  spectral resolution is ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' More details of the MMT observations  are provided in Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  In total, the Keck, LBT, and MMT data provides rest-UV spec-  tra for 16 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Our aim is to characterize CIII] emission fea-  ture, typically the strongest rest-UV emission line, along with other  rest-UV emission features in the whole sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Using redshift pre-  sented in Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2018) for Keck/ESI data and Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  (2013a) for MMT data, we first searched for CIII] emission in indi-  vidual galaxy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The line is spectrally resolved in the Keck  spectra, whereas it remained unresolved in the MMT spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' For  the resolved CIII] doublet, we measured emission line fluxes and  associated errors (for error spectrum) in individual components by  directly integrating the flux levels within ±1 Å of individual line  MNRAS 000, 1–20 (2022)  4  Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-141  CSWA-2  CSWA-40  CSWA-19  CSWA-128  CSWA-165  CSWA-13  CSWA-164  CSWA-163  CSWA-11  CSWA-128  CSWA-16  CSWA-139  CSWA-38  CSWA-39  CSWA-165  CSWA-116  CSWA-103  CSWA-141  CSWA-2  CSWA-19  CSWA-13  CSWA-164  CSWA-163  CSWA-11  CSWA-128  CSWA-16  CSWA-139  CSWA-38  CSWA-39  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Color images of sixteen gravitationally lensed galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The name of each sources is presented at the top of each image, along with the slit position  used for the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' For seven galaxies (CSWA-141, CSWA-13, CSWA-139, CSWA-116, CSWA-16, CSWA-165, CSWA-164) the color images are  created from g,r and i band images from LBT/MODS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The color images for four galaxies (CSWA-11,CSWA-163,CSWA-128,CSWA-103) are made with g,r  and i bands images from LBT/LBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Three color images (CSWA-38, CSWA-39, CSWA-40) are made using Keck/ESI images in V,R and I bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' For other two  galaxies (CSWA-2, CSWA-19), color images are created using archival HST images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' North is up and east is to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Each postage stamp is 25” × 25” in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The orientation and centroid of the ESI or MMT slit is overlaid in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  centers (rest-frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' When the line is unresolved, we computed  total CIII] emission line fluxes and associated line flux errors di-  rectly from the flux spectra and error spectra, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' If the  line remained undetected at 3𝜎 level, we calculated upper limits by  integrating error spectrum from 1905 Å to 1912 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The emission  line equivalent width is then computed by dividing the line fluxes  (or upper limits) by median continuum level measured on either side  of the CIII] line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We then searched for any other rest-UV emission  features in the spectra and characterized them when detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Our  sample includes 10 galaxies with CIII] detection where the equiva-  lent widths range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 Å to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 Å (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We will come  back to interpret the CIII] equivalent widths in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2, comparing the  line strengths to optical line ratios and gas physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  Near-Infrared Spectroscopy  Near-infrared (NIR) spectra supplement the optical spectra de-  scribed above, providing constraints on the metallicity, ionization  parameter, and electron density of the ionized gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Near-infrared  spectroscopic analysis is limited to the subset of sources from Ta-  ble 1 which are located at redshifts placing strong rest-optical emis-  sion lines in spectral windows in which atmospheric transmission  is near-unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' One of the sources in our sample (CSWA-2) was tar-  MNRAS 000, 1–20 (2022)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  Object  EW (Å)  Ly𝛼  [CIII]𝜆1907  CIII]𝜆1909  CIII]𝜆1908𝑎  CSWA-141  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9)  CSWA-13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9)  CSWA-139  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6)  CSWA-2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6)  CSWA-39  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2)  CSWA-19  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  CSWA-38  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  CSWA-128  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  CSWA-103  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  CSWA-164  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  CSWA-163  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='  < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1  CSWA-16  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3  CSWA-165  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9  CSWA-40  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7  CSWA-11  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4  CSWA-116  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1  𝑎 Total CIII] doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Equivalent width measurements of rest-UV emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The  numbers within parentheses represent uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The upper limits are 3𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Object  Dates  texp  PA  Observatory/  (ksec)  (deg)  Instrument  CSWA-165  2014 June 22  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  120  Magellen/FIRE  CSWA-164  2015 Nov 03  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8  58  Magellen/FIRE  CSWA-11  2015 Nov 04  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7  10  Magellen/FIRE  CSWA-141  2012 Feb 15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  280  Magellen/FIRE  CSWA-128  2012 Nov 7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6  190  LBT/LUCI  CSWA-163  2014 June 22  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  90  Magellen/FIRE  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Details of near-infrared spectroscopic observations obtained for this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' From left to right, the columns denote the object ID, observation dates,  exposure time, position angle of slit, and the observatory and instrument used  to acquire near-IR spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Further details are provided in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  geted with Keck near-infrared spectroscopy in Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  In observing runs between November 2012 and November 2015,  we have obtained near-infrared spectroscopic observations of five  additional sources (CSWA-141, CSWA-164, CSWA-165, CSWA-  163, CSWA-11) using the Folded-port InfraRed Echellette (FIRE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Simcoe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2008) on the Magellan Baade Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We used  FIRE in its echelle mode, providing continuous spectral coverage  spanning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='82-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='51 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We adopted a slit width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′75, delivering  a spectral resolution of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 in the J band, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 in the H band and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7  in the K band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Spectroscopic data reduction was performed using  standard routines in the FIREHOSE data reduction pipeline1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  One additional galaxy (CSWA-128) was observed on 2012  Nov 7 with the LUCI near-IR spectrograph on the Large Binocular  Telescope (LBT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We used the N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8 camera and 200_H+K grating in  longslit mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We first observed the lensed source with the HKSpec  filter centered at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='93 microns, providing spectral coverage from  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='50 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='30 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We also observed CSWA-128 with the zJSpec  filter centered at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 microns, providing spectral coverage between  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='95 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='40𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' A slit width of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′0 was used, resulting in a spectral  resolution of 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Spectroscopic data reduction was performed using  standard IDL long slit reduction packages (see Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2010 for  1 wikis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='edu/confluence/display/FIRE/FIRE+Data+Reduction  details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We summarize details of near-infrared spectroscopy of  CASSOWARY galaxies in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Example spectra are shown in  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Emission line fluxes in the NIR spectra are measured using the  IDL routine MPFITPEAK which computes line fluxes after fitting  a gaussian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In cases where the emission lines are partially  affected by a skyline, we mask the contaminated region before fit-  ting the emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Additionally, if one of the components of  [OIII]𝜆𝜆4959,5007 is strongly affected by an emission line, we as-  sume the theoretical line ratio of [OIII]𝜆5007/[OIII]𝜆4959=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='98  (Storey & Zeippen 2000) in calculating the total [OIII] line flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  We calculate the impact of dust on the nebular lines using the  Balmer decrement flux ratio of H𝛼/H𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The observed line ratio is  compared to the line ratio expected in absence of dust (H𝛼/H𝛽=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='86;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Osterbrock & Ferland 2006) for case B recombination assuming  T𝑒=10,000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In the two cases where the Balmer line ratios are not  available, we use the stellar reddening inferred from the broadband  data to estimate the nebular attenuation (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We note that  using stellar reddening for nebular attenuation correction doesn’t  impact our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We assume that the nebular gas attenuation  is similar to the stellar continuum attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We presented the  observed and dust-corrected emission line measurements in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  For one object (CSWA-141) where auroral [OIII]𝜆4363 line  is detected, we calculated electron temperature using PyNeb (ver-  sion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Luridiana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2015) using emission line flux ratio of  OIII]𝜆4363/[OIII]𝜆5007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' For our calculations, we adopted the de-  fault PyNeb atomic data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We assumed electron density of 250  cm−3, which is typical to 𝑧 ∼ 2 galaxies (Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' How-  ever, we note that our assumed electron density value has negligible  effect on the derived electron temperature in the low density regime  (<103 cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Since we don’t have T𝑒([OII]) sensitive emission line  measurements, we followed the relation given by Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2006)  for low metallicity to estimate the electron temperature in the O+2  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We use our electron temperature measurements to infer the  direct oxygen abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We only use O+/H+ and O+2/H+ to com-  pute oxygen abundance since the ionization states higher than O+2  contributes significantly low at less than 1 per cent (Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' O+/H+ and O+2/H+ are calculated from PyNeb using our  T𝑒([OIII], T𝑒([OII], 𝑛𝑒 and emission line fluxes of [OII], H𝛽 and  [OIII].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  We use PyNeb to calculate the electron density using the flux  ratio of the [OII], [SII], and [CIII], CIII] doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The typical  error in measurements is then calculated using the errors in emis-  sion line fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' When an electron temperature measurement is not  available, we assumed electron temperature of 10,000K following  Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2016a) to adopt temperature dependent effective col-  lision strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Adopting electron temperature of 7000K (15000K)  instead would overestimate (underestimate) electron density by 15-  20% which is effectively lower than density error from our line flux  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3  Imaging  Each of the CASSOWARY galaxies was discovered in SDSS imag-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In many cases, the photometric constraints from SDSS are  unreliable owing to blending with neighbors and low S/N detec-  tion of diffuse emission associated with the arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We have obtained  deeper optical multi-band imaging for each of the galaxies discussed  in this paper using cameras on the LBT and Keck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In order to better  characterize the stellar populations, we have also obtained near-IR  imaging sampling across the Balmer break for a subset of our tar-  gets using the LBT, MMT and Keck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In the following subsections,  MNRAS 000, 1–20 (2022)  6  Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Line  𝜆rest(Å)  𝜆obs (Å)  F(𝜆)/F(5007)  I(𝜆)/I(5007)  CSWA-141  [OII]  3727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='13  9039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='060 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='088 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='003  [OII]  3729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='92  9045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='075 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='110 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='006  [NeIII]  3869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='66  9384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='054 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='017  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='075 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='023  H𝛿  4102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='90  9950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='030 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='039 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='005  H𝛾  4341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='58  10529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='063 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='076 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='004  [OIII]  4365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='31  10586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='019 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='023 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='002  H𝛽  4862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='55  11792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='135 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='003  [OIII]  4960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='25  12029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='338 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='342 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='004  [OIII]  5008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='27  12146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='000  [SIII]  6310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='48  15304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='007 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='005 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='001  [NII]  6549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='84  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='004  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='002  H𝛼  6564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='61  15920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='529 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='386 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='002  [SII]  6717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='96  16292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='017 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='013 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='001  [SII]  6732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='34  16327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='016 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='61  21170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='04  [NII]  6585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='01  [SII]  6717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='96  21664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='01  [SII]  6732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='34  21711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='02  CSWA-164  [OII]  3727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='13  13090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='17  [OII]  3729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='92  13100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='07  [OIII]  5008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='27  17590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='00  H𝛼  6564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='61  23064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='06  CSWA-11  [OII]  3727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='13  8979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='16  [OII]  3729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='92  8986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='11  [NeIII]  3869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='66  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='43  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='53  [OIII]  5008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='27  12068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='00  H𝛼  6564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='61  15816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='05  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Rest-optical emission line measurements of six bright lensed  Cassowary galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Measurements are from Magellan/FIRE (CSWA-141,  CSWA-165, CSWA-163, CSWA-164, CSWA-11) and LBT/LUCI (CSWA-  128).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Line flux measurements are quoted relative to [OIII]𝜆5007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We adopt  [OIII]𝜆5007 as a reference line due to its high S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Three-sigma upper limits  are reported for emission lines that are not detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Object  Observatory /  Filters  Dates  texp  Instrument  Observed  (sec)  CSWA-165  LBT/MODS  g  2014 Jan 27  720  LBT/MODS  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' z  2014 Jan 27  360  MMT/MMIRS  J  2015 Sep 30  900  MMT/MMIRS  K  2015 Sep 30  1200  CSWA-116  LBT/MODS  g  2014 Jan 27  720  LBT/MODS  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' z  2014 Jan 27  360  MMT/MMIRS  J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='K  2015 Oct 01  600  CSWA-103  LBT/LBC  g  2015 Nov 15  600  LBT/LBC  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' i  2015 Nov 15  300  MMT/MMIRS  J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='K  2015 Oct 01  600  CSWA-164  LBT/MODS  g  2014 Jan 27  720  LBT/MODS  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' z  2014 Jan 27  360  CSWA-11  LBT/LBC  g  2015 Nov 15  600  LBT/LBC  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' z  2015 Nov 15  300  Keck/ MOSFIRE  K𝑠  2015 Nov 30  300  CSWA-139  LBT/MODS  g  2014 Jan 27  720  LBT/MODS  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' z  2014 Jan 27  360  MMT/MMIRS  H  2017 Jan 11  900  CSWA-141  LBT/ MODS  g  2014 Jan 27  720  LBT/ MODS  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' z  2014 Jan 27  360  Keck/ MOSFIRE  K𝑠  2014 Apr 11  300  Keck/ MOSFIRE  Y  2015 Nov 30  300  Keck/ MOSFIRE  J2  2015 Nov 30  300  CSWA-40  Keck/ESI  V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' I  2013 Mar 06  200  CSWA-16  LBT/MODS  g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' r  2014 Jan 27  360  CSWA-38  Keck/ESI  V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' I  2013 Mar 06  200  CSWA-13  LBT/MODS  g  2014 Jan 27  720  LBT/MODS  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' z  2014 Jan 27  360  LBT/LUCI  K𝑠  2014 Apr 13  1200  CSWA-39  Keck/ESI  V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' I  2013 Mar 06  200  CSWA-128  LBT/LBC  g  2015 Nov 15  600  LBT/LBC  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' i  2015 Nov 15  300  Keck/ MOSFIRE  K𝑠  2014 Apr 11  300  CSWA-163  LBT/LBC  g  2015 Nov 15  600  LBT/LBC  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' i  2015 Nov 15  300  MMT/MMIRS  J  2015 Sep 30  900  MMT/MMIRS  K  2015 Sep 30  1200  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' New optical and near-infrared imaging of CASSOWARY lensed  galaxies obtained with Keck, LBT, and MMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' From left to right, columns  denote the object ID, observatory and instrument, filters utilized, dates of  observations, and exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  we describe the optical and near-infrared imaging observations and  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Details of the new imaging observations are summarized  in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1  Optical imaging  We secured images of seven galaxies from Table 1 using the Multi  Object Double Spectrograph (MODS, Pogge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2010) on LBT in  January 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The dual channel mode of MODS provides images  in two separate filters simultaneously over a field of view of 6 ×  6 arc minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We used the blue and red channel to obtain g, r,  and z-band images for CSWA-13, CSWA-16, CSWA-116, CSWA-  139, CSWA-141, CSWA-164 and CSWA-165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Each arc was first  observed in the g and r-band for 360 seconds and then in the g  and z-band for an additional 360 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' For one source (CSWA  16), we obtained imaging only in the g and r-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The sky was  partly covered by clouds throughout the MODS observations and  the seeing varied between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Optical images of three  MNRAS 000, 1–20 (2022)  7  other galaxies (CSWA-39, CSWA-38, CSWA-40) were taken with  ESI on Keck, providing a field of view of 2×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 arc minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' These  three sources were observed with V, R and I filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' There was some  cloud during the observations, and the seeing was between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′8 and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' For the two other sources in Table 1, CSWA-2 and CSWA-19,  we make use of archival HST images (program 11974, PI: Allam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  We summarize the details of the imaging observations in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The optical images were reduced using standard routines for  flat fielding and image combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The 5𝜎 typical limiting magni-  tude in the g, r and z bands of MODS images are 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Similarly, for the ESI images the typical 5𝜎 limiting  magnitude in V, R and I bands are 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  For the archival HST images, the typical 5𝜎 limiting magnitudes in  F450W, F606W, and F814W filters are 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Each image was flux calibrated using stars that are isolated  and well detected in SDSS imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The mosaic in Figure 1 shows  the multi-color optical imaging obtained for each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Before performing photometry on the lensed galaxies, we mod-  eled the foreground lens using GALFIT (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2002) and  subtracted its contribution to the lensed source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' To calculate the ab-  solute magnitude for the galaxies in our sample, we first measured  the integrated flux in the g-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' After applying the appropriate  magnification corrections (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4), we arrive at the absolute UV  magnitudes shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The values span MUV = −23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 to  MUV = −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4, corresponding to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8-9 L★  UV at 𝑧 ≃ 2 − 3 (Reddy &  Steidel 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Parsa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  Near-infrared imaging  We obtained near-infrared images of CSWA-141, CSWA-11 and  CSWA-128 in the Ks-band using the Multi-Object Spectrometer for  Infra-Red Exploration (MOSFIRE, McLean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2012) on Keck I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Conditions were photometric with seeing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We obtained 10  dithered frames having 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 arc minute field of view, each with  exposure time of 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The 5𝜎 AB limiting magnitude in the  MOSFIRE Ks-band images is 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' With the goal of constraining  the equivalent width of [OIII] and H-beta emission, we observed  CSWA-141 with the MOSFIRE J2 medium band filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The filter  spans 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='11-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='25 𝜇m, covering H-beta and [OIII].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  We secured a Ks-band image of CSWA-13 using the LUCI  near-IR Spectrograph on the LBT when the seeing was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We  utilized the N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='75 camera providing 4×4 arc minutes of field of view  with a plate scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='12 arcseconds per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The near-IR imaging  reduction was performed using standard IDL routines designed to  flat field, sky-subtract and stack the dithered frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Finally, we  used isolated and unsaturated stars that are in the 2MASS catalog  to flux calibrate the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The 5𝜎 AB limiting magnitude in the  LUCI Ks-band image is 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' For another four galaxies (CSWA-  116, CSWA-165, CSWA-103, CSWA-163), near-IR imaging was  obtained using the MMIRS instrument on the MMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Each of the  four sources was observed in the J and K bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' MMIRS provides  imaging over a field of view of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 arc minutes with a plate  scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='20 arcsec per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The seeing was between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='′′9  throughout the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The typical 5𝜎 AB limiting magnitude  in MMIRS J and K-band images are 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4  Lensing Magnification  Derivation of stellar mass and intrinsic luminosity of lensed systems  requires magnification correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013b), magni-  fication factors were presented for eleven out of sixteen sources  given in Table 1 and typically range between 𝜇=5 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Three  out of the five remaining sources have a published lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We  adopted magnifications for CSWA-38 and CSWA-39 from Koester  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2010) and CSWA-11 from Leethochawalit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  For the two remaining sources (CSWA-13 & CSWA-16), we  follow a similar approach to Hezaveh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In brief, we  assumed a symmetric Gaussian light distribution for the lensed  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We also tested the impact of adding a second Gaussian  component to the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The foreground lens is modeled as a  Singular Isothermal Ellipsoid (SIE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Multiple lenses are allowed in  the modeling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' To obtain the best fit model, we perform  a Markov Chain Monte Carlo (MCMC) analysis to minimize 𝜒2  in the image plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In the case of CSWA-16, the observed data  are better fit after introducing a second Gaussian component to the  background source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The magnification factor is then calculated as  the ratio of the lensed to unlensed flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We derive a magnification  of 𝜇 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3 for CSWA-16 and 𝜇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2 for CSWA-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  Stellar Population Synthesis Modeling  Thirteen of the galaxies shown in Figure 1 have the necessary optical  and near-IR imaging to derive physical properties from broadband  SED fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' For these systems, we infer the stellar mass, specific  star formation rate and dust attenuation using the Bayesian galaxy  SED modeling and interpreting tool BEAGLE tool (version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Chevallard & Charlot 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' BEAGLE is based on the photoioniza-  tion models of star-forming galaxies in Gutkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2016), com-  bining the latest version of the Bruzual & Charlot (2003) stellar pop-  ulation synthesis models with the photoionization code CLOUDY  (Ferland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  We fit the broadband photometric fluxes as well as CIII] equiv-  alent widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' When relative fluxing between rest-UV and optical  emission lines is possible, we include all emission lines in the fit-  ting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The models assume constant star formation where  the maximum stellar age is allowed to vary freely between 5 Myr to  the Universe age at the given redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We use Chabrier (2003) initial  mass function and the Calzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2000) extinction curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The  metallicity is allowed to vary in the range of -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2≤log(Z/Z)⊙≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='25  assuming equivalent stellar and nebular metallicity (Z★=Z𝐼 𝑆𝑀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The redshift of all objects is fixed to their spectroscopic redshift  given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The ionization parameter (US;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' here defined as  the ratio of ionizing-photon to gas densities at the edge of the  Strömgren sphere) is varied in the range of -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0≤𝑈S≤-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0, and the  dust-to-metal mass ratio spanned within the range of 𝜉𝑑=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  We adopt models with hydrogen density (nH=100 cm−3) and C/O  abundance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 of solar value [(C/O)⊙ ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Finally, the prior  on the V-band dust attenuation optical depths ( ˆ𝜏𝑉 ) is taken as an  exponential distribution after fixing the fraction of attenuation op-  tical depth arising from the ambient ISM (𝜇) to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We note  that our assumption of constant star formation may underestimate  the stellar mass by as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 dex when the galaxy is dominated  by a young stellar population, but this would not affect the primary  conclusions of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We discuss the main results of this SED  fitting in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  3  RESULTS  All sixteen sources presented in this paper have optical spectra cov-  ering CIII] to quantify equivalent widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We report these values  (or 3𝜎 upper limits) in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Of the sixteen sources, six galaxies  have near-IR spectra enabling rest optical line flux ratios (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  MNRAS 000, 1–20 (2022)  8  Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='Lyα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='CIII]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='CIV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='SiIV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='OI CII  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='CSWA−164  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1890  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1900  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1910  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1920  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='CIII]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Rest-UV spectra of gravitationally lensed galaxies presented in Table 1 with CIII] detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The upper left side of each image contains the  CASSOWARY-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The dotted dashed line below and above the UV continuum represents absorption and emission features identified in the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The upper  right of each panel shows zoom in spectral coverage near CIII]𝜆𝜆1907,1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The vertical dashed lines in the inset show the location of CIII]𝜆𝜆1907,1909  doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The CIII] doublet remain unresolved in the MMT/BCS spectra of CSWA-13 and CSWA-139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  MNRAS 000, 1–20 (2022)  9  4850  4855  4860  4865  4870  4875  0  1  2  3  4  Flux(10−17erg−1s−1cm−2Å−1)  Hβ  CSWA−141  4980  4990  5000  5010  5020  5030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  [OIII]λ5007  6550  6555  6560  6565  6570  6575  6580  0  2  4  6  8  10  Hα  4855  4860  4865  4870  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  Flux(10−18erg−1s−1cm−2Å−1)  Hβ  CSWA−163  5000  5005  5010  5015  0  2  4  6  8  10  [OIII]λ5007  6550  6560  6570  6580  6590  0  1  2  3  4  5  6  Hα  [NII]  4855  4860  4865  4870  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='0  Flux(10−18erg−1s−1cm−2Å−1)  Hβ  CSWA−165  5000  5005  5010  5015  0  2  4  6  8  [OIII]λ5007  6550  6560  6570  6580  6590  0  1  2  3  4  5  6  7  [NII]  Hα  3720  3725  3730  3735  3740  0  2  4  6  8  10  Flux(10−18erg−1s−1cm−2Å−1)  [OII]λλ3727,3729  CSWA−164  5000  5005  5010  5015  0  2  4  6  8  10  [OIII]λ5007  6550  6560  6570  6580  6590  0  2  4  6  8  10  Hα  3720  3725  3730  3735  3740  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='0  Flux(10−18erg−1s−1cm−2Å−1)  [OII]λλ3727,3729  CSWA−11  4990  5000  5010  5020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  [OIII]λ5007  6550  6560  6570  6580  6590  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='0  Hα  4840  4850  4860  4870  4880  Rest Wavelength(Å)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  Flux(10−18erg−1s−1cm−2Å−1)  Hβ  CSWA−128  4980  4990  5000  5010  5020  5030  Rest Wavelength(Å)  0  5  10  15  20  [OIII]λ5007  6520  6540  6560  6580  6600  Rest Wavelength(Å)  0  2  4  6  8  10  12  Hα  [NII]  [NII]  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Optical emission lines in six gravitationally lensed galaxies discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Each row shows optical emission lines from a single source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The  object ID is given in the leftmost panel of each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The black line represents observed flux whereas blue line shows gaussian fit to the line profile calculated  using IDL routine MPFITPEAK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The region affected by skylines is shown as a grey swath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  MNRAS 000, 1–20 (2022)  10  Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  fν(µJy)  1  10  100  1000  CSWA−141  EWCIII]=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 Å  sSFR=31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2 Gyr−1  CSWA−13  EWCIII]=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 Å  sSFR=43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 Gyr−1  CSWA−139  EWCIII]=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 Å  sSFR=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 Gyr−1  CSWA−2  EWCIII]=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 Å  sSFR=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 Gyr−1  1  10  100  1000  CSWA−39  EWCIII]=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 Å  sSFR=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 Gyr−1  CSWA−19  EWCIII]=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 Å  sSFR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Gyr−1  CSWA−128  EWCIII]=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 Å  sSFR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0 Gyr−1  CSWA−103  EWCIII]=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Å  sSFR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 Gyr−1  1  10  100  1000  CSWA−163  EWCIII]<3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 Å  sSFR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8 Gyr−1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  λobs (µm)  CSWA−40  EWCIII]<1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 Å  sSFR=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 Gyr−1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  λobs (µm)  CSWA−165  EWCIII]<1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 Å  sSFR=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 Gyr−1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='0  λobs (µm)  CSWA−11  EWCIII]<1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 Å  sSFR=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='0  λobs (µm)  1  10  100  1000  CSWA−116  EWCIII]<1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 Å  sSFR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 Gyr−1  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' SEDs of galaxies in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The black circle represents broad band photometry data, blue line represents best fit population synthesis model to  the observed data (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5) and green square shows synthetic broad band flux from the best fit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The lower right of each panel shows object ID, sSFR  and CIII] equivalent widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='31  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='20  CSWA-116  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='27+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='34  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='18  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Physical properties inferred from BEAGLE fitting for subset of  CASSOWARY galaxies with optical and near-IR photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' From left to  right, the columns give the object name, C III] equivalent widths, magnifica-  tion corrected stellar mass, specific star formation rates and V-band optical  depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Ten sources have multi-wavelength photometry with SED infor-  mation enabling stellar mass and sSFR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Below we  briefly comment on individual galaxy properties where sources are  ordered by descending CIII] equivalent width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1  Notes on Individual Sources  CSWA-141 is an extreme equivalent width optical line emitting  galaxy at 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='425 with an integrated apparent magnitude of  𝑟 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The Magellan/FIRE spectrum reported in Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  (2013a) shows a large number of rest-frame optical emission  lines, including the temperature sensitive [OIII]𝜆4363 auroral line  and the density sensitive [SII]𝜆𝜆6717,6731 and [OII]𝜆𝜆3727,3730  lines (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We have since obtained deep Keck/ESI and  LBT/MODS optical spectra and optical and near-IR imaging, pro-  viding constraints on the rest-UV metal lines and the SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The optical to near-infrared SED of CSWA-141 (Figure 4) im-  plies a very large specific star formation rate (sSFR=31+61  −16 Gyr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  After correcting for magnification due to lensing (𝜇=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5), the stellar  mass and star formation rate of the best fitting model are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='98×108  M⊙ and 12+24  −7 M⊙ yr−1, respectively (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The flux in the  medium-band J2 MOSFIRE filter (covering 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='117 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='246 𝜇m) is  significantly in excess of that in adjacent filters, as expected given the  MNRAS 000, 1–20 (2022)  11  Line  𝜆rest  𝜆obs  Fline  EW  (Å)  (Å)  FCIII]1909  (Å)  Keck/ESI  [CIII]  1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='73  4624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='32  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='02(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='10)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3)  CIII]  1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='68  4628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='84  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4)  Fe II*  2626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='64  6370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='22(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='07)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5)  Mg II  2796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='36  6783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='42  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='30(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='07)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8)  2803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='53  6800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='65(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='04)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6)  [OII]  3727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='10  9039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='02  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='10(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='04)  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3(12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4)  3729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='86  9045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='68  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='62(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='09)  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2(19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9)  [NeIII]  3870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='16  9385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='28  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='79(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='23)  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8)  LBT/MODS  Line  𝜆rest  𝜆obs  Fline  EW  (Å)  (Å)  FCIII]1908  (Å)  [CIII]  1908  4622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='00  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5)  He II  1640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='52  3974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='16  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='08  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7  CIV  1549  3758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='75  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='11  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8  OIII]  1660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='81  4027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='09(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='02)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2)  1666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='15  4040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='23(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='03)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2)  Si III]  1882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='98  4566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='24(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='03)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  Si III]  1892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='03  4588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='11(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='02)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Rest-UV emission line measurements of CSWA-141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Emission line  fluxes are presented relative to the CIII]𝜆1909 line flux for the Keck/ESI  data, while the line fluxes are presented relative to unresolved combined  CIII] flux for LBT/MODS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The upper limits are 3𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  contamination by extremely strong [OIII] and H𝛽 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We use the  J2 flux excess to calculate the equivalent width from [OIII] and H𝛽,  following the methodology adopted at higher redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=',Stark  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2013b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This approach yields a rest-frame  equivalent width of W[OIII]+H𝛽 =730 Å, consistent with the very  young stellar populations (32 Myr for constant star formation) im-  plied by the population synthesis modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' While the optical line  EW is significantly in excess of what is seen in typical star forming  galaxies at 𝑧 ≃ 2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=',Boyett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2022), it is nearly identical to the  average [OIII]+H𝛽 EW at 𝑧 ≃ 7 − 8, as implied by flux excesses in  Spitzer/IRAC bandpasses (Labbé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' De  Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The continuum brightness  of CSWA-141 enables a unique and detailed view of this population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The ESI spectrum covers 4100-10000 Å, revealing promi-  nent emission from [CIII]𝜆1907, CIII]𝜆1909, Fe II★𝜆2626, Mg  II𝜆𝜆2797,2804, [OII]𝜆𝜆3726,3729, [Ne IIII]𝜆3869, and H𝛿 (Ta-  ble 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The [CIII], CIII] doublet is easily resolved by ESI (Figure 2)  with a total rest-frame equivalent width of WCIII]=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6Å, the largest  of the sixteen galaxies considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Nebular Mg II emis-  sion also exhibits a very large rest-frame equivalent width (WMgII  = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0 Å), as is common in lower mass galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=',Erb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Guseva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The MODS spectrum provides better  blue sensitivity than ESI, yielding detection of OIII]𝜆1661,1666  and Si III]𝜆𝜆1883,1892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The CIII] doublet is also detected, but it  is unresolved at the resolution of MODS (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The summed  equivalent widths of the OIII] and Si III] doublet (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0 Å and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0 Å,  respectively) are considerably lower than CIII] (see Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The  high ionization lines He II𝜆1640 and CIV𝜆𝜆1548,1550 are not de-  tected, implying rest-frame equivalent widths below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Å  (at 3𝜎), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' For emission lines detected by both ESI and  MODS, we will adopt whichever instrument provides the higher  S/N EW measurement for our subsequent analysis and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The rest-frame optical emission lines detected in the FIRE  spectrum provide an array of constraints on the nebular gas  physical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The detection of [OIII]𝜆4363 in the FIRE  spectrum enables a measure of the nebular electron temperature  (Te = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 × 104 K ) and the oxygen abundance via the direct  T𝑒 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Following the process discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3, we find that  12 + log(O/H) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='08, implying a gas-phase metallicity of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='18 Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Our measurement is very similar to Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2016b)  who reported oxygen abundance (12 + log(O/H)) of CSWA-141  based on our line flux measurements given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The derived  metallicity is also consistent with the metallicities derived from the  N2 and O3N2 indices with the calibration presented in Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  (2018) (12 + log(O/H) < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The O32 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6) and R23 indices (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4) point to a large ioniza-  tion parameter and gas excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The exact values depend on the  input ionizing spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Photoionization modeling using BEAGLE  resulted in a large ionization parameter of log 𝑈𝑠 = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1, which  is broadly consistent with O32 vs ionization parameter relationships  in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The  photoionization modeling further suggested a high ionizing photon  production efficiency of log (𝜉ion/Hz erg−1) = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This is  higher than the canonical value typically assumed for galaxy-driven  reionization model (Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2010) but consistent with other  strong CIII] emitters in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Nakajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  While the measured O32 and R23 are rare among more massive star-  forming galaxies at 𝑧 ≃ 2, they are consistent with the O32-sSFR  and O32-R23 trends that are observed at high redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Sanders  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Strom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The electron densities inferred from the flux ratios of the [OII]  and [SII] doublets using PyNeb (160+76  −74 cm−3 and 350+294  −206 cm−3,  respectively) are consistent with the median density of more massive  star forming galaxies at 𝑧 ≃ 2 (250 cm−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  In contrast, the resolved [CIII], CIII] doublet suggests gas at very  high density (16500+12100  −7800 cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Such an offset between densities  derived from CIII] and those inferred from [OII], [SII] have been  seen in other galaxies at high redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We  will come back to discuss this in more detail in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Finally, we note that the MODS and FIRE spectra reveal emis-  sion lines at 3827, 15304, 15612, 15763, 20663 Å which appear  nearly spatially coincident with CSWA-141 but are not associated  with the 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='425 galaxy (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We identify these as Ly𝛼,  H𝛽, [OIII]𝜆𝜆4959,5007, and H𝛼 in a second fainter gravitationally  lensed source at 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This higher redshift galaxy is unre-  solved from the 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='425 source in existing ground-based optical  and near infrared images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Given that the H𝛼 flux of the 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='425  galaxy is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4× larger than the 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='148 source, we expect that  the newly-discovered higher redshift source contributes negligibly  (at the 5% level) to the broadband flux and the rest-UV continuum  in the MODS and ESI spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Higher resolution imaging will be  required to disentangle the two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-13 is a bright galaxy at 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='87 that was first confirmed  in Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a) through the detection of Ly𝛼 emission and nu-  merous interstellar absorption lines in an MMT blue channel spec-  trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' CIII] emission is confidently detected (WCIII]=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 Å) in  the discovery spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' He II is also detected (WHeII=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 Å)  with a broad FWHM (2430 km s−1) that is indicative of a stellar  wind origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We do not detect the OIII] doublet, implying individual  components with equivalent widths less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' While this is  among the strongest CIII] emitters in our sample, the redshift places  the strong rest-optical lines in regions of poor atmospheric transmis-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We have obtained multi-wavelength imaging in the optical and  near-IR (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The SED reveals a large sSFR (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 Gyr−1), lit-  MNRAS 000, 1–20 (2022)  12  Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  tle dust attenuation ( ˆ𝜏𝑉 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='53), and a magnification-corrected stellar  mass of log (M★/M⊙) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-139 was confirmed to have a redshift of 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='54 in  Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a) based on the presence of Ly𝛼 absorption and in-  terstellar metal absorption lines in an MMT spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The spectrum  also shows emission from the [CIII],CIII]𝜆𝜆1907,1909 doublet with  a total equivalent width of WCIII] = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Here we present  new optical and near-IR imaging from the LBT and MMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The SED  is best fit with an sSFR of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 Gyr−1, ˆ𝜏𝑉 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='77, and a stellar mass  of log (M★/M⊙) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3 after magnification correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-2 (SDSS J1038+4849) was first reported in Belokurov  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2009), and the source redshift (𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='20) was subsequently  confirmed in Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013) through detection of rest-optical  emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The lens reconstruction described in Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  (2013) shows that CSWA-2 is a merger of two systems with a stel-  lar mass ratio (6 ± 3):1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' log (M★/M⊙) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 and one of the  largest sSFR (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 Gyr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The oxygen abundance 12+log(O/H)  of the system as calculated by using N2 index is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='25 (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 Z⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The Balmer decrement ratio (H𝛼/H𝛽=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='47) suggests little nebu-  lar extinction whereas O3 of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='80 imply relatively larger excita-  tion from ionized gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Using the metallicity calibration of Bian  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2018), we estimate the oxygen abundance 12+log(O/H) of  the system using the O3 index as 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Z⊙) In this paper, we  present new optical spectroscopy of this system obtained with ESI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The spectrum is dominated by continuum emission from the lower  mass source (denoted J1038 North in Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013) which  is considerably brighter in the optical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The spectrum shows emis-  sion from [CIII],CIII]𝜆𝜆1907,1909 with a total equivalent width of  WCIII]=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-39 (SDSS J1527+0652), a bright (𝑟 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5) 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='759  galaxy was first identified as part of the SDSS Giant Arcs Sur-  vey (SGAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Hennawi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2008) and was spectroscopically con-  firmed by Koester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2010) through detection of Ly𝛼 emission  and interstellar absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We have obtained an ESI opti-  cal spectrum and multi-color optical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Both components of  the [CIII], CIII] doublet are detected in the spectrum, with a total  S/N=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 for the summed doublet flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The rest-frame equivalent  width (WCIII]=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 Å) is typical of similarly luminous galaxies at  this redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Shapley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We also  detect Ly𝛼 emission with a moderate rest-frame equivalent width  (WLy𝛼=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' No other nebular UV lines are detected in the  ESI spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The upper limit on the OIII]𝜆𝜆1661,1666 compo-  nents imply equivalent widths below 1 Å, consistent with the OIII]  emission strengths in the Shapley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2003) composite of 𝑧 ≃ 3  galaxies with similar Ly𝛼 equivalent widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-38 (SDSS J1226+2152) was confirmed to lie at 𝑧 =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='923 by detection of Ly𝛼 and metal absorption lines in (Koester  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Like CSWA-39, this source was first identified in  Sloan Giant Arcs survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Here we present deep ESI spectroscopy  and imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The optical spectrum reveals a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1𝜎 detection of  the CIII]𝜆1909 component, but the [CIII]𝜆1907 component is situ-  ated on a skyline, precluding a useful limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The rest-frame equiv-  alent width of the CIII]𝜆1909 component (WCIII]𝜆1909=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 Å) is  comparable to CSWA-38 and CSWA-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Ly𝛼 emission is weak  (WLy𝛼=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 Å), and no other nebular UV lines are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-19 (SDSS J0900+2234) was first confirmed in (Diehl  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2009) at 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='03 via detection of Ly𝛼 emission and several  metal absorption features in a spectrum from the dual imaging spec-  trograph (DIS) on the Astrophysical Research Consortium (ARC)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 meter telescope at the Apache Point Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We have since  obtained a deep ESI spectrum and MMT near-infrared imaging of  this source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The combined optical and near-infrared SED is best-  fit by a model with a (magnification-corrected) stellar mass of log  (M★/M⊙) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5, a specific star formation rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Gyr−1, and  V-band attenuation optical depth of ˆ𝜏𝑉 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The ESI spectrum  of the source shows weak detections of [CIII], CIII] 𝜆𝜆1907, 1909  with an integrated S/N=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 across both components of the doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The total rest-frame equivalent width (WCIII]=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 Å) is among the  lowest in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' No other nebular rest-UV lines are detected,  as Ly𝛼 is situated to the blue of the ESI coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-128 (SDSS J1958+5950) is a bright (r=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7) z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='22  galaxy that was spectroscopically confirmed in Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a)  through detection of rest-optical emission lines in an LBT/LUCI  near-infrared spectrum and interstellar metal absorption lines in  an MMT optical spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We have since obtained optical and  near-infrared imaging and a deep ESI optical spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The  magnification-corrected SED (Figure 4) implies a stellar mass of log  (M★/M⊙) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3 and an sSFR of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0 Gyr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The ESI spectrum  reveals a faint 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8𝜎 detection of [CIII], CIII] emission at the sys-  temic redshift (𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='225) defined by the rest-optical lines, implying  a rest-frame equivalent width of WCIII]=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 Å for the doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' No  other nebular lines are detected in the rest-UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The flux ratios of rest-optical lines (Table 4) constrain the gas  physical conditions, providing a framework to understand the weak  UV nebular line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We infer the gas-phase oxygen abun-  dance using the R23, O32 and O3 calibration presented in Bian  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The three indices suggest metallicities of 12+log O/H  = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2 (O32), 12+log O/H=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 (R23) and 12+log O/H=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 (O3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' As  discussed in Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2020), O32 metallicity calibration in  Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2018) may provide a better estimate of oxygen abun-  dance for high redshift galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This suggests that the metallicity of  the ionized gas of CSWA-128 is ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The values of O32  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3) and R23 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0) are a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 smaller than that of  the strong CIII] emitter CSWA-141, consistent with expectations  for nebular gas with a lower ionization parameter and excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  In contrast, the electron density implied by the [SII]𝜆𝜆6717,6731  doublet (200 cm−3) is similar to that found in both strong CIII]  emitters such as CSWA-141 and typical 𝑧 ≃ 2−3 galaxies (Sanders  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-103 (SDSS J0145-0455) is a 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='96 galaxy that was  confirmed in Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a) through detection of metal absorp-  tion lines and weak Ly𝛼 emission in an MMT blue channel spec-  trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We have since obtained a moderate resolution Keck/ESI spec-  trum and optical and near-infrared imaging with LBT and MMT,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The broadband SED (Figure 4) is best fit by a stel-  lar synthesis model with a stellar mass (magnification corrected)  of log (M★/M⊙) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4, sSFR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 Gyr−1, and ˆ𝜏𝑉 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The ESI  spectrum constrains rest-frame wavelengths 1350-3425 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Positive  emission is detected from both components of the [CIII], CIII] dou-  blet, implying a total rest-frame equivalent width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' No other  nebular emission lines are observed in the ESI spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-164 (SDSS J0232-0323) was spectroscopically con-  firmed in Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a) based on the presence of Ly𝛼 emission  and interstellar absorption lines in an MMT blue channel spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a) also presented detection of [OII], [OIII]𝜆5007,  and H𝛼 in a Magellan/FIRE near-infrared spectrum of CSWA-164,  revealing a systemic nebular redshift of 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We have since  obtained a deep moderate resolution optical spectrum with ESI  and optical broadband imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The ESI spectrum reveals the Ly𝛼  emission (W𝐿𝑦𝛼 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0 Å) detected previously together with weak  emission (S/N=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1) from the [CIII],CIII]𝜆𝜆1907,1909 doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This  corresponds to a total CIII],CIII] equivalent width of just 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 Å, the  smallest measured value in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  MNRAS 000, 1–20 (2022)  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='48  λobs(µm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  Relative Fλ  Lyα (z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='148)  CIII]1908 (z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='425)  Si III]1883 (z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='425)  OIII]1666 (z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='425)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='54  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='56  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='58  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='60  λobs(µm)  0  2  4  6  8  Hα (z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='425)  [OIII]λ4959 (z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='147)  Hβ (z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='147)  [OIII]λ5007 (z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='147)  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Optical and NIR spectra of CSWA-141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' A secondary source (CSWA-141b) is identified in the Optical+NIR spectra (see §3) (Left:) Ly𝛼 emission at  𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='148 from CSWA-141b in the LBT/MODS spectrum alongside [SiIII]𝜆1883 and CIII]𝜆1908 from CSWA-141 at 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (Right:) Rest optical lines  ([OIII]𝜆𝜆4959,5007, H𝛽) from CSWA-141b at 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='147 observed in the FIRE spectrum alongside the H𝛼 emission line from CSWA-141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The FIRE spectrum provides insight into the ionized gas phys-  ical conditions of CSWA-164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Unfortunately both H𝛽 and [NII] are  obscured by skylines, precluding a robust determination of the oxy-  gen abundance through standard strong line calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We can  however estimate oxygen abundance using O32 metallicity calibra-  tions from Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We find that O32=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 corresponds  to the oxygen abundance of 12+log O/H=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Thus the ionized gas  appear to be reasonably metal rich in CSWA-164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The value of  ionization-sensitive line ratios (O32 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7) is among the lowest in  our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In contrast to the metallicity and ionization parameter,  the electron density derived from the [OII] doublet (165 cm−3) is  consistent with the range spanned by other galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-40 (SDSS J0952+3434) is a 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='190 galaxy identified  through the Sloan Bright Arcs Survey and spectroscopically con-  firmed by Kubo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2010) using the DIS on the ARC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 meter  and the RC Spectrograph on the Mayall 4 meter telescope at Kitt  Peak National Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The spectra reveal Ly𝛼 in absorption  along with several other metal absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We have obtained a  moderate resolution optical spectrum and imaging with Keck/ESI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The continuum S/N of the CSWA-40 ESI spectrum is lower than  other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This is the only ESI spectrum that does not reveal  detection of the [CIII], CIII] doublet, implying a rest-frame equiv-  alent width below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 Å for the sum of both components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' No other  nebular emission lines are observed in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-163 (SDSS J2158+0257) was spectroscopically con-  firmed in Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a) based on identification of metal ab-  sorption lines and Ly𝛼 absorption in an MMT blue channel spec-  trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' A Magellan/FIRE near-infrared spectrum was also obtained  in that paper, revealing a systemic nebular redshift of 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='079  based on detection of [OII], H𝛾, H𝛽, [OIII], H𝛼, and [NII].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We  have obtained new optical and near-infrared imaging of this source,  allowing constraints to be placed on the broadband SED (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The data are best fit by stellar synthesis models with stellar mass  log (M★/𝑀⊙) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 after magnification correction, V-band at-  tenuation optical depth of ˆ𝜏𝑉 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='67, and an sSFR of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8 Gyr−1,  suggesting CSWA-163 is a relatively massive galaxy with a fairly  evolved stellar population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Perhaps not surprisingly given the low  sSFR and lack of Ly𝛼 emission, the MMT blue channel spectrum  does not show any CIII] emission at the systemic redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The 3𝜎  flux upper limit suggests that the rest-frame equivalent width of the  double must be lower than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The FIRE spectrum provides useful constraints on the ionized  gas of CSWA-163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The oxygen abundance can be derived from O32  and O3 calibration from Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Both suggest moderately  enriched gas: 12+log O/H=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 (O32) and12+log O/H=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 (O3) im-  plying a nebular oxygen abundance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The ionization-  sensitive line ratios (O32=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1, O3=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3) are consistent with a rel-  atively low ionization parameter and moderate gas-excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The  electron density inferred from [OII] is 144 cm−3, consistent with  the other systems studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-16 (SDSS J1111+5308) was confirmed to have a red-  shift of 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='95 based on the presence of numerous metal absorp-  tion lines in an MMT blue channel spectrum (Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2013a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The discovery spectrum shows no emission from the blended [CIII],  CIII] doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The 3𝜎 upper limit on the total flux from the doublet  requires the rest-frame equivalent width to be smaller than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  We have obtained multi-band optical imaging with LBT, allowing  more robust constraints to be placed on the apparent magnitude  (r=21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8) and the magnification-corrected UV absolute magnitude  (MUV=−20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-165 (SDSS J0105+0144) is a 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='13 galaxy that  was first confirmed in Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013b) through detection of  strong metal absorption lines and weak Ly𝛼 emission in an MMT  blue channel spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We have since obtained a Magellan/FIRE  near-infrared spectrum and optical and near-infrared imaging from  LBT and MMT respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The FIRE spectrum reveals detection  of [OII], H𝛽, [OIII]𝜆5007, H𝛼, and [NII], indicating a systemic  redshift of 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The MMT shows no emission from the  [CIII], CIII] doublet at the systemic redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The 3𝜎 upper limit  on the flux of the doublet indicates that the total CIII] equivalent  width must be lower than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The broadband SED is best-fit by  a synthesis model with stellar mass of log(M★/M⊙) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0, sSFR  of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 Gyr−1, and ˆ𝜏𝑉 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The gas-phase metallicity can be inferred from O32, R23, and  O3 strong line calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' All three indicate metal rich ionized  gas: 12+log O/H = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 (O32) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 (R23), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 (O3), consistent with  a gas-phase metallicity in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8 Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The ionization-  sensitive ratios (O32=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6, O3=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1) suggest an ionization parameter  that is lower than average among 𝑧 ≃ 2 − 3 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In contrast,  MNRAS 000, 1–20 (2022)  14  Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  the electron density derived from the [OII] doublet flux ratio (220  cm−3) is similar to that found in other systems at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-11 (SDSS J0800+0812) was confirmed at 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='41 via  detection of metal absorption lines and [OII] emission in MMT  blue and red channel spectra (Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2013a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The CIII] dou-  blet is not detected in either the blue or red channel MMT spectra,  implying a rest-frame equivalent width below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9 Å at 3𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We  have obtained multi-wavelength imaging and near-infrared spec-  troscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The magnification-corrected SED suggests a stellar mass  of log(M★/M⊙) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3, an sSFR of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 Gyr−1, and V-band  attenuation optical depth of ˆ𝜏𝑉 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The FIRE spectrum reveals  detections of [OII], [OIII]𝜆5007, and H𝛼, but strong skylines ob-  scure detections of H𝛽, [OIII]𝜆4959 and [NII]𝜆6586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Using the  theoretically-expected flux ratio of [OIII]𝜆5007/[OIII]𝜆4959, we  infer that this system has O32 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='92, relatively low for our sam-  ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Using the O32 calibration, we estimate metallicity of 12+log  O/H=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The electron density implied by the [OII] doublet flux  ratio is 469+416  −238 cm−3, consistent with the range spanned by other  galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-116 (SDSS J0143+1607) is a 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='50 galaxy con-  firmed in Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013a) through detection of rest-UV metal  absorption lines and [OII] in MMT blue and red channel spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  We do not detect the CIII] doublet in the MMT spectra, indicating  that the rest-frame equivalent width is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 Å at 3𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We have  obtained optical and near-infrared imaging of CSWA-116, provid-  ing constraints on the broadband SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The data are best fit by a  stellar model with log(M★/M⊙) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4, an sSFR of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3 Gyr−1, and  V-band attenuation optical depth of ˆ𝜏𝑉 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  Physical Properties of CASSOWARY galaxy sample  The data obtained for this paper provide new constraints on the  ionized gas properties and the stellar populations of lensed galaxies  at 𝑧 ≃ 1 − 3 identified by the CASSOWARY selection in SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Here we briefly describe what these data reveal about the aver-  age properties in this sample, providing a concise summary of the  source-by-source description presented in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Following correc-  tion for magnification, the rest-frame UV absolute magnitudes are  found to range between MUV=−20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2 and MUV=−23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0, with a me-  dian value (MUV=−21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9) that is roughly three times the value of  L★  UV at 𝑧 ≃ 2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Reddy & Steidel 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Both optical and near-  infrared imaging exist for 11 of the 16 galaxies considered in this  paper, allowing the stellar content to be characterized through SED  fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The median stellar mass and sSFR of this subset is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3×1010  M⊙ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1 Gyr−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The latter is similar to the average  sSFR of 𝑧 ≃ 2 − 3 galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Reddy & Steidel 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Rest-optical line measurements exist for seven of the CAS-  SOWARY galaxies shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The median gas-phase oxy-  gen abundance derived from the rest-optical line ratios is 12+log  O/H = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='33, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', ionized gas metallicity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The ionization-  sensitive ratio O32 ranges between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 with a median value  of O32=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' While this is slightly lower than the median O32 in the  KBSS and MOSDEF surveys, it is well within the range spanned  by galaxies in these samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Strom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Steidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The Balmer decrements range between  H𝛼/H𝛽=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='47 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The median electron density derived from  the [OII] or [SII] doublet flux ratios (198 cm−3) is close to the aver-  age electron density of 𝑧 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3 galaxies from the MOSDEF survey  (Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  All sixteen galaxies in our sample have constraints on rest-UV  line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Most sources show very weak [CIII], CIII] emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The median equivalent width for the summed doublet is 1 Å, similar  to that seen in composite spectra of 𝑧 ≃ 3 LBGs (Shapley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Llerena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The two strongest [CIII], CIII] emitters (CSWA  141, CSWA -13) have rest-frame equivalent widths of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 Å and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 Å, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The rest-UV spectrum of CSWA-141 also shows  prominent nebular emission from OIII], Si III] and Mg II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' While  these lines are absent in CSWA-13, its spectrum does reveal broad  He II emission, indicative of a significant Wolf Rayet population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  No high ionization nebular lines are detected in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Our sample allows to compare and contrast ISM conditions ex-  pected in strong and weak CIII] emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' A typical CIII] equivalent  widths of star-forming galaxies at 𝑧 ∼ 2−3 is ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 Å (Shapley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' From here on, we refer to those galaxies with  CIII] equivalent widths > 2× typically seen at 𝑧 ∼ 2−3 (≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Å) as  strong CIII] emitters, whereas objects with CIII] equivalent widths  below this threshold are described as weak CIII] emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Together  with galaxies presented in this paper, we compile sources from the  literature having constraints from both CIII] equivalent widths and  optical spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The majority of the literature objects are lower red-  shifts galaxies (Giavalisco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Leitherer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Berg  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Senchyna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Ravindranath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2020), while majority of sample at 𝑧 > 1 are comprised of gravita-  tionally lensed systems (Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' de Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Vanzella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Bayliss  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Vanzella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Erb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Quider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Hain-  line et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Pettini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Teplitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We present a compilation of 𝑧 > 1 sources in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  In Figure 6, we plot empirical relationships between CIII]  equivalent width and oxygen abundance (left panel), and CIII]  equivalent width and O32 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The oxygen abundance  of the strong CIII] emitters are mostly measured using direct metal-  licity measurements, while strong optical line calibrations are used  for the weaker CIII] emitters (see Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' As can be seen in the  Figure 6, the gas-phase metallicity of the strong CIII] emitters is  consistently below 12+log(O/H)=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0, suggesting metallicities be-  low 20% Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In contrast, most of the weaker CIII] emitters imply  higher metallicities than this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This result is consistent with pre-  vious investigations in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Jaskot  & Ravindranath 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Maseda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Nakajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Schaerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Senchyna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Ravindranath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The right panel of Figure 6 shows  that the O32 values of the stronger CIII] emitters (≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Å) are on  an average six times larger than the those of the weaker CIII] emit-  ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The median O32 value of the weaker CIII] emitters are similar  to typical galaxies at 𝑧 ∼ 2 − 3 (O32=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3,Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' A  variety of factors can influence the O32 value of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The ele-  vated values seen in the strongest of CIII] emitters tend to be found  in galaxies dominated by the light of a very young stellar popula-  tion (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020), as expected in galaxies  undergoing a burst of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Overall these empirical rela-  tionships support a physical picture where galaxies with metal poor  gas and young stellar populations are able to power strong CIII]  emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Senchyna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The scatter seen in the CIII] EW at a given metallicity or  O32 value may stem from differences in ISM conditions (relative  carbon abundances (C/O), ionization parameters) and stellar age  and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The compilation of strong CIII] emitters further provides in-  formation on the global spectral properties expected from typical  reionization era systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In Figure 7, we show strong CIII] emit-  ters (denoted by red square) and weak CIII] emitters (denoted by  blue square) in log([OIII]/H𝛽) vs sSFR (left panel) and log(O32)  MNRAS 000, 1–20 (2022)  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  EWCIII] (Å)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  12+log(O/H)  This work  Mainali+2020  Ravindranath+2020  Stark+2014  James+2014  Erb+2010  Stark+2015  deBarros+2014  Christensen+2012  Bayliss+2014  Vanzella+2016/17  Rigby+2015/21  Berg+2016/19  Leitherer+2011  Giavalisco+1996  Quider+2009  Pettini+2000  Berg+2018  Senchyna+2017/19  1  10  EWCIII] (Å)  1  10  O32  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (Left:) Empirical relationship between oxygen abundance (12+log(O/H)) and rest frame CIII] equivalent width (EWCIII]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (Right:) Empirical  relationship between the line ratio of [OIII]𝜆𝜆4959,5007 to [OII]𝜆𝜆3727,3729 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' O32) and rest-frame CIII] EW (EWCIII]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The red symbol represents bright  lensed galaxies presented in this paper, while other data points are compilation from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The strong (blue) and weak (red) CIII] emitters (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2) in O3 vs sSFR (left) and O32 vs R23 (right) plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Star forming galaxies at 𝑧 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3 from  KBSS survey (Strom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016) are shown in cyan circle whereas MOSDEF survey (Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016a) are shown in violet triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Local galaxies from  SDSS survey are shown in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Galaxies with properties similar to those at z>6 tend to have strongest C III] emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  vs log(R23) (right panel) plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We compare them to typical line  ratios expected at 𝑧 ∼ 2 using dataset from KBSS survey (Steidel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Strom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016) and MOSDEF survey (Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2016a), as well as at 𝑧 ∼ 0 (local SDSS galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The strong CIII]  emitting galaxies appears to be distinct from local SDSS galaxies  as well as typical galaxies at 𝑧 ∼ 2 in both the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' However, the  position of weaker CIII] emitters on both the diagrams is similar to  typical galaxies at 𝑧 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Taken together, this further demonstrates  that strong CIII] emitters show highly ionized gas conditions from  a large sSFR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Assuming these strong CIII] emitters repre-  sentative of a typical reionization era systems, we might expect a  similar ISM conditions in galaxies at 𝑧 > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  4  DISCUSSION  In this paper, we present spectra of some of the brightest-known  gravitationally-lensed galaxies at 𝑧 ≃ 2−3, discovered over the foot-  print of SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Included in this sample are CSWA-13 and CSWA-  141, two exceptionally bright systems with sSFRs (> 20 Gyr−1)  that are similar to those of the reionization-era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Their spectra reveal  rest-frame CIII] equivalent widths more than twice what is typical  at 𝑧 ≃ 2 − 3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Shapley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Llerena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In this section, we explore the ionized gas conditions and  the properties of the outflowing gas, taking advantage of emission  and absorption lines that are often too faint to be detected in indi-  vidual high redshift galaxies with similarly intense emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  MNRAS 000, 1–20 (2022)  1og([OIII]25008 / Hβ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  Strong CIIIl emitters  Weak CIIIl emitters  KBSS z~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3  SDSS z~0  2  1  0  2  1  log(sSFR /Gyr-1)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  Strong CIIIl emitters  Weak CIIIl emitters  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  KBSS z~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3  MOSDEF z~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='3  log(O32)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  ■SDSS z～0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  log(R23)16  Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  EWFeII λ2374 (Å)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  EWFeII λ2382 (Å)  CSWA−141  optically  thin  optically  thick  Erb+2012 (Composite)  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' A comparison between equivalent widths of Fe II 𝜆2374 and  Fe II 𝜆2382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The red square represents CSWA-141 data point whereas black  circles are composite data points presented in Erb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The two  dashed black lines represent optically thin and optically thick cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This  reflects that Fe II is not optically thick in CSWA-141 and therefore has a  lower column density c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' black points  Our analysis will primarily focus on CSWA-141, as the redshift of  CSWA-13 places the strong rest-optical lines in regions of low at-  mospheric transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' For CSWA 141, the rest-optical lines are  similar in equivalent width to those found in the reionization era  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', De Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Boyett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2022),  classifying this galaxy as an EELG and revealing a stellar population  dominated by a recent burst or upturn in star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The gas properties in such intense line emitting galaxies have  been the subject of a number of spectroscopic investigations in  recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' These studies demonstrate that the nebular gas is gen-  erally under extreme ionization conditions in EELGs (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2018), with the ionization parameter reaching its largest values in the  galaxies powered by the youngest stellar populations (or the largest  equivalent width rest-optical nebular lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The gas in CSWA-141  is consistent with this picture, showing a large O32 ratio (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7) as  expected given its elevated sSFR (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2 Gyr−1) and [OIII]+H𝛽 EW  (730 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Its large ionization parameter implied by the photoion-  isaiton models (log U = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1) suggests a large photon density is  impingent on the ionised gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This can be driven be a variety of fac-  tors, including efficient ionizing photon production (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Chevallard  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2021) and a compact configuration of ion-  ized gas around the sources of ionizing radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The latter is the  norm for galaxies dominated by very young stellar clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=',  Whitmore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Hannon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2023), and  thus may be expected in EELGs like CSWA-141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Further insight into the gas conditions of EELGs is made possi-  ble by detection of three density-sensitive emission lines in CSWA-  141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' As we discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1, the [CIII], CIII] doublet implies  very high gas densities (16500+12100  −7800 cm−3), perhaps again reflect-  ing a very compact or concentrated gas geometry surrounding the  extremely young star clusters that power EELGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Although the un-  certainty in CIII] density is currently large, such high densities are  routinely seen in 𝑧 ≃ 2 − 4 galaxies with spectrally-resolved CIII]  measurements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA-141  EWMgII = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5Å  F2797  F2803 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Mg II 2797,2803 line profile in CSWA-141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The black curve  represents continuum subtracted flux level while the green solid line is  gaussian fit to the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Both the Mg II doublet are well fitted by a  single component gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Mg II 2797 is clearly stronger than Mg II 2803  line, suggesting a Mg II doublet intensity ratio (𝐹2797/𝐹2803) of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The doublet ratio is consistent with the intrinsic recombination value (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0)  indicating minimal scattering by low-ionization ISM along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Based on the observed Mg II doublet ratio, CSWA-141 has implied escape  fraction of 𝑓esc(LyC) = 27±4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Bayliss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Acharyya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2019) and are also starting to be  seen in the first handful of 𝑧 ∼> 7 galaxies with CIII] doublet mea-  surements (Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' As such, the high  C III] density is less likely due to a statistical fluctuation, although  a larger sample should confirm this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In most of the cases, the CIII]  density is also significantly in excess of that inferred from lower-  ionization species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The same trend is apparent in CSWA-141, with  the CIII] density being nearly two orders of magnitude higher than  those derived from [SII] and [OII].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This is consistent with a picture  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Acharyya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2021)  dictated by the ionization structure of the nebulae, with the lower  ionization rest-optical lines (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', [OII] and [SII]) probing primarily  the outer layers which are preferentially dominated by lower density  gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The higher ionization lines (like CIII]) probe the inner regions  of the nebula, where densities are expected to be higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Variations  in the critical density of the ions will additionally contribute to CIII]  probing higher density gas than [OII] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' A key  question is whether very young EELGs like CSWA-141 (and many  of the 𝑧 > 7 galaxies) might have a large fraction of their ionized  gas in very dense clumps, as is expected at the earliest evolution-  ary phases following the formation of star clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Rigby & Rieke 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Currently samples with CIII] densities  tend to be those that have at least moderately large CIII] EW, gener-  ally indicative of a relatively young stellar population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Larger CIII]  density samples are required to test whether the presence of very  large densities is at all sensitive to the luminosity-weighted stellar  population age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  CSWA 141 also provides a unique window on the kinematics  and covering fraction of the outflowing gas in EELGs, a population  which becomes commonplace at 𝑧 > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' As it is this outflowing  gas which regulates the escape of ionising radiation, systems like  MNRAS 000, 1–20 (2022)  4  3  itive  Rel  2792  2794  2796  2798  2800  2802  2804  2806  Wavelenqth (A)17  CSWA 141 offer potential for understanding the likely contribu-  tion of EELGs to reionisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' At low redshift, (Jaskot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017) find that the most extreme [OIII] emitting  galaxies (the Green Peas) tend to have low outflow velocities and  suggest that this is consistent with models of suppressed superwinds,  where catastrophic cooling prevents the development of large scale  outflows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Silich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Silich & Tenorio-Tagle 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Gray  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The low ionization absorption lines in the CSWA 141  spectra are all blueshifted with respect to the systemic redshift, in-  dicating the presence of outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In the ESI spectrum, we detect  Fe II and Mg II absorption lines, while the MODS spectrum probes  slightly bluer wavelengths, allowing detection of Al II𝜆1670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The  average velocity of low ionization outflowing gas in CSWA 141 is  76 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This is slightly lower than outflow velocities of 100-300  km s−1 that are commonly observed in galaxies at 𝑧 ∼ 1 − 3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=',  Shapley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Weiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Steidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Jones  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' However given the low stellar mass of CSWA-141, the  measurement is consistent with expectations from known trends be-  tween galaxy mass and outflow velocity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Martin 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Weiner  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Erb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The strength of the absorption lines in CSWA 141 provide  insight into the opacity of neutral gas along the line of sight to the  young star clusters which dominate the light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The Fe II 𝜆2374 and  Fe II 𝜆2382 absorption lines are very weak with equivalent widths  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Å and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Å, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2018) measured a  column density N(Fe II) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2 × 1014 cm−2, which is among  the lowest of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In Figure 8, we compare these equivalent  widths with those measured from composite spectra of more typical  star forming galaxies at 1 < 𝑧 < 2 (Erb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' As is clear in  the figure, most star forming galaxies at these redshifts show similar  Fe II 𝜆2374 and Fe II 𝜆2382 equivalent widths, as expected for  optically thick neutral gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In contrast, the Fe II 𝜆2374 absorption in  CSWA-141 is significantly weaker than Fe II 𝜆2382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' With an Fe II  𝜆2374 EW that is roughly three times weaker than that found in the  𝑧 ≃ 1 − 2 composites, the lines are much closer to expectations for  optically thin neutral gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We additionally note that the Fe II 𝜆2382  line is susceptible to emission filling (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Erb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2012) which  tends to weaken the observed Fe II 𝜆2382 equivalent widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This  is particularly likely to be the case for EELGs like CSWA-141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  In such a scenario, the line ratio would move even closer to that  expected for optically thin conditions, implying a low covering frac-  tion and potentially low column density of neutral gas in the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Further indications that CSWA 141 has a low column density  of neutral gas comes from the resonant Mg II𝜆𝜆 2797,2803 emis-  sion line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Both components of the doublet are confidently detected  (S/N>10) in emission with total equivalent widths of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Å (Fig-  ure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This is not only one of the highest measured Mg II EWs at  z>1, but it is also one of the brightest Mg II emission lines, making  detailed (and resolved) study of the line uniquely possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Recent  studies have pointed out that the flux ratio of the doublet is strongly  sensitive to the neutral gas column density in the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' As such  it is thought to correlate closely with the ionizing photon escape  fraction (Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Seive et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2023), perhaps providing one of the best  indirect indicators of photon leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' At low HI column densities  (<1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2 cm−2), galaxies become optically thin to the resonant Mg  II line photons (Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This leads to strong nebu-  lar Mg II emission, and it drives the doublet ratio (R=F2797/F2803)  to its intrinsic value of 𝑅 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' If the line photons are resonantly  scattered by Mg+ ions in the neutral gas along the line of sight, the  doublet ratio will decrease, asymptoting to a value of R=1 if the gas  is optically thick to Mg II photons (see Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020 for a  detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The measured Mg II doublet ratio in CSWA-141 is R=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2,  consistent with the intrinsic value produced in the HII regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This  suggests that optically thin channels along the line of sight to the  young star clusters are powering the nebular emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Both compo-  nents of the doublet are well-fit by single Gaussians, suggesting that  the line profile is not significantly impacted by resonant scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The very large EW of the line also points to minimal attenuation of  line photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We note that the Mg II profile does show very weak  blue-shifted absorption (see Figure 9), suggesting the presence of  some neutral gas along the line of sight, but this gas must be either  optically thin or clumpy with low density channels to result in the  observed line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Given the derived oxygen abundance of CSWA-141 (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1)  and nominal assumptions on the Mg/O ratio (Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020),  this can be converted to an estimate of the hydrogen column den-  sity (See equation 14 of Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Given the measured  doublet ratio is consistent with the intrinsic value, CSWA-141 is for-  mally consistent with a negligible hydrogen column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Within  the measurement errors of the flux ratio, we find a 1𝜎 upper limit on  the HI column density of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8×1016 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This value is well below  the HI column density at which galaxies become optically thin to  LyC radiation (<1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2 cm−2), suggesting that CSWA-141 may be  a likely candidate for LyC leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' While more realistic geometries  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', clumpy gas) would alter the derived column densities, the ob-  served line profile requires there to be low density channels along  the line of sight where resonant line photons (and potentially LyC  emission) are transmitted (Gazagnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Saldana-Lopez  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Because of the extreme brightness of CSWA-141, it  presents a unique opportunity to spatially resolve the absorbing gas  in an extreme line emitter that is very similar in its properties to  those systems at 𝑧 > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' An upcoming HST UVIS grism observa-  tions (GO-16710, PI: Mainali) will provide more direct constraints  on the escape of ionizing radiation from the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  5  SUMMARY  We present new spectroscopic and photometric observations of six-  teen bright gravitationally lensed galaxies originally identified in  SDSS via the CASSOWARY program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Observations were con-  ducted using LBT, Keck, MMT and Magellan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Included in this  sample is the 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='42 galaxy CSWA-141, one of the brightest  known EELGs at high redshift, with an [OIII]+H𝛽 EW (730 Å)  nearly identical to the average value seen at 𝑧 ≃ 7 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' In this  paper, we focus on the rest-UV spectral properties of the sample,  leveraging high quality Keck/ESI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Owing to the brightness of  our targets (𝑔 ≃ 19-21), we are able to detect rest-UV metal line  emission in the Keck spectra down to very low EW values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' While  most systems have weak line emission (median CIII] EW =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7 Å),  CSWA 141 shows relatively strong emission (CIII] EW 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6 Å) to-  gether with detections of a variety of UV lines (OIII], Si III], Fe  II★, Mg II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  We compare the properties of the strong (∼> 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Å) and weak  (∼< 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5 Å) CIII] emitters in our sample and in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We  find that the stronger CIII] emitters have larger sSFR and lower gas-  phase oxygen abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We find that the strong CIII] emitters  can be easily separated by their rest-optical line ratios, with larger  values of O32 at roughly fixed R23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Overall, these results suggest  that CIII] tends to be strong in galaxies dominated by young stellar  populations with low metallicity and extreme ionization conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  MNRAS 000, 1–20 (2022)  18  Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Object  EWCIII]  R23  O32  O3  Electron Density  12 + log (O/H)  Reference  ()  (cm−3)  CSWA-141  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  160+76  −74  𝑎, 350+294  −206  𝑏, 16500+12100  −7800  𝑐  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='08𝑑  This work  CSWA-2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8𝑑  9  A1689 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1  7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7  330𝑎, 2900𝑐  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='76𝑑  10  MACS 0451-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7  < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0 𝑓  3  SDSS J1723+3411  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4  47𝑎, 1950𝑐  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4 𝑓  11  MACS 0451-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  3  Cosmic Horseshoe  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='31  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='4  840-6900𝑏  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='65 𝑓  12,13  MS 1512-cB58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='36  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='47 𝑓  14,15  𝑎Derived from [OII] doublet, 𝑏Derived from [SII] doublet, 𝑐Derived from CIII] doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  𝑑Direct (T𝑒) method, 𝑒O32 method, 𝑓 R23 method, 𝑔O3 method,  ℎ Using HII-CHI-mistry code (Perez-Montero 2014)  1Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2020),2de Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2016),3 Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2014),4Vanzella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2017),5Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2018),6Bayliss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2014),7Vanzella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  (2016),8James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2014), 9Erb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2010),10Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2012), 11Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2021), 12Quider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2009), 13Hainline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2009), 14Pettini  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2000), 15Teplitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2000), 16Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' (2013)  Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Rest-optical emission line properties of galaxies at high redshift with CIII] emission constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The upper half shows data from this paper whereas  lower half shows measurements from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  This is consistent with trends found in observations at low and high  redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Senchyna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Maseda  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Nakajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Schaerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Ravindranath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2021) and in photoionization  models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', Jaskot & Ravindranath 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Nakajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The brightness of CSWA-141 enables a detailed investigation  of an EELG with properties similar to that which become common  at 𝑧 > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This galaxy is characterized by low stellar mass (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0×108  M⊙), large sSFR (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2 Gyr−1), low gas-phase metallicity (12+log  O/H=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='95) and relatively highly ionized gas (O32=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='7) and has  likely undergone a recent upturn or burst of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We  find that the electron density traced by the CIII] doublet (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='65×104  cm−3) is higher than that traced by [OII] and [SII] doublet (160  and 350 cm−3, respectively), a discrepancy that is also found in  other systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=', James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Acharyya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' While  this is likely to reflect the ionization structure of the HII regions  powering the lines (Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2019), it may also indicate that  CSWA-141 contains a significant fraction of its ionized gas in very  dense clumps, as is expected in the earliest stages following the  formation of star clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  The spectra of CSWA-141 provide several probes of the neutral  gas opacity in the galaxy, including both low ionization absorption  lines and the resolved Mg II doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The Fe II𝜆𝜆2374, 2382 ab-  sorption lines indicate the presence of outflowing gas with average  velocity of 76 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The lines are much weaker than in typical star  forming galaxies at 𝑧 ≃ 1 − 2, implying a low covering fraction and  potentially low column density of neutral gas in the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The res-  onant Mg II𝜆𝜆 2797,2803 emission line supports this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The  Mg II doublet ratio in CSWA-141 ( R= F2797/F2803) is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='2, con-  sistent with the intrinsic value produced in the HII regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' When  combined with the very large EW of Mg II and the near-Gaussian  profiles of the doublet components, this suggests minimal resonant  scattering, consistent with a very low column density of neutral  hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' These indirect indicators suggest CSWA-141 may be a  likely candidate for LyC leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We would like to thank Ryan Endsley for helping with MMT ob-  servations of some of the sources presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We also  thank Stephane Charlot and Jacopo Charlot for making the BEA-  GLE population synthesis tool available to us for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  DPS acknowledges support from the National Science Founda-  tion through the grant AST-2109066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' TJ acknowledges support from  the National Science Foundation through the grant AST-2108515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  RSE acknowledges funding from the European Research Council  (ERC) under the European Union’s Horizon 2020 research and in-  novation 0programme (grant agreement No 669253).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' acknowl-  edges support from the National Sciences and Engineering Council  of Canada grant RGPIN-2020-05102, the Fonds de recherche du  Québec grant 2022-NC-301305, and the Canada Research Chairs  Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='  Observations presented in this paper were obtained from the  MNRAS 000, 1–20 (2022)  19  Keck Observatory, which was made possible by the generous fi-  nancial support of the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Keck Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The material  is based upon work supported by NASA under award number  80GSFC21M0002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The authors acknowledge the very significant  cultural role that the summit of Mauna Kea has always had within  the indigenous Hawaiian community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' We are most fortunate to have  the opportunity to conduct observations from this mountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Some  of the observations reported here were obtained at the MMT Obser-  vatory, a joint facility of the University of Arizona and the Smith-  sonian Institution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' This paper includes data gathered with the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='5  meter Magellan Telescopes located at Las Campanas Observatory,  Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Some of the data presented in this paper were obtained using  the Large Binocular Telescope (LBT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The LBT is an international  collaboration among institutions in the United States, Italy and Ger-  many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The LBT Corporation partners are: The University of Ari-  zona on behalf of the Arizona university system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' Istituto Nazionale  di Astrofisica, Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' LBT Beteiligungsgesellschaft, Germany, rep-  resenting the Max Planck Society, the Astrophysical Institute Pots-  dam, and Heidelberg University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The Ohio State University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content=' The  Research Corporation, on behalf of The University of Notre Dame,  University of Minnesota and University of Virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.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/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
+page_content='04087  de Barros S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+page_content='  MNRAS 000, 1–20 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFIT4oBgHgl3EQfcCt7/content/2301.11264v1.pdf'}
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+UniHD at TSAR-2022 Shared Task:
+Is Compute All We Need for Lexical Simplification?
+Dennis Aumiller and Michael Gertz
+Institute of Computer Science
+Heidelberg University
+{aumiller, gertz}@informatik.uni-heidelberg.de
+Abstract
+Previous state-of-the-art models for lexical
+simplification consist of complex pipelines
+with several components, each of which re-
+quires deep technical knowledge and fine-
+tuned interaction to achieve its full potential.
+As an alternative, we describe a frustratingly
+simple pipeline based on prompted GPT-3 re-
+sponses, beating competing approaches by a
+wide margin in settings with few training in-
+stances.
+Our best-performing submission to
+the English language track of the TSAR-2022
+shared task consists of an “ensemble” of six
+different prompt templates with varying con-
+text levels. As a late-breaking result, we fur-
+ther detail a language transfer technique that
+allows simplification in languages other than
+English.
+Applied to the Spanish and Por-
+tuguese subset, we achieve state-of-the-art re-
+sults with only minor modification to the orig-
+inal prompts. Aside from detailing the imple-
+mentation and setup, we spend the remainder
+of this work discussing the particularities of
+prompting and implications for future work.
+Code for the experiments is available online.1
+1
+Introduction
+With recent advancements in Machine Learning
+(ML) research coming largely from increasing
+compute budgets, Richard Sutton coined the idea
+of a “bitter lesson”, wherein more computational
+power will ultimately supersede a hand-crafted
+solution (Sutton, 2019). More recently, increas-
+ing compute power on a general purpose archi-
+tecture has also shown to be wildly successful in
+the Natural Language Processing (NLP) commu-
+nity (Vaswani et al., 2017; Wei et al., 2022). In
+particular, emergent capabilities in very large lan-
+guage models (vLLMs) have made it possible to
+approach a variety of tasks wherein only few (if
+any) samples are labeled, and no further fine-tuning
+1https://github.com/dennlinger/
+TSAR-2022-Shared-Task
+on task-specific data is required at all.
+In stark contrast to the complex pipelines in modern
+lexical simplification systems (Ferrés et al., 2017;
+Qiang et al., 2020; Štajner et al., 2022), we present
+a simplistic approach utilizing few-shot prompts
+based on a vLLM with basic instructions on sim-
+plification, which returns frustratingly good results
+considering the overall complexity of the approach,
+which utilizes a grand total of four hand-labeled in-
+stances. We present our results on the TSAR-2022
+shared task (Saggion et al., 2022), which evaluates
+lexical simplification systems in three available lan-
+guages (English, Spanish and Portuguese), with
+ten labeled instances and around 350 unlabeled
+test samples provided per language. For the En-
+glish subset, official results rank our model as the
+best-performing submission, indicating that this ap-
+proach may be another instance of the bitter lesson.
+While the initial findings are indeed promising, we
+want to carefully evaluate erroneous instances on
+the test set to analyze potential pitfalls, and further
+detail some of our experiences in hand-crafting
+prompts. We also acknowledge the technical chal-
+lenges in reproducing (and deploying) systems
+based on vLLMs, especially given that suitable
+models exceed traditional computing budgets.
+2
+Prompt-based Lexical Simplification
+With the public release of the GPT-3 language
+model (Brown et al., 2020), OpenAI has started the
+run on a series of now-available vLLMs for general-
+purpose text generation (Thoppilan et al., 2022;
+BigScience, 2022; Zhang et al., 2022). Across
+these models, a general trend in scaling beyond a
+particular parameter size can be observed, while
+keeping the underlying architectural design close to
+existing smaller models. Through exhibiting zero-
+shot transfer capabilities, such models have also
+become more attractive for lower-resourced tasks;
+oftentimes, models are able to answer questions
+formulated in natural language with somewhat sen-
+arXiv:2301.01764v1  [cs.CL]  4 Jan 2023
+
+sible results. Particular template patterns (so-called
+prompts) are frequently used to guide models to-
+wards predicting a particularly desirable output or
+answer format, without requiring a dedicated train-
+ing on labeled examples.
+Utilizing this paradigm shift, we experimented
+with different prompts issued to OpenAI’s largest
+available model, text-davinici-002, which to-
+tals 176B parameters. Our first approach uses a
+singular prompt template in a zero-shot setting to
+obtain predictions for the shared task; we further
+improve upon these results by combining predic-
+tions from different prompt templates later on.
+2.1
+Run 1: Zero-shot Prediction
+Upon inspecting the provided trial data, we noted
+that the simplification operations required a vastly
+different contextualization within the provided sam-
+ple sentence.
+Whereas some instances can be
+solved with pure synonym look-ups (e.g., “com-
+pulsory” and “mandatory”), others require a more
+nuanced look at the context sentence (e.g., replac-
+ing “disguised” with “dressed”). To avoid bias-
+ing system predictions by providing samples as
+a prompt template, we provide a baseline that is
+entirely based on a single zero-shot query; it pro-
+vides the context sentence and identifies the com-
+plex word, asking the model for ten simplified syn-
+onyms of the complex word in the given context.
+Given that no additional knowledge is provided
+to the model, the zero-shot contextual query also
+provides a reasonable lower-bound for the task set-
+ting. A secondary advantage of minimal provided
+context in zero-shot settings is the reduced com-
+putational cost, which will be discussed in more
+detail in Section 3.4.
+2.2
+Filtering Predictions
+Model suggestions are returned as free-form text
+predictions, generally in the form of comma-
+separated lists or enumerations. This requires the
+additional step of parsing the output prediction into
+the more structured ranked predictions required for
+the shared task, which varies between the mod-
+els used. In our experience, no clear pattern can
+be expected from the model and seems to be non-
+deterministic even with set template structures. We
+additionally employ a list of simple filters to en-
+sure the quality of predictions, as detailed in Ap-
+pendix C. The resulting model suggestions are con-
+sidered in ranked order, and no prediction confi-
+dence scores or similar information was used to
+re-rank single-prompt predictions.
+2.3
+Run 2: Ensemble Predictions
+Upon inspecting the results from the first run,
+we noticed that in some instances, predictions
+were almost fully discarded due to filtering. Si-
+multaneously, we had already previously encoun-
+tered strong variability in system generations when
+changing the prompt template or altering the con-
+text setting. To this extent, an ensemble of pre-
+dictions from multiple different prompt templates
+was utilized to broaden the spectrum of possible
+generations, as well as ensuring that a minimum
+number of suggestions survives the filtering step.
+2.3.1
+Prompt Variations
+The exact prompts are detailed in Table 3. Utilized
+variations can be grouped into with context (the
+context sentence is provided), or without context
+(synonyms are generated from the complex word
+alone). Simultaneously, different prompts also con-
+tain between zero and two examples taken from the
+trial data, including their expected outputs. This
+can be interpreted as a few-shot setting in which
+the model is demonstrated on what correct answers
+may look like for the particular task. We further
+vary the generation temperature, where a higher
+value increases the likelihood of a more creative
+(but not always correct) prediction, enabling a more
+diverse candidate set.
+2.3.2
+Combining Predictions
+For each of the six prompts p, we ask the model
+to suggest ten alternative simplified expressions
+Sp and filter them with the exact same rules as the
+single prompt system in Run 1. In order to combine
+and re-rank suggestions s, we assign a combination
+score V to each distinct prediction s ∈ �
+p Sp:
+V (s) =
+�
+p
+max(5.5 − 0.5 · rankSp(s), 0),
+(1)
+where rankSp(s) is the ranked position of sugges-
+tion s in the resulting ranking from prompt p. If
+s /∈ Sp, we set rankSp(s) = ∞. The scaling pa-
+rameters are chosen arbitrarily and can be adjusted
+to account for the expected number of suggestions
+per prompt. We estimate that the biggest perfor-
+mance improvement is coming simply from pro-
+viding enough predictions post filtering. As a sec-
+ondary gain, we see more consistent behavior in
+the top-most prediction slots, boosting especially
+the @1 performance of the ensemble.
+
+Acc@k@Top1
+MAP@k
+Potential@k
+Run
+ACC@1 k = 1
+k = 2
+k = 3
+k = 3
+k = 5 k = 10 k = 3
+k = 5 k = 10
+Ensemble (Ours)
+0.8096
+0.4289 0.6112 0.6863 0.5834 0.4491 0.2812 0.9624 0.9812 0.9946
+Single (Ours)
+0.7721
+0.4262 0.5335 0.5710 0.5090 0.3653 0.2092 0.8900 0.9302 0.9436
+MANTIS-1
+0.6568
+0.319 0.4504 0.5388
+0.473 0.3599 0.2193 0.8766 0.9463 0.9785
+UoM&MMU-1
+0.6353
+0.2895 0.4530 0.5308 0.4244 0.3173 0.1951 0.8739 0.9115 0.9490
+LSBert
+0.5978
+0.3029 0.4450 0.5308 0.4079 0.2957 0.1755 0.8230 0.8766 0.9463
+TUNER
+0.3404
+0.1420 0.1689 0.1823 0.1706 0.1087 0.0546 0.4343 0.4450 0.4450
+Table 1: Results on the English language test set of the TSAR-2022 shared task, ranked by ACC@1 scores.
+Listed are our own results (Ensemble and Single), the two best-performing competing systems (MANTIS and
+UoM&MMU), as well as provided baselines (LSBert (Qiang et al., 2020) and TUNER (Ferrés et al., 2017)).
+3
+Results and Limitations
+3.1
+Results for English
+For the official runs, we initially only submitted
+predictions for the English subset; an excerpt of the
+results can be seen in Table 1. While the zero-shot
+single prompt run has consistently better results on
+most metrics, it does not outperform all systems
+for large candidate sets; e.g., Potential@10 is lower
+than that of competing approaches, including the
+LSBert baseline. We attribute this to the previously
+mentioned issue of filtering predictions, and can
+see a consequent improvement especially for larger
+k by using the proposed ensemble method. Here,
+the Potential@10 scores indicate that at least one
+viable prediction is present in all but three samples.
+3.2
+Results for Spanish and Portuguese
+Given the surprisingly good results on the English
+subset, we decided to extend our experiments to
+the Spanish and Portuguese tracks as well. Trans-
+ferring the prompts to Spanish or Portuguese is sur-
+prisingly simple. We alter the prompt to: “Given
+the above context, list ten alternative Spanish
+words for ‘complex_word’ that are easier to un-
+derstand.” (bold highlight indicates change).
+Without this adaption, returned suggestions gen-
+erally tend to be in English, which could be an
+attractive opportunity to mine cross-lingual sim-
+plifications in future work. By adding the output
+language explicitly, we ensure that the suggestions
+match the expected results. For Portuguese, the
+prompt can be adapted accordingly.
+We find that our system also outperforms all com-
+peting submitted approaches in the shared task;
+result comparisons can be found in Table 4 and 5
+in the Appendix, respectively. Notably, predictions
+for Portuguese perform slightly better, which goes
+against intuition, given that Spanish is usually a
+highly represented language in multilingual cor-
+pora. We suspect that a more literal wording of
+synonyms in Portuguese, compared to multi-word
+expressions in Spanish, could be the cause.
+3.3
+Error Analysis
+As is common for sequence-to-sequence tasks,
+crafting an approach centered around a LM requires
+consideration of the particular challenges arising.
+We detail some of the errors we have encountered
+in our predictions that are unlikely to appear in
+more stringently designed pipelines. Instances for
+particular failure cases can be found in Table 2.
+Unstable Prompts
+One of the primary chal-
+lenges, particularly for zero-shot prompt settings, is
+the unreasonable variance observed in results based
+on even just slightly altered prompt templates. For
+example, when removing the explicit mention of
+Context:, Question: and Answer: in the prompt
+template, the model is frequently predicting fewer
+than the ten requested answers. Practical limita-
+tions in our computational budget also mean that
+we have no guarantee that these prompts are yield-
+ing the best possible results; given the variability,
+multiple runs should be compared for a thorough
+pattern of a “best” prompt.
+Lack of Context
+Instances with longer (or sub-
+tly enforced) context cues show issues where these
+hints are not properly recognized. In Table 2, we
+can see the model changing the term “collision”
+to a particular mode of transportation, such as
+“car crash”, while an explicit context clue is given
+through the word “flight” in the original sentence.
+Enforcing Language
+While the transfer to Span-
+ish and Portuguese is largely successful, the
+model’s capabilities seem to be still limited in
+maintaining the language throughout all samples.
+
+Error Type
+Context (complex word in bold)
+Model Predictions
+Lack of Context #7-8 Despite the fog, other flights are reported
+to have landed safely leading up to the collision.
+car crash, train wreck, ...
+Hallucinations
+The larva grows to about 120-130 mm,
+and pupates in an underground chamber.
+Transforms into a pupa, ...
+Language
+[...] propiciado la decadencia de la Revolución francesa. decline, deterioration, ...
+Table 2: Instances of observed failure classes in our system’s predictions.
+For instances with particularly rare complex terms,
+the predictions are sometimes still in English, de-
+spite the specific prompt request to return Span-
+ish/Portuguese results.
+Hallucinations
+The necessity for post-filtering
+of suggestions stems largely from the spontaneous
+occurrence of hallucinations in responses. While
+hallucinations in vLLMs are less about invalid vo-
+cabulary terms, we observe instances where unnec-
+essary multi-word suggestions were chosen over
+a simple synonymous single-word expression, or
+random inflections (such as the infinitive form with
+an additional “to”) were generated.
+Similar to the issues with language enforcing, this
+occurs more frequently with particularly complex
+words; in this sense, the system conversely fails
+at instances that are most in need of simplifica-
+tion. However, we note that some of the generated
+multi-word expressions are actually more helpful
+for understanding, even though the generations are
+not precisely matching expected outputs.
+3.4
+Computational Limitations
+Running a vLLM in practice, even for inference-
+only settings, is non-trivial and requires compute
+resources that are far beyond many public institu-
+tion’s hardware budget. For the largest models with
+publicly available checkpoints2, a total of around
+325GB GPU memory is required, assuming ef-
+ficient storage in bfloat16 or similar precision
+levels. The common alternative is to obtain pre-
+dictions through a (generally paid) API, as was
+the case in this work. Especially for the ensemble
+model, which issues six individual requests to the
+API per sample, this can further bloat the net cost
+of a single prediction. To give context of the total
+cost, we incurred a total charge of slightly over $7
+for computing predictions across the entire test set
+of 373 English samples, which comes out to about
+2At the time of writing, this would be the 176B Bloom
+model (BigScience, 2022), which has a similar parameter
+count to OpenAI’s davinci-002 model.
+1000 tokens per sample, or around $0.02 at the cur-
+rent OpenAI pricing scheme.3 For the Spanish sub-
+set and language-dependent prompt development,
+the total cost came to about $10, primarily due
+to longer sample contexts. Costs for Portuguese
+processing were around $6.50. While the singu-
+lar prompt approach is cheaper at around 1/6 of
+the total cost, even then a continuously deployed
+model has to be supplied with a large enough bud-
+get. Aside from monetary concerns, environmental
+impacts are also to be considered for larger-scale
+deployments of this kind (Lacoste et al., 2019).
+4
+Conclusion and Future Work
+Utilizing prompted responses from vLLMs seems
+to be a promising direction for lexical simplifica-
+tion; particularly in the constrained setting with
+pre-identified complex words the model performs
+exceptionally well, even when presented with a
+severely restricted budget of labeled training data.
+While the approach also offers promising directions
+for multi- and cross-lingual approaches, obtaining
+state-of-the-art results in other languages, we are
+presented with a prohibitive amount of computa-
+tion per sample instance. It would therefore be an
+interesting addition to deal with resource-constraint
+systems, putting the prediction power into a slightly
+different perspective. Finally, we are reminded of
+the unstable nature of neural LMs; given similar
+inputs, quality can vary greatly between samples,
+including a complete breakdown in performance.
+For future work, we are considering approaches
+to generate static resources from vLLMs (Schick
+and Schütze, 2021), which may require only a one-
+time commitment to spending on datasets, which
+can then used as training data for cheaper systems.
+Exploration of prompt tuning approaches for auto-
+mated search of suitable prompt templates would
+also greatly accelerate the development process of
+domain-specific applications (Lester et al., 2021).
+3https://openai.com/api/pricing/,
+last accessed:
+2022-10-01
+
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+Todor Mihaylov, Myle Ott, Sam Shleifer, Kurt Shus-
+ter, Daniel Simig, Punit Singh Koura, Anjali Srid-
+har, Tianlu Wang, and Luke Zettlemoyer. 2022.
+OPT: open pre-trained transformer language models.
+CoRR, abs/2205.01068.
+
+A
+Prompt Templates
+Table 3 provides the exact prompt templates used
+in the submission. Notably, the zero-shot with con-
+text prompt is included twice, but with different
+generation temperatures; with this we increase the
+likelihood of strong candidates being retained. For
+few-shot prompts, we have taken samples from the
+previously published trial set for the respective lan-
+guage. In instances where less than 10 distinctly
+different suggestions were provided by annotators,
+we manually extended the list of examples to match
+exactly ten results based on our own judgment. For
+instances with more provided suggestions, we limit
+ourselves to the ten most frequently occurring ones.
+The reason for this is that GPT-3 otherwise tended
+to return an inconsistent number of suggestions in
+our preliminary testing. The exact prompts for the
+Spanish and Portuguese runs can be found in our
+repository.
+B
+Hyperparameters
+We use the OpenAI Python package4 version
+0.23.0 for our experiments. For generation, the
+function openai.Completion.create() is used,
+where most hyperparameters remain fixed across
+all prompts. We explicitly list those hyperparame-
+ters below that differ from their respective default
+values.
+1. model="text-davinci-002", which is the
+latest and biggest available model for text
+completion.
+2. max_tokens=256, to ensure sufficient room
+for generated outputs. In practice, most com-
+pletions are vastly below the limit.
+3. frequency_penalty=0.5,
+as
+well
+as
+presence_penalty=0.3,
+which
+jointly
+penalize present tokens and token repetitions.
+The values are well below the maximum
+(values can range from -2 to 2), since
+individual subword tokens might indeed be
+present several times across multiple (valid)
+predictions. A more detailed computation can
+be found in the documentation of OpenAI.5
+Outside of the repetition penalties, the most influen-
+tial parameter choice for generation is the sampling
+4https://github.com/openai/openai-python
+5https://beta.openai.com/docs/api-reference/
+parameter-details
+temperature. We generally take a more measured
+approach than the default (temperature=1.0), but
+vary temperature across our ensemble prompts to
+ensure a more diverse result set overall. We list
+the used temperatures in Table 3. Zero-shot with
+context is used twice in the ensemble, once with
+a more conservative temperature, and once with a
+more “creative” (higher) temperature. For the sin-
+gular prompt run, we use the conservative zero-shot
+with context variant.
+C
+Post-Filtering Operations
+Given the uncertain nature of predictions by a lan-
+guage model, we employ a series of post-filtering
+steps to ensure high quality outputs. This includes
+stripping newlines/spaces/punctuation (\n :;.?!),
+lower-casing, removing infinitive forms (in some
+instances, we observed predictions in the form of
+“to deploy” instead of simply “deploy”), as well as
+removing identity predictions (e.g., the prediction
+being the same as the original complex word) and
+deduplicating suggestions. Additionally, we no-
+ticed that for some instances, generated synonyms
+resemble more of a “description” rather than truly
+synonymous expressions (example: “people that
+are crazy” as a suggestion for “maniacs”). Given
+the nature of provided data, we removed extreme
+multi-word expressions (for English, any sugges-
+tion with more than two words, for Spanish and
+Portuguese more than three words in a single ex-
+pression).
+
+Prompt Type
+Template
+Zero-shot /w context
+Context: {context_sentence}\n
+temperature:
+Question: Given the above context, list ten alternatives for
+0.3 (conservative),
+“{complex_word}” that are easier to understand.\n
+0.8 (creative)
+Answer:
+Single-shot /w context
+Context: A local witness said a separate group of attackers
+temperature: 0.5
+disguisedin burqas — the head-to-toe robes worn by conservative
+Afghan women — then tried to storm the compound.\n
+Question: Given the above context, list ten alternative words
+for “disguised” that are easier to understand.\n
+Answer:\n1. concealed\n2. dressed\n3. hidden\n4. camouflaged\n
+5. changed\n6. covered\n7. masked\n8. unrecognizable\n
+9. converted\n10. impersonated\n\n
+Context: {context_sentence}\n
+Question: Given the above context, list ten alternatives for
+“{complex_word}” that are easier to understand.\n
+Answer:
+Two-shot /w context
+Context: That prompted the military to deploy its largest
+temperature: 0.5
+warship, the BRP Gregorio del Pilar, which was recently
+acquired from the United States.\n
+Question: Given the above context, list ten alternative words
+for “deploy” that are easier to understand.\n
+Answer:\n1. send\n2. post\n3. use\n4. position\n5. send out\n
+6. employ\n7. extend\n8. launch\n9. let loose\n10. organize\n\n
+Context: The daily death toll in Syria has declined as the
+number of observers has risen, but few experts expect the
+U.N. plan to succeed in its entirety.\n
+Question: Given the above context, list ten alternative words
+for “observers” that are easier to understand.\n
+Answer:\n1. watchers\n2. spectators\n3. audience\n4. viewers\n
+5. witnesses\n6. patrons\n7. followers\n8. detectives\n
+9. reporters\n10. onlookers\n\n
+Context: {context_sentence}\n
+Question: Given the above context, list ten alternatives for
+“{complex_word}” that are easier to understand.\n
+Answer:
+Zero-shot w/o context
+Give me ten simplified synonyms for the following word:
+temperature: 0.7
+{complex_word}
+Single-shot w/o context Question: Find ten easier words for “compulsory”.\n
+temperature: 0.6
+Answer:\n1. mandatory\n2. required\n3. essential\n4. forced\n
+5. important\n6. necessary\n7. obligatory\n8. unavoidable\n
+9. binding\n10. prescribed\n\n
+Question: Find ten easier words for “{complex_word}”.\n
+Answer:
+Table 3: The English prompt templates used for querying the OpenAI model, including associated generation
+temperatures. Only written out “\n” symbols indicate newlines, visible line breaks are inserted for better legibility.
+Only top-most prompt template with conservative temperature was used in the single prompt (Run 1), as well as in
+the ensemble run (Run 2). All other prompts were only included in the ensemble submission.
+
+Acc@k@Top1
+MAP@k
+Potential@k
+Run
+ACC@1 k = 1
+k = 2
+k = 3
+k = 3
+k = 5 k = 10 k = 3
+k = 5 k = 10
+Ensemble (Ours)
+0.6521
+0.3505 0.5108 0.5788 0.4281 0.3239 0.1967 0.8206 0.8885 0.9402
+Single (Ours)
+0.5706
+0.3070 0.3967 0.4510 0.3526 0.2449 0.1376 0.6902 0.7146 0.7445
+PresiUniv-1
+0.3695
+0.2038 0.2771 0.3288 0.2145 0.1499 0.0832 0.5842 0.6467 0.7255
+UoM&MMU-3
+0.3668
+0.1603 0.2282 0.269
+0.2128 0.1506 0.0899 0.5326 0.6005 0.6929
+LSBert
+0.2880
+0.0951 0.1440 0.1820 0.1868 0.1346 0.0795 0.4945 0.6114 0.7472
+TUNER
+0.1195
+0.0625 0.0788 0.0842 0.0575 0.0356 0.0184
+0.144 0.1467 0.1494
+Table 4: Results on the Spanish language test set of the TSAR-2022 shared task, ranked by ACC@1 scores.
+Listed are our own results (Ensemble and Single), the two best-performing competing systems (PresiUniv and
+UoM&MMU), as well as provided baselines (LSBert (Qiang et al., 2020) and TUNER (Ferrés et al., 2017)).
+Acc@k@Top1
+MAP@k
+Potential@k
+Run
+ACC@1 k = 1
+k = 2
+k = 3
+k = 3
+k = 5 k = 10 k = 3
+k = 5 k = 10
+Ensemble (Ours)
+0.7700
+0.4358 0.5347 0.6229 0.5014 0.3620 0.2167 0.9171 0.9491 0.9786
+Single (Ours)
+0.6363
+0.3716 0.4625 0.5160 0.4105 0.2889 0.1615 0.7860 0.8181 0.8422
+GMU-WLV-1
+0.4812
+0.2540 0.3716 0.3957 0.2816 0.1966 0.1153 0.6871 0.7566 0.8395
+Cental-1
+0.3689
+0.1737 0.2433 0.2673 0.1983 0.1344 0.0766
+0.524 0.5641 0.6096
+LSBert
+0.3262
+0.1577 0.2326 0.286
+0.1904 0.1313 0.0775 0.4946 0.5802 0.6737
+TUNER
+0.2219
+0.1336 0.1604 0.1604 0.1005 0.0623 0.0311 0.2673 0.2673 0.2673
+Table 5: Results on the Portuguese language test set of the TSAR-2022 shared task, ranked by ACC@1 scores.
+Listed are our own results (Ensemble and Single), the two best-performing competing systems (GMU-WLV and
+Cental), as well as provided baselines (LSBert (Qiang et al., 2020) and TUNER (Ferrés et al., 2017)).
+
diff --git a/NNAzT4oBgHgl3EQfzP4S/content/tmp_files/load_file.txt b/NNAzT4oBgHgl3EQfzP4S/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,523 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf,len=522
+page_content='UniHD at TSAR-2022 Shared Task:  Is Compute All We Need for Lexical Simplification?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Dennis Aumiller and Michael Gertz  Institute of Computer Science  Heidelberg University  {aumiller, gertz}@informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='uni-heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='de  Abstract  Previous state-of-the-art models for lexical  simplification consist of complex pipelines  with several components, each of which re-  quires deep technical knowledge and fine-  tuned interaction to achieve its full potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  As an alternative, we describe a frustratingly  simple pipeline based on prompted GPT-3 re-  sponses, beating competing approaches by a  wide margin in settings with few training in-  stances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Our best-performing submission to  the English language track of the TSAR-2022  shared task consists of an “ensemble” of six  different prompt templates with varying con-  text levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' As a late-breaking result, we fur-  ther detail a language transfer technique that  allows simplification in languages other than  English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Applied to the Spanish and Por-  tuguese subset, we achieve state-of-the-art re-  sults with only minor modification to the orig-  inal prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Aside from detailing the imple-  mentation and setup, we spend the remainder  of this work discussing the particularities of  prompting and implications for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Code for the experiments is available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='1  1  Introduction  With recent advancements in Machine Learning  (ML) research coming largely from increasing  compute budgets, Richard Sutton coined the idea  of a “bitter lesson”, wherein more computational  power will ultimately supersede a hand-crafted  solution (Sutton, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' More recently, increas-  ing compute power on a general purpose archi-  tecture has also shown to be wildly successful in  the Natural Language Processing (NLP) commu-  nity (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' In  particular, emergent capabilities in very large lan-  guage models (vLLMs) have made it possible to  approach a variety of tasks wherein only few (if  any) samples are labeled, and no further fine-tuning  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='com/dennlinger/  TSAR-2022-Shared-Task  on task-specific data is required at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  In stark contrast to the complex pipelines in modern  lexical simplification systems (Ferrés et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Qiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Štajner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2022), we present  a simplistic approach utilizing few-shot prompts  based on a vLLM with basic instructions on sim-  plification, which returns frustratingly good results  considering the overall complexity of the approach,  which utilizes a grand total of four hand-labeled in-  stances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' We present our results on the TSAR-2022  shared task (Saggion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2022), which evaluates  lexical simplification systems in three available lan-  guages (English, Spanish and Portuguese), with  ten labeled instances and around 350 unlabeled  test samples provided per language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' For the En-  glish subset, official results rank our model as the  best-performing submission, indicating that this ap-  proach may be another instance of the bitter lesson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  While the initial findings are indeed promising, we  want to carefully evaluate erroneous instances on  the test set to analyze potential pitfalls, and further  detail some of our experiences in hand-crafting  prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' We also acknowledge the technical chal-  lenges in reproducing (and deploying) systems  based on vLLMs, especially given that suitable  models exceed traditional computing budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  2  Prompt-based Lexical Simplification  With the public release of the GPT-3 language  model (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2020), OpenAI has started the  run on a series of now-available vLLMs for general-  purpose text generation (Thoppilan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  BigScience, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Across  these models, a general trend in scaling beyond a  particular parameter size can be observed, while  keeping the underlying architectural design close to  existing smaller models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Through exhibiting zero-  shot transfer capabilities, such models have also  become more attractive for lower-resourced tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  oftentimes, models are able to answer questions  formulated in natural language with somewhat sen-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='01764v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='CL]  4 Jan 2023  sible results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Particular template patterns (so-called  prompts) are frequently used to guide models to-  wards predicting a particularly desirable output or  answer format, without requiring a dedicated train-  ing on labeled examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Utilizing this paradigm shift, we experimented  with different prompts issued to OpenAI’s largest  available model, text-davinici-002, which to-  tals 176B parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Our first approach uses a  singular prompt template in a zero-shot setting to  obtain predictions for the shared task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' we further  improve upon these results by combining predic-  tions from different prompt templates later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='1  Run 1: Zero-shot Prediction  Upon inspecting the provided trial data, we noted  that the simplification operations required a vastly  different contextualization within the provided sam-  ple sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Whereas some instances can be  solved with pure synonym look-ups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', “com-  pulsory” and “mandatory”), others require a more  nuanced look at the context sentence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', replac-  ing “disguised” with “dressed”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' To avoid bias-  ing system predictions by providing samples as  a prompt template, we provide a baseline that is  entirely based on a single zero-shot query;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' it pro-  vides the context sentence and identifies the com-  plex word, asking the model for ten simplified syn-  onyms of the complex word in the given context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Given that no additional knowledge is provided  to the model, the zero-shot contextual query also  provides a reasonable lower-bound for the task set-  ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' A secondary advantage of minimal provided  context in zero-shot settings is the reduced com-  putational cost, which will be discussed in more  detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='2  Filtering Predictions  Model suggestions are returned as free-form text  predictions, generally in the form of comma-  separated lists or enumerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' This requires the  additional step of parsing the output prediction into  the more structured ranked predictions required for  the shared task, which varies between the mod-  els used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' In our experience, no clear pattern can  be expected from the model and seems to be non-  deterministic even with set template structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' We  additionally employ a list of simple filters to en-  sure the quality of predictions, as detailed in Ap-  pendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' The resulting model suggestions are con-  sidered in ranked order, and no prediction confi-  dence scores or similar information was used to  re-rank single-prompt predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='3  Run 2: Ensemble Predictions  Upon inspecting the results from the first run,  we noticed that in some instances, predictions  were almost fully discarded due to filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Si-  multaneously, we had already previously encoun-  tered strong variability in system generations when  changing the prompt template or altering the con-  text setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' To this extent, an ensemble of pre-  dictions from multiple different prompt templates  was utilized to broaden the spectrum of possible  generations, as well as ensuring that a minimum  number of suggestions survives the filtering step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='1  Prompt Variations  The exact prompts are detailed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Utilized  variations can be grouped into with context (the  context sentence is provided), or without context  (synonyms are generated from the complex word  alone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Simultaneously, different prompts also con-  tain between zero and two examples taken from the  trial data, including their expected outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' This  can be interpreted as a few-shot setting in which  the model is demonstrated on what correct answers  may look like for the particular task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' We further  vary the generation temperature, where a higher  value increases the likelihood of a more creative  (but not always correct) prediction, enabling a more  diverse candidate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='2  Combining Predictions  For each of the six prompts p, we ask the model  to suggest ten alternative simplified expressions  Sp and filter them with the exact same rules as the  single prompt system in Run 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' In order to combine  and re-rank suggestions s, we assign a combination  score V to each distinct prediction s ∈ �  p Sp:  V (s) =  �  p  max(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='5 · rankSp(s), 0),  (1)  where rankSp(s) is the ranked position of sugges-  tion s in the resulting ranking from prompt p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' If  s /∈ Sp, we set rankSp(s) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' The scaling pa-  rameters are chosen arbitrarily and can be adjusted  to account for the expected number of suggestions  per prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' We estimate that the biggest perfor-  mance improvement is coming simply from pro-  viding enough predictions post filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' As a sec-  ondary gain, we see more consistent behavior in  the top-most prediction slots, boosting especially  the @1 performance of the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Acc@k@Top1  MAP@k  Potential@k  Run  ACC@1 k = 1  k = 2  k = 3  k = 3  k = 5 k = 10 k = 3  k = 5 k = 10  Ensemble (Ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
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+page_content='9463  TUNER  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
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+page_content='4450  Table 1: Results on the English language test set of the TSAR-2022 shared task, ranked by ACC@1 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Listed are our own results (Ensemble and Single), the two best-performing competing systems (MANTIS and  UoM&MMU), as well as provided baselines (LSBert (Qiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2020) and TUNER (Ferrés et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  3  Results and Limitations  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='1  Results for English  For the official runs, we initially only submitted  predictions for the English subset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' an excerpt of the  results can be seen in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' While the zero-shot  single prompt run has consistently better results on  most metrics, it does not outperform all systems  for large candidate sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', Potential@10 is lower  than that of competing approaches, including the  LSBert baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' We attribute this to the previously  mentioned issue of filtering predictions, and can  see a consequent improvement especially for larger  k by using the proposed ensemble method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Here,  the Potential@10 scores indicate that at least one  viable prediction is present in all but three samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='2  Results for Spanish and Portuguese  Given the surprisingly good results on the English  subset, we decided to extend our experiments to  the Spanish and Portuguese tracks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Trans-  ferring the prompts to Spanish or Portuguese is sur-  prisingly simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' We alter the prompt to: “Given  the above context, list ten alternative Spanish  words for ‘complex_word’ that are easier to un-  derstand.” (bold highlight indicates change).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Without this adaption, returned suggestions gen-  erally tend to be in English, which could be an  attractive opportunity to mine cross-lingual sim-  plifications in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' By adding the output  language explicitly, we ensure that the suggestions  match the expected results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' For Portuguese, the  prompt can be adapted accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  We find that our system also outperforms all com-  peting submitted approaches in the shared task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  result comparisons can be found in Table 4 and 5  in the Appendix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Notably, predictions  for Portuguese perform slightly better, which goes  against intuition, given that Spanish is usually a  highly represented language in multilingual cor-  pora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' We suspect that a more literal wording of  synonyms in Portuguese, compared to multi-word  expressions in Spanish, could be the cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='3  Error Analysis  As is common for sequence-to-sequence tasks,  crafting an approach centered around a LM requires  consideration of the particular challenges arising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  We detail some of the errors we have encountered  in our predictions that are unlikely to appear in  more stringently designed pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Instances for  particular failure cases can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Unstable Prompts  One of the primary chal-  lenges, particularly for zero-shot prompt settings, is  the unreasonable variance observed in results based  on even just slightly altered prompt templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' For  example, when removing the explicit mention of  Context:, Question: and Answer: in the prompt  template, the model is frequently predicting fewer  than the ten requested answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Practical limita-  tions in our computational budget also mean that  we have no guarantee that these prompts are yield-  ing the best possible results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' given the variability,  multiple runs should be compared for a thorough  pattern of a “best” prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Lack of Context  Instances with longer (or sub-  tly enforced) context cues show issues where these  hints are not properly recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' In Table 2, we  can see the model changing the term “collision”  to a particular mode of transportation, such as  “car crash”, while an explicit context clue is given  through the word “flight” in the original sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Enforcing Language  While the transfer to Span-  ish and Portuguese is largely successful, the  model’s capabilities seem to be still limited in  maintaining the language throughout all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Error Type  Context (complex word in bold)  Model Predictions  Lack of Context #7-8 Despite the fog, other flights are reported  to have landed safely leading up to the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  car crash, train wreck, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Hallucinations  The larva grows to about 120-130 mm,  and pupates in an underground chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Transforms into a pupa, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Language  [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='] propiciado la decadencia de la Revolución francesa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' decline, deterioration, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Table 2: Instances of observed failure classes in our system’s predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  For instances with particularly rare complex terms,  the predictions are sometimes still in English, de-  spite the specific prompt request to return Span-  ish/Portuguese results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Hallucinations  The necessity for post-filtering  of suggestions stems largely from the spontaneous  occurrence of hallucinations in responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' While  hallucinations in vLLMs are less about invalid vo-  cabulary terms, we observe instances where unnec-  essary multi-word suggestions were chosen over  a simple synonymous single-word expression, or  random inflections (such as the infinitive form with  an additional “to”) were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Similar to the issues with language enforcing, this  occurs more frequently with particularly complex  words;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' in this sense, the system conversely fails  at instances that are most in need of simplifica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' However, we note that some of the generated  multi-word expressions are actually more helpful  for understanding, even though the generations are  not precisely matching expected outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='4  Computational Limitations  Running a vLLM in practice, even for inference-  only settings, is non-trivial and requires compute  resources that are far beyond many public institu-  tion’s hardware budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' For the largest models with  publicly available checkpoints2, a total of around  325GB GPU memory is required, assuming ef-  ficient storage in bfloat16 or similar precision  levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' The common alternative is to obtain pre-  dictions through a (generally paid) API, as was  the case in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Especially for the ensemble  model, which issues six individual requests to the  API per sample, this can further bloat the net cost  of a single prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' To give context of the total  cost, we incurred a total charge of slightly over $7  for computing predictions across the entire test set  of 373 English samples, which comes out to about  2At the time of writing, this would be the 176B Bloom  model (BigScience, 2022), which has a similar parameter  count to OpenAI’s davinci-002 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  1000 tokens per sample, or around $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='02 at the cur-  rent OpenAI pricing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='3 For the Spanish sub-  set and language-dependent prompt development,  the total cost came to about $10, primarily due  to longer sample contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Costs for Portuguese  processing were around $6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' While the singu-  lar prompt approach is cheaper at around 1/6 of  the total cost, even then a continuously deployed  model has to be supplied with a large enough bud-  get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Aside from monetary concerns, environmental  impacts are also to be considered for larger-scale  deployments of this kind (Lacoste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  4  Conclusion and Future Work  Utilizing prompted responses from vLLMs seems  to be a promising direction for lexical simplifica-  tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' particularly in the constrained setting with  pre-identified complex words the model performs  exceptionally well, even when presented with a  severely restricted budget of labeled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  While the approach also offers promising directions  for multi- and cross-lingual approaches, obtaining  state-of-the-art results in other languages, we are  presented with a prohibitive amount of computa-  tion per sample instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' It would therefore be an  interesting addition to deal with resource-constraint  systems, putting the prediction power into a slightly  different perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Finally, we are reminded of  the unstable nature of neural LMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' given similar  inputs, quality can vary greatly between samples,  including a complete breakdown in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  For future work, we are considering approaches  to generate static resources from vLLMs (Schick  and Schütze, 2021), which may require only a one-  time commitment to spending on datasets, which  can then used as training data for cheaper systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Exploration of prompt tuning approaches for auto-  mated search of suitable prompt templates would  also greatly accelerate the development process of  domain-specific applications (Lester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  3https://openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='com/api/pricing/,  last accessed:  2022-10-01  References  Workshop BigScience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  BLOOM (revision  4ab0472).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Tom Brown, Benjamin Mann, Nick Ryder, Melanie  Subbiah,  Jared  D  Kaplan,  Prafulla  Dhariwal,  Arvind Neelakantan, Pranav Shyam, Girish Sastry,  Amanda Askell, Sandhini Agarwal, Ariel Herbert-  Voss, Gretchen Krueger, Tom Henighan, Rewon  Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu,  Clemens Winter, Chris Hesse, Mark Chen, Eric  Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess,  Jack Clark, Christopher Berner, Sam McCandlish,  Alec Radford, Ilya Sutskever, and Dario Amodei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Language models are few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' In  Advances in Neural Information Processing Systems,  volume 33, pages 1877–1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Curran Associates,  Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Daniel  Ferrés,  Horacio  Saggion,  and  Xavier  Gómez Guinovart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  An adaptable lexical  simplification architecture for major Ibero-Romance  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' In Proceedings of the First Workshop on  Building Linguistically Generalizable NLP Systems,  pages 40–47, Copenhagen, Denmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Association  for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Alexandre  Lacoste,  Alexandra  Luccioni,  Victor  Schmidt, and Thomas Dandres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Quantifying  the carbon emissions of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' CoRR,  abs/1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='09700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Brian Lester, Rami Al-Rfou, and Noah Constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  The power of scale for parameter-efficient prompt  tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' In Proceedings of the 2021 Conference on  Empirical Methods in Natural Language Processing,  pages 3045–3059, Online and Punta Cana, Domini-  can Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Association for Computational Lin-  guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Jipeng Qiang, Yun Li, Zhu Yi, Yunhao Yuan, and Xin-  dong Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Lexical simplification with pre-  trained encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Thirty-Fourth AAAI Conference  on Artificial Intelligence, page 8649–8656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Horacio  Saggion,  Sanja  Štajner,  Daniel  Ferrés,  Kim Cheng Sheang, Matthew Shardlow, Kai North,  and Marcos Zampieri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Findings of the tsar-  2022 shared task on multilingual lexical simplifica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' In Proceedings of TSAR workshop held in con-  junction with EMNLP 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Timo Schick and Hinrich Schütze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Generating  datasets with pretrained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' In Pro-  ceedings of the 2021 Conference on Empirical Meth-  ods in Natural Language Processing, pages 6943–  6951, Online and Punta Cana, Dominican Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Richard Sutton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' The bitter lesson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Incomplete  Ideas (blog), 13:12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Romal Thoppilan, Daniel De Freitas, Jamie Hall,  Noam Shazeer, Apoorv Kulshreshtha, Heng-Tze  Cheng, Alicia Jin, Taylor Bos, Leslie Baker, Yu Du,  YaGuang Li, Hongrae Lee, Huaixiu Steven Zheng,  Amin Ghafouri, Marcelo Menegali, Yanping Huang,  Maxim Krikun, Dmitry Lepikhin, James Qin, Dehao  Chen, Yuanzhong Xu, Zhifeng Chen, Adam Roberts,  Maarten Bosma, Yanqi Zhou, Chung-Ching Chang,  Igor Krivokon, Will Rusch, Marc Pickett, Kath-  leen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Meier-Hellstern, Meredith Ringel Morris,  Tulsee Doshi, Renelito Delos Santos, Toju Duke,  Johnny Soraker, Ben Zevenbergen, Vinodkumar  Prabhakaran, Mark Diaz, Ben Hutchinson, Kristen  Olson, Alejandra Molina, Erin Hoffman-John, Josh  Lee, Lora Aroyo, Ravi Rajakumar, Alena Butryna,  Matthew Lamm, Viktoriya Kuzmina, Joe Fenton,  Aaron Cohen, Rachel Bernstein, Ray Kurzweil,  Blaise Aguera-Arcas, Claire Cui, Marian Croak,  Ed H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Chi, and Quoc Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Lamda:  Lan-  guage models for dialog applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  CoRR,  abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='08239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob  Uszkoreit, Llion Jones, Aidan N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Gomez, Lukasz  Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Attention is all  you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' In Advances in Neural Information Pro-  cessing Systems 30: Annual Conference on Neural  Information Processing Systems 2017, December 4-  9, 2017, Long Beach, CA, USA, pages 5998–6008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Sanja Štajner, Daniel Ferrés, Matthew Shardlow, Kai  North, Marcos Zampieri, and Horacio Saggion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Lexical simplification benchmarks for En-  glish, Portuguese, and Spanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Frontiers in Artifi-  cial Intelligence, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Jason Wei, Yi Tay, Rishi Bommasani, Colin Raf-  fel, Barret Zoph, Sebastian Borgeaud, Dani Yo-  gatama, Maarten Bosma, Denny Zhou, Donald Met-  zler, Ed H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Chi, Tatsunori Hashimoto, Oriol Vinyals,  Percy Liang, Jeff Dean, and William Fedus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Emergent abilities of large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' CoRR,  abs/2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='07682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Susan Zhang, Stephen Roller, Naman Goyal, Mikel  Artetxe, Moya Chen, Shuohui Chen, Christopher  Dewan, Mona T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Diab, Xian Li, Xi Victoria Lin,  Todor Mihaylov, Myle Ott, Sam Shleifer, Kurt Shus-  ter, Daniel Simig, Punit Singh Koura, Anjali Srid-  har, Tianlu Wang, and Luke Zettlemoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  OPT: open pre-trained transformer language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  CoRR, abs/2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='01068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  A  Prompt Templates  Table 3 provides the exact prompt templates used  in the submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Notably, the zero-shot with con-  text prompt is included twice, but with different  generation temperatures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' with this we increase the  likelihood of strong candidates being retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' For  few-shot prompts, we have taken samples from the  previously published trial set for the respective lan-  guage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' In instances where less than 10 distinctly  different suggestions were provided by annotators,  we manually extended the list of examples to match  exactly ten results based on our own judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' For  instances with more provided suggestions, we limit  ourselves to the ten most frequently occurring ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  The reason for this is that GPT-3 otherwise tended  to return an inconsistent number of suggestions in  our preliminary testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' The exact prompts for the  Spanish and Portuguese runs can be found in our  repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  B  Hyperparameters  We use the OpenAI Python package4 version  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='0 for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' For generation, the  function openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='Completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='create() is used,  where most hyperparameters remain fixed across  all prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' We explicitly list those hyperparame-  ters below that differ from their respective default  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' model="text-davinci-002", which is the  latest and biggest available model for text  completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' max_tokens=256, to ensure sufficient room  for generated outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' In practice, most com-  pletions are vastly below the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' frequency_penalty=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='5,  as  well  as  presence_penalty=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='3,  which  jointly  penalize present tokens and token repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  The values are well below the maximum  (values can range from -2 to 2), since  individual subword tokens might indeed be  present several times across multiple (valid)  predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' A more detailed computation can  be found in the documentation of OpenAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='5  Outside of the repetition penalties, the most influen-  tial parameter choice for generation is the sampling  4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='com/openai/openai-python  5https://beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='com/docs/api-reference/  parameter-details  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' We generally take a more measured  approach than the default (temperature=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='0), but  vary temperature across our ensemble prompts to  ensure a more diverse result set overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' We list  the used temperatures in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Zero-shot with  context is used twice in the ensemble, once with  a more conservative temperature, and once with a  more “creative” (higher) temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' For the sin-  gular prompt run, we use the conservative zero-shot  with context variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  C  Post-Filtering Operations  Given the uncertain nature of predictions by a lan-  guage model, we employ a series of post-filtering  steps to ensure high quality outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' This includes  stripping newlines/spaces/punctuation (\\n :;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='.?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' ),  lower-casing, removing infinitive forms (in some  instances, we observed predictions in the form of  “to deploy” instead of simply “deploy”), as well as  removing identity predictions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=', the prediction  being the same as the original complex word) and  deduplicating suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Additionally, we no-  ticed that for some instances, generated synonyms  resemble more of a “description” rather than truly  synonymous expressions (example: “people that  are crazy” as a suggestion for “maniacs”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' Given  the nature of provided data, we removed extreme  multi-word expressions (for English, any sugges-  tion with more than two words, for Spanish and  Portuguese more than three words in a single ex-  pression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Prompt Type  Template  Zero-shot /w context  Context: {context_sentence}\\n  temperature:  Question: Given the above context, list ten alternatives for  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='3 (conservative),  “{complex_word}” that are easier to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='\\n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='8 (creative)  Answer:  Single-shot /w context  Context: A local witness said a separate group of attackers  temperature: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='5  disguisedin burqas — the head-to-toe robes worn by conservative  Afghan women — then tried to storm the compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='\\n  Question: Given the above context, list ten alternative words  for “disguised” that are easier to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='\\n  Answer:\\n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' concealed\\n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' dressed\\n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' hidden\\n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' camouflaged\\n  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' changed\\n6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' covered\\n7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' masked\\n8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' unrecognizable\\n  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' converted\\n10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' impersonated\\n\\n  Context: {context_sentence}\\n  Question: Given the above context, list ten alternatives for  “{complex_word}” that are easier to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='\\n  Answer:  Two-shot /w context  Context: That prompted the military to deploy its largest  temperature: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='5  warship, the BRP Gregorio del Pilar, which was recently  acquired from the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='\\n  Question: Given the above context, list ten alternative words  for “deploy” that are easier to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='\\n  Answer:\\n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' send\\n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' post\\n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' use\\n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' position\\n5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' send out\\n  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content=' employ\\n7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
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+page_content='2673  Table 5: Results on the Portuguese language test set of the TSAR-2022 shared task, ranked by ACC@1 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
+page_content='  Listed are our own results (Ensemble and Single), the two best-performing competing systems (GMU-WLV and  Cental), as well as provided baselines (LSBert (Qiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAzT4oBgHgl3EQfzP4S/content/2301.01764v1.pdf'}
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+MNRAS 000, 1–19 (2022)
+Preprint 11 January 2023
+Compiled using MNRAS LATEX style file v3.0
+A deep radius valley revealed by Kepler short cadence observations
+Cynthia S. K. Ho★ and Vincent Van Eylen
+Mullard Space Science Laboratory, University College London, Dorking RH5 6NT, UK
+Accepted 2022 December 19. Received 2022 December 2; in original form 2022 October 17
+ABSTRACT
+The characteristics of the radius valley, i.e., an observed lack of planets between 1.5-2 Earth radii at periods shorter than about
+100 days, provide insights into the formation and evolution of close-in planets. We present a novel view of the radius valley
+by refitting the transits of 431 planets using Kepler 1-minute short cadence observations, the vast majority of which have not
+been previously analysed in this way. In some cases, the updated planetary parameters differ significantly from previous studies,
+resulting in a deeper radius valley than previously observed. This suggests that planets are likely to have a more homogeneous
+core composition at formation. Furthermore, using support-vector machines, we find that the radius valley location strongly
+depends on orbital period and stellar mass and weakly depends on stellar age, with 𝜕 log �𝑅𝑝,valley
+�/𝜕 log 𝑃 = −0.096+0.023
+−0.027,
+𝜕 log �𝑅𝑝,valley
+�/𝜕 log 𝑀★ = 0.231+0.053
+−0.064, and 𝜕 log �𝑅𝑝,valley
+�/𝜕 log (age) = 0.033+0.017
+−0.025. These findings favour thermally-driven
+mass loss models such as photoevaporation and core-powered mass loss, with a slight preference for the latter scenario. Finally,
+this work highlights the value of transit observations with short photometric cadence to precisely determine planet radii, and we
+provide an updated list of precisely and homogeneously determined parameters for the planets in our sample.
+Key words: planets and satellites: composition – planets and satellites: formation – planets and satellites: fundamental parameters
+1 INTRODUCTION
+The ‘radius valley’, also known as the ‘radius gap’, is the relative
+paucity of planets with sizes between about 1.5 and 2 Earth radii
+at orbital periods less than about 100 days. This phenomenon has
+been predicted theoretically due to the heavy radiation these close-in
+planets receive from their host star (e.g. Owen & Wu 2013; Lopez &
+Fortney 2013) and was subsequently seen observationally (e.g. Fulton
+et al. 2017; Van Eylen et al. 2018; Fulton & Petigura 2018). Several
+theories have been suggested to explain the physical origin of the
+radius valley. On one hand, thermally-driven mass loss scenarios have
+been proposed, which include photoevaporation (e.g. Owen & Wu
+2013; Lopez & Fortney 2013; Owen & Wu 2017) and core-powered
+mass loss (e.g. Ginzburg et al. 2018; Gupta & Schlichting 2019, 2020)
+models. In these scenarios, the valley separates planets that have lost
+their atmosphere from those that have retained it. Alternatively, late
+gas-poor formation, where planets below the valley have formed
+atmosphere-free, may also be able to explain the origin of the valley
+(e.g. Lee et al. 2014; Lee & Chiang 2016; Lopez & Rice 2018;
+Cloutier & Menou 2020).
+Observed characteristics of the radius valley can therefore reveal
+the properties of these close-in planets and their formation history.
+For example, in photoevaporation models, the location of the radius
+valley and its slope as a function of orbital period depend on the
+planetary composition and photoevaporation physics (Owen & Wu
+2017; Mordasini 2020). The valley’s location and relative emptiness
+can therefore be used to infer the composition of planets surrounding
+it and their relative homogeneity (e.g. Van Eylen et al. 2018). Planets
+★ E-mail: sze.ho.20@ucl.ac.uk
+located inside the radius valley may have a different composition or
+could be undergoing the final stages of atmospheric loss by thermally-
+driven mechanisms and hence may be important targets for further
+studies (e.g. Owen & Wu 2017; Gupta & Schlichting 2019; Petigura
+2020). The valley’s location as a function of orbital period can be
+used to distinguish between thermal mass-loss models, which exhibit
+a negative slope as a function of orbital period, and late gas-poor
+formation models which have the opposite slope (e.g. Van Eylen
+et al. 2019; Cloutier & Menou 2020; Van Eylen et al. 2021). Within
+thermal mass-loss models, photoevaporation and core-powered mass
+loss models predict a different dependence of the valley’s location
+on stellar mass and age (e.g. Rogers et al. 2021).
+Observationally, these valley characteristics have been challeng-
+ing to reliably ascertain. A deficit of planets with sizes around 1.5-2
+Earth radii (𝑅⊕) was first observed by Fulton et al. (2017) in a sam-
+ple of 2025 planets, with stellar radii determined spectroscopically
+as part of the California-Kepler survey (CKS). These planets were
+about a factor of two rarer than planets both smaller and larger.
+Independently, Van Eylen et al. (2018) (V18 hereafter) analysed a
+subset of this sample (117 planets), incorporating higher-precision
+stellar parameters using asteroseismology and refitting transit light
+curves to achieve a median uncertainty on planet sizes of 3.3%. This
+study revealed the valley’s slope as a function of orbital period for
+the first time, and suggested the radius valley may be very deep or
+even entirely empty.The tension between the valley’s views of Fulton
+et al. (2017) and V18 was further exacerbated when the precision
+of stellar parameters of the former study were further improved by
+Fulton & Petigura (2018) (F18 hereafter). Despite improving stellar
+uncertainties from 11% to 3% by incorporating Gaia parallaxes, the
+valley remained partially filled in, with its depth largely unchanged.
+© 2022 The Authors
+arXiv:2301.04062v1  [astro-ph.EP]  10 Jan 2023
+
+2
+Ho & Van Eylen
+Petigura (2020) investigated the discrepancy in the valley’s depth
+between V18 and F18 and concluded it is unlikely to be caused by
+differing sample sizes or differing values or uncertainties in stellar
+radii. The study argued that the 6.9% dispersion in planetary radii
+is instead primarily caused by a discrepancy in the ratio of planet
+to stellar radii (𝑅𝑝/𝑅★)) determined from the transit fits. F18 used
+radius ratios from Mullally et al. (2015), which fitted Kepler 30-
+minute ’long cadence’ observations, whereas V18 used Kepler 1-
+minute ’short cadence’ observations, also used for orbital eccentricity
+determination and described in Van Eylen & Albrecht (2015) and Van
+Eylen et al. (2019).
+Here, we seek to refit planet transits for the full subset of F18
+for which short cadence observations are available. This increases
+the sample of planets relevant for the radius valley for which short
+cadence transit fits are used from 60 in V18 to 431 here. Furthermore,
+we will apply the methods to determine the valley’s location and
+slope used by V18, notably the use of support vector machines, to
+this larger sample, and we expand to other dimensions such as stellar
+mass and age.
+In Section 2, we describe the sample and methodology used to
+analyse the radius valley. In Section 3, we present the results of this
+analysis, such as revised planetary sizes, the depth of the valley, and
+its dependence on parameters such as the orbital period and stellar
+mass and age. These findings are compared to other observational
+studies and theoretical models in Section 4. Finally, we provide con-
+clusions in Section 5.
+2 METHODS
+2.1 Sample selection
+We use the sample of planets for which stellar parameters are avail-
+able from F18 as a starting point. To focus on the radius valley, we
+limit the sample to planets with radii 1 ≤ 𝑅𝑝/𝑅⊕ ≤ 4 and orbital
+periods 1 ≤ 𝑃/days ≤ 100, resulting in a sample of 1272 planets (for
+comparison, applying the same period and radius cuts to the sample
+studied by V18 leaves 74 planets). As Kepler 1-minute short cadence
+observations may yield superior precision (Petigura 2020), we fur-
+ther limit our sample to those planets for which at least 6 months of
+Kepler short cadence data are available.
+To avoid issues with transit fitting related to transit timing varia-
+tions (TTVs), we also remove planets with known TTVs based on the
+catalogue by Holczer et al. (2016). We further exclude KOI-1576.03,
+as we find that the short cadence data suggested an orbital period dif-
+ferent to the one recorded in the archive. Furthermore, we exclude any
+planets that are classified as potential false positives in Petigura et al.
+(2017). The results in a total sample size of 431 planets, 60 of which
+have parameters previously analysed by V18 and 371 which have not
+(a further 14 planets in V18 have TTVs and are not reanalysed here).
+2.2 Data reduction
+The 1-minute Kepler short cadence Pre-search Data Conditioning
+SAP (PDCSAP, Stumpe et al. 2012; Smith et al. 2012) light curves
+of these targets are downloaded from the NASA Mikulski Archive
+for Space Telescopes (MAST) database using the lightkurve
+package (Lightkurve Collaboration et al. 2018), which incorporates
+astroquery (Ginsburg et al. 2019) and astropy (Astropy Collab-
+oration et al. 2013, 2018) dependencies. We only retain data within
+0.2 days before the estimated ingress and after the estimated egress of
+the transits of the planets of interest, using the transit durations and
+mid-times in Mullally et al. (2015) as the expected transit locations.
+For multi-planet systems, we only retain transits of planets that are
+within our sample. We remove data outliers that lie beyond 6𝜎 from
+the median after masking the transits. We then flatten the transits by
+dividing the data points with the slope obtained by performing linear
+regression on the data points immediately before ingress and after
+egress, to remove long-term systematic trends present in the transits.
+We then again remove data outliers with 𝜎 = 5 to further clean the
+data.
+2.3 Stellar multiplicity
+Around 46% of solar-type stars have at least one stellar companion
+(Raghavan et al. 2010). When a planet orbits a single star, the transit
+depth 𝛿 is approximately given by
+𝛿 = Δ𝐹
+𝐹tot
+≈
+𝑅2𝑝
+𝑅2
+★
+(1)
+where 𝐹tot is the total stellar flux, Δ𝐹 is the change in stellar flux, and
+𝑅𝑝 and 𝑅★ are the planetary and stellar radius respectively. However,
+in a multi-stellar system, the total flux is the sum of fluxes of all stars
+in the system, but the change in flux during transit is only relative to
+the star(s) which the planet transits (Furlan et al. 2017). Therefore
+it is important to take into account the effect of nearby stars on the
+light curve flux.
+Furlan et al. (2017) compiled a catalogue of Kepler Objects of
+Interest (KOI) observations with adaptive optics, speckle interfer-
+ometry, lucky imaging, and imaging from space with the Hubble
+Space Telescope. The typical point spread function (PSF) widths
+and sensitivities (Δ𝑚) are different for every observation method,
+target and bandpass, hence whether stellar companions are detected
+is dependent on the above factors. For example, Furlan et al. (2017)
+were able to detect a median Δ𝑚 ∼ 8mag with Keck in the K band at
+a separation of ∼0.5”, but only at ∼2.5” at Lick in the J or H bands.
+About 30% of KOIs observed in Furlan et al. (2017) have at least one
+companion detected within 4” (Furlan et al. 2017), and given a mean
+distance of 616pc for the 431 planets in our sample computed from
+distances reported in Mathur et al. (2017), corresponds to ∼2464AU.
+Here, we adopt the ‘radius correction factor’ (RCF), given in Furlan
+et al. (2017) as
+RCF =
+𝑅𝑝,corr
+𝑅𝑝,uncorr
+(2)
+and multiply the normalised Kepler light curve fluxes by RCF2, and
+subtract
+�
+RCF2 − 1
+�
+to re-normalise, to obtain the corrected light
+curve reflecting the transit of one planet orbiting around one star.
+137 of the 431 planets in our sample (32%) have RCF measurements
+from Furlan et al. (2017).
+2.4 Transit fitting
+We use the exoplanet package (Foreman-Mackey et al. 2021) to
+generate a transit light curve model with quadratic stellar limb dark-
+ening, and then run a Hamiltonian Monte Carlo (HMC) algorithm
+implemented in PyMC3 (Salvatier et al. 2016) to perform fitting and
+determine orbital parameter posteriors. We also implement a Gaus-
+sian Process (GP) model (Rasmussen & Williams 2006) to account
+for correlated noise in the light curves. However, for Kepler-65 and
+Kepler-21 A, we do not fit for a GP model due to convergence
+constraints. The parameters fitted for each planet are orbital period
+(𝑃), transit mid-time (𝑡0), ratio between planetary and stellar radii
+MNRAS 000, 1–19 (2022)
+
+Deep radius valley with Kepler short cadence
+3
+(𝑅𝑝/𝑅★), impact parameter (𝑏), eccentricity (𝑒), argument of peri-
+apsis (𝜔), and stellar density (𝜌★). For each light curve, we further
+include two quadratic stellar limb darkening parameters (𝑢0 and 𝑢1)
+for the host star, with bounds 0 < 𝑢0, 𝑢1 < 1 and implemented with
+the Kipping (2013) reparameterisation in exoplanet, the transit jit-
+ter (𝜎lc), and two parameters describing the GP contribution (𝜎gp,
+𝜌gp).
+We initialise the HMC chains by using values presented in the
+Kepler Q1-16 dataset (Mullally et al. 2015) for 𝑃, 𝑡0, 𝑅𝑝/𝑅★, and 𝑏.
+We set the system to begin with near-circular orbits, with 𝑒 = 0.01 and
+𝜔 = 0.01 rad. We take initial stellar densities from Fulton & Petigura
+(2018). We use the Exoplanet Characterization ToolKit (ExoCTK)
+(Bourque et al. 2021) to estimate the initial 𝑢0 and 𝑢1, which takes
+the stellar temperature, surface gravity, and metallicity, which we
+use values from the Kepler Q1-16 dataset (Mullally et al. 2015), as
+inputs.
+We apply Gaussian priors to 𝑃, 𝑡0, 𝑢0, 𝑢1, and 𝜌★, using the initial
+guesses as the mean, and 𝜎𝑃 = 2 × 10−5 days, 𝜎𝑡0 = 10−3 days,
+𝜎𝑢 = 0.2, and the 𝜌★ uncertainty from Fulton & Petigura (2018) if
+available, and Mullally et al. (2015) otherwise. A beta distribution
+prior according to Van Eylen et al. (2019) is placed on 𝑒, which is
+PDF(𝑒, 𝛼, 𝛽) ∝ 𝑒𝛼(1 − 𝑒)𝛽
+(3)
+with 𝛼 = 1.58 and 𝛽 = 4.4 for system with only one transiting planet,
+and 𝛼 = 1.52 and 𝛽 = 29 for a multi-transiting-planet system.
+3 RESULTS
+3.1 Revised planet parameters
+We report the updated orbital periods (𝑃), planetary-to-stellar-radii
+ratio (𝑅𝑝/𝑅★), planetary radii (𝑅𝑝), the number of transits in the
+fitted light curve (𝑁tr), and their uncertainties of the 431 planets fitted
+in this sample in Table 1. The full list of parameters are provided in
+Appendix A. We convert our 𝑅𝑝/𝑅★ to 𝑅𝑝 using the updated stellar
+parameters available: values used in V18 from asteroseismology (i.e.
+taken from Huber et al. 2013; Silva Aguirre et al. 2015; Lundkvist
+et al. 2016) if the planets are included in the V18 samples, and F18
+otherwise. Full homogeneity is lost by using stellar radii from two
+sources. To investigate the consequences of this, we compute the
+difference, 𝛿𝑅𝑝, between the planetary radii obtained by converting
+𝑅𝑝/𝑅★ to 𝑅𝑝 using 𝑅★ from F18 and V18, and found the mean
+𝛿, ¯𝛿 = 0.03 ± 0.11, hence 𝛿 = 0 (no difference) is well within
+1𝜎, and we conclude that there is no substantial drawbacks of using
+multiple sources. This sample of 431 planets with updated parameters
+is plotted on the radius-orbital period plot as shown in Figure 1.
+We present the typical uncertainties of 𝑅𝑝/𝑅★ and 𝑅𝑝 of planets
+fitted in this work, compared with F18 and V18 in Table 2. For our
+newly fitted results, we find that 𝑅𝑝/𝑅★ has a mean uncertainty of
+4.76%, and a median uncertainty of 3.44%. This is smaller when
+compared to the mean and median uncertainties of 8.73% and 4.07%
+respectively for the F18 sample. For 𝑅𝑝, we find a mean and median
+uncertainty of 6.09% and 4.70% in our work, again smaller compared
+to 10.00% and 5.22% for F18. However, they are larger than that of
+V18, possibly due to V18 analysing brighter stars, hence the light
+curves are less noisy. This can be seen by comparing the mean
+photometric flux error of the transit light curves fitted: 1007 parts
+per million (ppm) for this work, and 271 ppm for V18.
+We select the planets in F18 that are in common with the planets in
+our sample, and examine the change in 𝑅𝑝. We find that 217 planets
+have a larger 𝑅𝑝 after refitting, and smaller for 214 planets. Of the
+planets whose sizes have increased, the mean change is 9.79%, and
+Figure 1. Radius-period plot of all 431 planets refitted in this work. The
+black dotted lines indicate the upper and lower boundaries of the radius
+valley defined in V18.
+8.69% for planets with reduced sizes. Considering all 431 planets,
+our new results change 𝑅𝑝 by a mean and median of only 0.62%
+and 0.02% respectively, indicating that our refitting results do not
+systematically alter the planet sizes. We observe that 120 planets
+(28%) have a revised 𝑅𝑝 > 2𝜎 away from their corresponding values
+from F18, and 62 planets (14%) > 3𝜎 away.
+3.2 Radius valley dependence on orbital period
+To obtain the most precise planetary sample, we apply the following
+conservative cuts to our sample for subsequent analyses in the rest of
+this paper:
+(i) Precision on 𝑅𝑝: We exclude planets with a planet radius
+precision 𝜎𝑅𝑝 > 10%.
+(ii) Radius correction factor (RCF): We exclude planets with an
+RCF > 5%, as reported in Furlan et al. (2017). RCFs may themselves
+be uncertain due to observational challenges in detecting nearby stel-
+lar companions and measuring their brightness (Furlan et al. 2017).
+Of the 137 planets with RCF measurements from Furlan et al. (2017),
+18 (13%) have RCF > 5%. The remaining 294 planets do not have
+RCF measurements currently available.
+(iii) Number of transits: We exclude planets with fewer than 3
+transits in the Kepler short cadence data to limit the risk that corre-
+lated noise during an individual transit strongly affects the resulting
+fit.
+After implementing these filters, 375 planets remain in our sample
+which we will use throughout the remainder of this work.
+We first use this sample to investigate the location of the radius val-
+ley as a function of orbital period. Following the procedure outlined
+in Van Eylen et al. (2018), we calculate the position of the radius
+valley by determining the hyperplane of maximum separation. We
+perform this with a linear support-vector machine (SVM). To ini-
+tialise our model, we initially classify our sample into two groups,
+‘above’ and ‘below’ the radius valley on the radius-orbital period
+plane, by applying a Gaussian Mixture Model with two components
+in the 𝑅𝑝-𝑃 plane. Since the orders of magnitude of 𝑅𝑝 and 𝑃 are
+different, we divide 𝑃 by 5 before applying the above clustering al-
+gorithm to allow the model to separate the planetary population into
+MNRAS 000, 1–19 (2022)
+
+4
+HO
+D
+0
+D
+3
+TO
+DT
+D
+!
+HO
+D
+TO
+O
+2
+R
+D
+O
+D
+勇
+DI
+D
+D
+D
+3
+10
+30
+100
+P (days)4
+Ho & Van Eylen
+Table 1. Table showing estimates of the orbital periods (𝑃), planetary-to-stellar-radii ratio (𝑅𝑝/𝑅★), planetary radii (𝑅𝑝), the number of transits in the fitted
+light curve (𝑁tr) of 431 planets refitted in this work. The source of 𝑅★ to convert 𝑅𝑝/𝑅★ to 𝑅𝑝 is listed in the References column: (1) Fulton & Petigura
+(2018), (2) Van Eylen et al. (2018).‘Flag’ refers to whether the planet passes the filter checks and is included in the smaller subset for further analyses (1 for
+True and 0 for False). The complete list of parameters are provided in Appendix A. Only the first 10 planets are shown here; the full table is available online in
+a machine-readable format.
+KOI
+Kepler name
+𝑃 (days)
+𝑅𝑝/𝑅★
+𝑅𝑝 (𝑅⊕)
+𝑅★ Source
+𝑁tr
+Flag
+K00041.01
+Kepler-100 c
+12.815893 ± 0.000008
+0.0138 ± 0.0002
+2.28+0.03
+−0.03
+(2)
+93
+1
+K00041.02
+Kepler-100 b
+6.887062 ± 0.000007
+0.0082 ± 0.0001
+1.36+0.03
+−0.03
+(2)
+173
+1
+K00041.03
+Kepler-100 d
+35.333093 ± 0.000019
+0.0104 ± 0.0002
+1.71+0.04
+−0.04
+(2)
+35
+1
+K00046.02
+Kepler-101 c
+6.029792 ± 0.000020
+0.0073 ± 0.0007
+1.32+0.14
+−0.14
+(1)
+50
+0
+K00049.01
+Kepler-461 b
+8.313784 ± 0.000015
+0.0287 ± 0.0010
+4.03+0.17
+−0.17
+(1)
+34
+1
+K00069.01
+Kepler-93 b
+4.726739 ± 0.000001
+0.0151 ± 0.0001
+1.50+0.04
+−0.04
+(2)
+275
+1
+K00070.01
+Kepler-20 A c
+10.854089 ± 0.000003
+0.0290 ± 0.0002
+2.79+0.07
+−0.07
+(1)
+116
+1
+K00070.02
+Kepler-20 A b
+3.696115 ± 0.000001
+0.0182 ± 0.0002
+1.75+0.05
+−0.04
+(1)
+336
+1
+K00070.03
+Kepler-20 A d
+77.611598 ± 0.000019
+0.0263 ± 0.0003
+2.52+0.07
+−0.07
+(1)
+15
+1
+K00070.05
+Kepler-20 A f
+19.577627 ± 0.000020
+0.0091 ± 0.0004
+0.88+0.04
+−0.04
+(1)
+62
+1
+...
+...
+...
+...
+...
+...
+...
+...
+Table 2. Mean and median values radii determined in this work, F18, and
+V18. Values are listed as percentages (%).
+Parameter
+This work
+F18
+V18
+Sample Size
+431
+1901
+117
+𝑅𝑝/𝑅★
+Mean
+4.76
+8.73
+3.20
+Median
+3.44
+4.07
+2.44
+𝑅★
+Mean
+3.23
+3.17
+2.49
+Median
+2.78
+2.74
+2.20
+𝑅𝑝
+Mean
+6.09
+10.00
+3.96
+Median
+4.70
+5.22
+3.36
+two groups above and below the radius valley. Otherwise, the pop-
+ulation would be clustered in a way dominated by the difference in
+period. Following Van Eylen et al. (2018) and David et al. (2021),
+we select an SVM penalty parameter of 𝐶 = 10 for the hyperplane,
+to minimise misclassification of data points above or below the ra-
+dius valley, but still allow the hyperplane location to be determined
+by a sufficient number of data points. To determine accurate uncer-
+tainties on the location of the valley, we then perform a bootstrap
+by generating 1000 new sample sets on which we repeat the above
+procedure. Each bootstrap sample is generated by generating a new
+sample of the same size from the original sample, allowing replace-
+ment. Each bootstrapped sample is then categorised into two groups
+with a Gaussian Mixture Model, and the SVM procedure is repeated.
+Reporting the median value and taking the 16th and 84th percentiles
+as the upper and lower uncertainties, we find
+log10
+�𝑅𝑝/𝑅⊕
+� = 𝑚 log10 (𝑃/days) + 𝑐
+(4)
+with 𝑚 = −0.11±0.02, and 𝑐 = 0.37+0.02
+−0.03. The location of the radius
+valley is plotted in Fig. 2.
+We also implement the method adopted by Petigura et al. (2022),
+which involves computing the planet density in the radius-period
+plane with a Gaussian kernel density estimate (KDE), and fitting the
+planet radius at the KDE minima between log10 (𝑃/days) = 0.5−1.5
+(i.e. 𝑃 ≈ 3.16 − 31.6 days) and log10
+�𝑅𝑝/𝑅⊕
+� = 0.15 − 0.35 (i.e.
+𝑅𝑝 ≈ 1.41 − 2.24𝑅⊕) according to equation 4, and performing
+bootstrap with 1000 sample sets to find the uncertainties. The result
+is illustrated in Figure 2. We use a KDE bandwidth of 0.467 in
+log10 𝑃 and 0.075 in log10 𝑅𝑝, which is based on the bandwidth used
+by Petigura et al. (2022), but scaled up linearly based on the ratios
+between the two sample sizes. We note that this method fits a narrower
+period range compared to the SVM method, which fits for the full
+𝑃 = 1 − 100 days. With this method, we obtain 𝑚 = −0.12+0.03
+−0.05
+and 𝑐 = 0.37+0.05
+−0.03. We find this value to be slightly steeper than that
+determined with the SVM, however the two values are well within
+1𝜎 of each other. Here, we do not correct for detection completeness,
+and we note that this method is highly sensitive to the choice of the
+KDE bandwidth, and using a bandwidth five times larger results with
+a slope approximately twice as steep.
+Some studies have opted to study the radius valley location as a
+function of incident flux 𝑆, in addition to, or instead of, 𝑃 (e.g. Rogers
+et al. 2021; Petigura et al. 2022). We calculate 𝑆 according to the
+formula
+𝑆 =
+𝐿★
+4𝜋𝑎2
+(5)
+where 𝐿★ is the host star’s luminosity. 𝐿★ can itself be calculated
+using
+𝐿★ = 4𝜋𝑅2
+★𝜎sb𝑇4
+eff
+(6)
+where 𝑅★ is the stellar radius, 𝜎sb is the Stefan-Boltzmann constant,
+and 𝑇eff is the effective temperature of the star. Finally, 𝑎 is the orbital
+semi-major axis, which is given by
+� 𝑎
+𝑅★
+�3
+= 𝐺𝑃2𝜌★
+3𝜋
+(1 + 𝑒 sin 𝜔)3
+�1 − 𝑒2�1.5
+(7)
+according to Kepler’s third law of planetary motion (e.g. Van Eylen &
+Albrecht 2015; Petigura 2020). Here, 𝐺 is the gravitational constant,
+𝑃 is the orbital period, 𝜌★ is the stellar density, 𝑒 and 𝜔 are the orbital
+eccentricity and argument of periapsis respectively. As before, we
+obtain the stellar properties (𝑅★, 𝜌★, 𝑇eff) from V18 when available,
+and from F18 otherwise. 𝑃, 𝑒, and 𝜔 are obtained from the transit
+fitting results of this work. Fitting the valley with the SVM, we obtain
+MNRAS 000, 1–19 (2022)
+
+Deep radius valley with Kepler short cadence
+5
+Table 3. Dependencies of the radius valley in 𝑛 dimensions (𝑚𝑖), given by the equation log10
+�𝑅𝑝/𝑅⊕
+� = �𝑛
+𝑖=1 𝑚𝑖 𝑥𝑖, with different parameter combinations
+𝑥𝑖. The methods used to obtain the equation of the radius valley hyperplanes are also given, where SVM and KDE stand for the support-vector machine and
+fitting the minima of the kernel density estimates respectively.
+Dimensions
+log10 (𝑃/days)
+log10
+�𝑆/𝑆⊕
+�
+log10
+�𝑀/𝑀⊙
+�
+log10 (Age/Gyr)
+[Fe/H]
+Intercept
+Method
+2
+−0.11+0.02
+−0.02
+0.37+0.02
+−0.03
+SVM
+−0.12+0.03
+−0.05
+0.37+0.05
+−0.03
+KDE
+0.07+0.02
+−0.01
+0.11+0.03
+−0.04
+SVM
+0.23+0.09
+−0.08
+0.27+0.01
+−0.01
+SVM
+0.02+0.01
+−0.02
+0.26+0.01
+−0.01
+SVM
+0.06+0.06
+−0.08
+0.26+0.01
+−0.01
+SVM
+3
+−0.09+0.02
+−0.03
+0.21+0.06
+−0.07
+0.35+0.02
+−0.03
+SVM
+0.07+0.02
+−0.02
+−0.01+0.07
+−0.09
+0.11+0.04
+−0.05
+SVM
+−0.10+0.02
+−0.02
+0.03+0.02
+−0.03
+0.34+0.03
+−0.02
+SVM
+−0.10+0.03
+−0.03
+0.03+0.03
+−0.04
+0.36+0.02
+−0.03
+SVM
+4
+−0.096+0.023
+−0.027
+0.231+0.053
+−0.064
+0.033+0.017
+−0.025
+0.339+0.026
+−0.018
+SVM
+Figure 2. Left: Radius valley location determined with the support-vector machine (SVM), with 𝑚 = −0.11 ± 0.02, and 𝑐 = 0.36+0.02
+−0.03, indicated by the black
+solid line. The dashed lines represent the median boundaries passing through the supporting vectors determining the position of the solid line. We define the area
+between the two lines as the radius valley region. The green and blue points show planets above and below the radius valley respectively. The grey shaded regions
+represent the ±1𝜎 uncertainties of the lines determined using the bootstrap method. The red dotted lines show the plot divided into multiple orbital-period
+dependent bins, which is then used for plotting the adjusted histogram in Fig. 4. Right: Radius valley position determined by fitting a line through the region
+where the kernel density estimate is minimum. With this method, we obtain 𝑚 = −0.12+0.03
+−0.05, and 𝑐 = 0.37+0.0
+−0.03, indicated by the black solid line with ±1𝜎
+uncertainty shaded in grey.
+log10 𝑅𝑝/𝑅⊕ = 𝑚 log10 𝑆/𝑆⊕ + 𝑐
+(8)
+with 𝑚 = 0.07+0.02
+−0.01, and 𝑐 = 0.11+0.03
+−0.04. The location of the radius
+valley in terms of 𝑆 is shown in Figure 3.
+3.3 Depth of the radius valley
+We investigate the depth of the radius valley. As we find the position
+of the radius valley is dependent on orbital period, we divide the
+radius valley into multiple tilted bins (as shown in Fig. 2 left) and
+plot an adjusted histogram in logarithmic scale. We shift planets
+along the slope of the radius valley obtained with the SVM method
+in Section 3.2, i.e. 𝑚 = −0.11, and plot a histogram of ‘expected’
+planetary radii at an orbital period of 10 days, shown in Fig. 4. We
+choose to fit the histogram with a Gaussian Mixture Model of two
+clusters, as opposed to a Gaussian kernel density estimate, as the
+former is independent of the sizes and locations of the histogram
+bins, as well as the Gaussian bandwidth. Also, with the Gaussian
+Mixture Model, we are able to force the planets to fall into two
+groups only, matching the bimodal distribution of small planets.
+MNRAS 000, 1–19 (2022)
+
+4
+3
+2
+3
+10
+30
+100
+P (days)Relative density
+YD
+2
+1
+3
+TOI
+R
+2
+R
+3
+10
+30
+100
+P (days)6
+Ho & Van Eylen
+Figure 3. Same as Figure 2 left, but as a function of incident flux 𝑆 instead
+of orbital period 𝑃.
+Figure 4. Histogram of planet radii, adjusted to equivalent radii at 𝑃 = 10
+days, according to the radius valley slope calculated in Section 3.2 with the
+SVM. Note that completeness corrections are not performed. Here, 𝐸avg =
+2.98+0.60
+−0.47.
+Here, we propose the metric 𝐸, defined as
+𝐸SN = 𝑁sub-Neptune,peak/𝑁valley
+(9)
+and
+𝐸SE = 𝑁super-Earth,peak/𝑁valley
+(10)
+to compare the number of planets inside the valley and the peak num-
+ber outside the valley. A higher 𝐸 indicates a deeper radius valley.
+𝑁sub-Neptune,peak and 𝑁super-Earth,peak are the number of planets at
+the sub-Neptune and super-Earth Gaussian peaks respectively, and
+𝑁valley is the number of planets at the lowest point between the two
+Gaussian peaks. As 𝑁sub-Neptune, peak, 𝑁super-Earth, peak, and 𝑁valley
+are determined directly from the curve resulting from the Gaussian
+Mixture Model, this 𝐸 metric is also independent of the histogram
+bin sizes or locations. To calculate the uncertainties of 𝐸, we again
+perform a bootstrap with 1000 sample sets, where each bootstrap
+Figure 5. Plot of planetary radius against mass of host star. The solid line
+shows the location of the radius valley determined with the SVM, with slope
+𝑚 = 0.23+0.09
+−0.08, and 𝑐 = 0.27 ± 0.01. The dashed lines show the boundaries
+of the radius valley, given by the same 𝑚, and 𝑐upper = 0.33 ± 0.01, 𝑐lower =
+0.22 ± 0.01. The shaded regions depict the ±1𝜎 uncertainties of the lines
+determined with bootstrapping.
+sample is generated by generating a new sample of the same size
+from the original, allowing replacements. We also replace the ra-
+dius of each selected planet by randomly drawing from a normal
+distribution with the reported planet radius as the mean, and the
+uncertainties on the radius as the variance. We report the median
+of this bootstrap distribution as our results, and the 84th and 16th
+percentiles as our ±1𝜎 uncertainties. We test this metric on known
+planetary samples on F18 and V18, which we know the difference
+in the radius valley depth, and find that their corresponding 𝐸 value
+differ, hence demonstrating the reliability of such metric. The dif-
+ference in the depth of the radius valley is further discussed in Sec-
+tion 4.3. We find that for our new short cadence results, the ratios are
+𝐸SN = 3.59+0.77
+−0.62 and 𝐸SE = 2.40+0.61
+−0.41. Averaging the two numbers
+gives us 𝐸avg = 2.98+0.60
+−0.47.
+3.4 Radius valley dependence on stellar mass
+We investigate the radius valley location as a function of stellar
+mass. We first implement a 2-dimensional SVM, using the same
+method as in Section 3.2. The result is shown in Figure 5. We find
+d log 𝑅𝑝/d log 𝑀★ = 0.23+0.09
+−0.08, and the intercept 𝑐 = 0.27 ± 0.01.
+Rogers et al. (2021) suggested the degeneracy between the photo-
+evaporation and core-powered mass loss scenarios could be broken
+with an analysis of the radius valley in 3 dimensions. Hence, we
+implement an SVM in 3 dimensions: planet radius 𝑅𝑝, orbital period
+𝑃, and mass of the host star 𝑀★, to fit the radius valley in the form
+of a plane. We perform bootstrapping with 1000 sample sets as per
+previous. We obtain the relation
+log10
+�𝑅𝑝/𝑅⊕
+� = 𝐴 log10 (𝑃/days) + 𝐵 log10
+�𝑀★/𝑀⊙
+� + 𝐶
+(11)
+with 𝐴 = −0.09+0.02
+−0.03, 𝐵 = 0.21+0.06
+−0.07, 𝐶 = 0.35+0.02
+−0.02. An illustration
+of the SVM plane is shown in Figure 6.
+We also investigate the radius valley location in the 𝑅𝑝–𝑆–𝑀★
+space. Figure 6 right shows the radius valley location in this space.
+MNRAS 000, 1–19 (2022)
+
+4
+甲
+TH
+3
+®
+R
+2
+R
+88
+由
+申
+1000
+100
+10
+1
+S (S)70
+60
+S
+f planet
+50
+40
+Number of
+30
+20
+N
+10
+0
+1
+2
+3
+4
+5
+6
+Rp at P= 10 days (R)4
+3
+2
+0.7
+1.0
+1.4
+M* (Mo)Deep radius valley with Kepler short cadence
+7
+Figure 6. Plot of planet radius against orbital period and mass of host star (left), and planet radius against incident flux and stellar mass (right). The grey plane
+shows the radius valley location determined with the SVM in 3 dimensions, with the ±1𝜎 uncertainties shown in pink. The uncertainties on individual planet
+parameters are not displayed in the interest of clarity.
+We find
+log10
+�𝑅𝑝/𝑅⊕
+� = 𝐴 log10
+�𝑆/𝑆⊕
+� + 𝐵 log10
+�𝑀★/𝑀⊙
+� + 𝐶
+(12)
+with 𝐴 = 0.07 ± 0.02, 𝐵 = −0.01+0.07
+−0.09, and 𝐶 = 0.11+0.04
+−0.05.
+3.5 Radius valley dependence on stellar age
+We investigate the location of the radius valley as a function of stellar
+age. We obtain the stellar ages from F18. Kepler-174 does not have
+stellar age data from this source, hence we omit Kepler-174 b and
+Kepler-174 c in our analysis with stellar age.
+Fitting the valley with the SVM, we obtain
+log10
+�𝑅𝑝/𝑅⊕
+� = 𝑚 log10 (Age/Gyr) + 𝑐
+(13)
+with 𝑚 = 0.02+0.01
+−0.02, 𝑐 = 0.26+0.01
+−0.01, showing no significant corre-
+lation with the radius valley location in 2-dimensional space. The
+result is displayed in Figure 7 (left panel).
+We further investigate whether the radius valley depth may be a
+function of stellar age. To do so we generate histograms, similar to
+Figure 4, split into different stellar age subsamples. Figure 8 (left
+panel) shows that for older stars, the radius valley location shifts to
+higher 𝑅𝑝, and the radius valley becomes shallower. The change in
+the 𝐸 metric is reported in Table 4. These findings suggest that the
+radius valley has a dependence on the age of the host stars.
+We further plot the radius valley in 𝑅𝑝–𝑃–age space, and fit the
+valley with the SVM, as shown in Figure 9 (left panel). We find
+log10
+�𝑅𝑝/𝑅⊕
+� = 𝐴 log10 (𝑃/days) + 𝐵 log10 (Age/Gyr) + 𝐶
+(14)
+with 𝐴 = −0.10 ± 0.02, 𝐵 = 0.03+0.02
+−0.03, and 𝐶 = 0.34+0.03
+−0.02.
+We can also combine 𝑅𝑝, 𝑃, 𝑀★, and stellar age, and determine
+the radius valley with a 4-dimensional SVM, as shown in Figure 10.
+The resulting equation representing the radius valley is in the form
+of a 4-dimensional hyperplane
+log10
+�𝑅𝑝/𝑅⊕
+� = 𝐴 log10 (𝑃/days) + 𝐵 log10
+�𝑀★/𝑀⊙
+�
++ 𝐶 log10 (Age/Gyr) + 𝐷
+(15)
+Table 4. 𝐸 values of the radius valley for different ages of the host stars.
+𝐸avg = (𝐸SN + 𝐸SE)/2.
+log10 (Age/yr)
+Age (Gyr)
+𝑁planets
+𝐸SN
+𝐸SE
+𝐸avg
+< 9.25
+< 1.78
+53
+4.93
+3.64
+4.28
+9.25 − 9.5
+1.78 − 3.16
+95
+4.69
+3.08
+3.89
+9.5 − 9.75
+3.16 − 5.62
+114
+3.22
+3.32
+3.27
+> 9.75
+> 5.62
+111
+2.21
+4.40
+3.30
+with 𝐴 = −0.096+0.023
+−0.027, 𝐵 = 0.231+0.053
+−0.064, 𝐶 = 0.033+0.017
+−0.025, 𝐷 =
+0.339+0.026
+−0.018. These results imply there is strong evidence the radius
+valley location is dependent on 𝑃 and 𝑀★, and weak evidence for its
+dependence on stellar age (> 1𝜎). These values are also consistent
+within 1𝜎 with their corresponding dependencies in two and three
+dimensions (see Table 3).
+3.6 Radius valley dependence on stellar metallicity
+We perform a similar analysis in terms of the stellar metallicity. As
+for age, we obtain the stellar metallicity from V18 if available, and
+F18 otherwise. We find
+log10
+�𝑅𝑝/𝑅⊕
+� = 𝑚[Fe/H] + 𝑐
+(16)
+with 𝑚 = 0.06+0.06
+−0.08, 𝑐 = 0.26+0.01
+−0.01, again displaying no significant
+correlation with the radius valley location in 2-dimensional space.
+We divide the planet population into two groups, based on the
+median [Fe/H] = 0.06. The adjusted histograms in Figure 8 (right
+panel) show that the super-Earth peak is lower for metal-poor stars.
+The 𝐸 values are reported in Table 5.
+We perform a similar SVM analysis in 𝑅𝑝–𝑃–[Fe/H] space (shown
+in Figure 9 right panel), and find
+log10
+�𝑅𝑝/𝑅⊕
+� = 𝐴 log10 (𝑃/days) + 𝐵[Fe/H] + 𝐶
+(17)
+with 𝐴 = −0.10 ± 0.03, 𝐵 = 0.03+0.03
+−0.04, and 𝐶 = 0.36+0.02
+−0.03. These
+MNRAS 000, 1–19 (2022)
+
+Sub-Neptunes
+Super-Earths
+4
+Rp (R)
+2
+1
+1.4
+1
+3
+1.0
+10
+30
+0.7
+100
+P (days)
+M* (Mo)Sub-Neptunes
+Super-Earths
+4
+Rp (R)
+2
+1
+1.4
+1000
+100
+1.0
+10
+0.7
+S (S)
+1
+M* (Mo)8
+Ho & Van Eylen
+Figure 7. Same as Figure 5, but for stellar age (left, 𝑚 = 0.01+0.01
+−0.02, 𝑐 = 0.26+0.01
+−0.01), and stellar metallicity [Fe/H] (right, 𝑚 = 0.06+0.06
+−0.08, 𝑐 = 0.26+0.01
+−0.01).
+Figure 8. Same as Figure 4, but separated into different stellar ages (left), and metallicities (right). Here, 𝑇 = log10 (Age/yr). The histograms are normalised
+such that the relative density of the sub-Neptune peak equals to unity.
+Table 5. 𝐸 values of the radius valley for different stellar metallicities. 𝐸avg =
+(𝐸SN + 𝐸SE)/2.
+[Fe/H]
+𝑁planets
+𝐸SN
+𝐸SE
+𝐸avg
+< 0.06
+181
+4.10
+2.20
+3.15
+≥ 0.06
+194
+3.28
+2.78
+3.03
+values imply that we find no evidence that the radius valley location
+depends on stellar metallicity.
+4 DISCUSSION
+4.1 𝑅𝑝-𝑃 relation suggests a thermally-driven mass loss model
+As presented in Section 3.2, we observe the radius valley scales
+as 𝑚 = d log 𝑅𝑝/d log 𝑃 = −0.11 ± 0.02. This negative period-
+dependence is a robust finding which remains roughly similar even
+when other parameters are included in the fit (see Table 3).
+Different theoretical mechanisms to create the radius valley result
+in a different slope as a function of orbital period. For example,
+Lopez & Rice (2018) predicted that if the rocky planets are core
+remnants of sub-Neptunes with evaporated atmospheres, the radius
+valley location should decrease with increasing orbital period, with
+𝑚 = −0.09, whereas if those rocky planets were formed after disk
+dissipation (i.e., late gas-poor formation), the radius valley location
+tends to larger planetary radii at longer orbital periods, with 𝑚 =
+0.11. Similarly, Owen & Wu (2017) predicted a negative period-
+radius valley slope for a photoevaporation model, with −0.25 ≤
+𝑚 ≤ −0.16 depending on the photoevaporation efficiency. If the
+radius valley is thermally driven but powered by the core rather than
+photoevaporation, the slope would be similarly negative, with e.g.,
+Gupta & Schlichting (2019) predicting that 𝑚 ≈ −0.11 in this case.
+Theoretically predicted slopes for different formation mechanisms
+are summarised in Table 6.
+Our observed negative slope is consistent with thermally driven
+mass-loss models but inconsistent with late gas-poor formation mod-
+els. We can also compare our observed slope with other observational
+MNRAS 000, 1–19 (2022)
+
+4
+3
+2
+=0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+Fe/H/(dex)1.6
+T<9.25
+1.4
+9.25 ≤T<9.5
+9.5 ≤ T< 9.75
+Relative density
+1.2
+T≥ 9.75
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+3
+4
+2
+5
+6
+Rp at P=
+:10 days (R@)1.4
+[Fe/H] < 0.06
+[Fe/H] ≥ 0.06
+1.2
+Relative density
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+2
+3
+4
+5
+6
+Rp at P= lO days (R)4
+3
+④
+R
+p
+R
+1.0
+3.0
+10.0
+Age (Gyr)Deep radius valley with Kepler short cadence
+9
+Figure 9. Same as Figure 6, but in 𝑅𝑝–𝑃–age space (left) and 𝑅𝑝–𝑃–[Fe/H] space (right).
+Table 6. Slope of the radius valley on the radius-period plane from various sources.
+Source
+𝑚 = d log 𝑅𝑝/d log 𝑃
+Stellar type
+Observations
+This work
+−0.11+0.02
+−0.02
+FGK
+Van Eylen et al. (2018)
+−0.09+0.02
+−0.04
+FGK
+Martinez et al. (2019)
+−0.11+0.02
+−0.02
+FGK
+MacDonald (2019)
+−0.319+0.088
+−0.116
+FGK
+Cloutier & Menou (2020)
+0.058+0.022
+−0.022
+M
+Van Eylen et al. (2021)
+−0.11+0.05
+−0.04
+M
+Petigura et al. (2022)
+−0.11+0.02
+−0.02
+FGKM
+Luque & Pallé (2022)
+−0.02+0.05
+−0.05
+M
+Source
+𝑚 = d log 𝑅𝑝/d log 𝑃
+Model
+Theory
+Owen & Wu (2017)
+−0.25 ≤ 𝑚 ≤ −0.16
+Photoevaporation
+Lopez & Rice (2018)
+-0.09
+Photoevaporation
+0.11
+Gas-poor formation
+Gupta & Schlichting (2019)
+−0.11
+Core-powered mass loss
+Rogers et al. (2021)
+−0.16
+Photoevaporation
+−0.11
+Core-powered mass loss
+studies (see again Table 6). The period-radius slope was first observed
+by Van Eylen et al. (2018), who used the SVM approach that we
+adopted here and who found 𝑚 = −0.09+0.02
+−0.04. A different approach
+was followed by Martinez et al. (2019), who divided their planetary
+sample into 10 bins with equal number of planets, determined the
+minimum radius in each bin, and fitted a linear relationship to obtain
+equation 4. These two approaches led to a consistent result, with
+𝑚 = −0.11±0.02. MacDonald (2019) adopted machine learning ap-
+proaches, and report 𝑚 = −0.319+0.088
+−0.116. The above studies all focus
+on samples of FGK stars, where various approaches to model the
+valley’s location appear to result in negative slopes with consistent
+magnitude, matching thermally-driven atmospheric loss models.
+For smaller and cooler (M type) stars, Cloutier & Menou (2020)
+found a positive slope (𝑚 = 0.058 ± 0.022) using a method similar
+to Martinez et al. (2019), suggesting for these stars the valley may be
+the result of gas-poor formation rather than being thermally driven.
+Van Eylen et al. (2021) used the SVM approach to measure the
+M dwarf valley and found a negative slope instead, of −0.11+0.05
+−0.04.
+Luque & Pallé (2022) used the gapfit package (Loyd et al. 2020)
+and found 𝑚 = −0.02±0.05. A recent study by Petigura et al. (2022)
+also included M type stars in addition to FGK stars, and they found
+𝑚 = −0.11 ± 0.02 for this sample. Our sample does not include M
+type stars but does span a mass range from about 0.6 to 1.4 𝑀★.
+To investigate whether the slope of 𝑚 changes with stellar mass
+within our sample, we split our planetary sample into two groups:
+𝑀★ ≥ 1𝑀⊙, and 𝑀★ < 1𝑀⊙. We determine the 𝑅𝑝 − 𝑃 relation
+separately for these two groups with the same methods as above.
+We find for 𝑀★ ≥ 1𝑀⊙, 𝑚 = −0.07+0.02
+−0.04 and 𝑐 = 0.35+0.03
+−0.02; for
+MNRAS 000, 1–19 (2022)
+
+Sub-Neptunes
+Super-Earths
+4
+Rp (R)
+2
+1
+1
+3
+10
+30
+0.3 1.0 3.010.0
+100
+P (days)
+Age (Gyr)Sub-Neptunes
+Super-Earths
+4
+Rp (R)
+2
+1
+0.5
+¥3
+1
+0
+10
+30
+-0.5
+100
+P (days)
+[Fe/H]10
+Ho & Van Eylen
+Figure 10. Radius valley location in terms of orbital period, stellar mass,
+and stellar age. The colour bar represents the age of the planetary host stars,
+and planes of different colours indicate the radius valley location for different
+stellar ages.
+𝑀★ < 1𝑀⊙, 𝑚 = −0.11+0.02
+−0.07 and 𝑐 = 0.35+0.05
+−0.02. These results are
+shown in Figure 11. The two values are in agreement within 1𝜎,
+suggesting that within our sample the radius valley location as a
+function of orbital period is inconsistent with the gas-poor formation
+scenario.
+We can also look at the slope of the valley as a function of incident
+flux (𝑆) rather than orbital period. By Kepler’s third law (as shown in
+equation 7), planets at longer orbital periods are located further away
+from the planet, thus the incident flux 𝑆 is lower for planets with
+larger star-planet distances as shown in equation 5. Hence, we expect
+for a thermally-driven planetary mass loss scenario, the radius valley
+location tends to larger planetary radii for higher 𝑆. We observe this
+positive relationship in this work, in agreement with other previous
+observations as shown in Table 7, and consistent with thermally-
+driven mass loss models which is also shown in radius-period space.
+4.2 𝑅𝑝-𝑀★ relation supports a thermally-driven mass loss
+model
+As presented in Section 3.4, we find that in two dimensions, 𝑚 =
+d log 𝑅𝑝/d log 𝑀★ = 0.23+0.09
+−0.08.
+A stellar mass dependence has been predicted by radius valley
+models. Both thermally driven mass-loss models predict a similar
+dependence of the valley on stellar mass. For example, Rogers et al.
+(2021) predicted 𝑚 = 0.29 and 𝑚 = 0.32 for photoevaporation
+(Owen & Wu 2017) and core-powered mass loss models (Gupta &
+Schlichting 2019, 2020) respectively. Our results are consistent with
+both sets of models within 1𝜎.
+A stellar mass dependence was observed by Berger et al. (2020),
+who find 𝑚 = 0.26+0.21
+−0.16 by fitting the minima of the 2-dimensional
+KDE in 𝑅𝑝 − 𝑀★ space. A recent study by Petigura et al. (2022),
+similarly following a binning approach and incorporating data from
+Data Release 2 (DR2) of the California-Kepler Survey (CKS) for
+cooler stars, estimated 𝑚 = 0.18+0.08
+−0.07. It is therefore reassuring to
+see that despite the different method adopted here, the slope derived
+in this work is consistent with both of these studies within 1𝜎. For
+lower mass stars, Luque & Pallé (2022) found 𝑚 = 0.08 ± 0.12; this
+may be inconsistent with our results at 1𝜎, however the stellar mass
+range they studied is significantly lower than that in our sample with
+no overlaps. The results are summarised in Table 8.
+When extending our analysis to 3 dimensions as a function of
+𝑃 and 𝑀★, we obtain 𝐴 = �𝜕 log 𝑅𝑝/𝜕 log 𝑃�
+𝑀★ = −0.09+0.02
+−0.03,
+𝐵 = �𝜕 log 𝑅𝑝/𝜕 log 𝑀★
+�
+𝑃 = 0.21+0.06
+−0.08 from determining the radius
+valley location in the 𝑅𝑝–𝑃–𝑀★ space. Note that this is different to
+the total derivative d log 𝑅𝑝/d log 𝑀★ in two dimensions (shown in
+Table 3). Based on the models of photoevaporation (Owen & Wu
+2017; Owen & Adams 2019; Mordasini 2020) and core-powered
+mass loss (Gupta & Schlichting 2019, 2020), Van Eylen et al. (2021)
+predicted 𝐵 = 0.19 for a photoevaporation model, and 𝐵 = 0.33
+for a core-powered mass-loss model. Our resulting posterior distri-
+bution of 𝐴 and 𝐵 determined from the bootstrapping presented in
+Section 3.4, as shown in Figure 12, is consistent with both the pho-
+toevaporation and core-powered mass loss cases at 2𝜎, hence we
+are unable to distinguish between the two models in this particular
+parameter space.
+Rogers et al. (2021) proposed an analysis of the radius valley in
+𝑅𝑝–𝑆–𝑀★ space that could distinguish between the two different
+thermally-driven mass-loss mechanisms. Using theoretical models,
+they predicted the radius valley scales as a function of 𝑆 and 𝑀★
+as equation 11, with 𝐴 = �𝜕 log 𝑅𝑝/𝜕 log 𝑆�
+𝑀★ ≃ 0.12 and 𝐵 =
+�𝜕 log 𝑅𝑝/𝜕 log 𝑀★
+�
+𝑆 ≃ −0.17 for a photoevaporation model, and
+𝐴 ≃ 0.08 and 𝐵 ≃ 0.00 for a core-powered mass loss model. Again,
+we plot the posterior distributions of 𝐴 and 𝐵 as shown in Figure 13,
+and observe that our results are consistent with the core-powered
+mass loss case well within 1𝜎. For the photoevaporation scenario,
+our values overlap with the theoretical predictions at the edge of the
+2𝜎 confidence interval. Rogers et al. (2021) also measured the planet
+density of the California-Kepler Survey (CKS, Fulton & Petigura
+2018), and the Gaia-Kepler Survey (GKS, Berger et al. 2020), in
+𝑅𝑝–𝑆–𝑀★ space. They found for the CKS data, 𝐴 = 0.13+0.03
+−0.05, 𝐵 =
+−0.21+0.33
+−0.39, and for the GKS data, 𝐴 = 0.10+0.03
+−0.02, 𝐵 = −0.03+0.10
+−0.12.
+Our results are in agreement with both the CKS and GKS values,
+and our measurements have smaller uncertainties.
+There are some caveats to this comparison between our observation
+results and theoretical models. Firstly, the thermally-driven mass loss
+models predict the slope of the bottom of the valley (Van Eylen
+et al. 2018; Rogers et al. 2021), whereas our SVM finds the slope
+for the middle of the radius valley. Some studies have suggested
+a different planet size dependence with orbital period for super-
+Earths and sub-Neptunes (e.g. Petigura et al. 2022), hence these two
+slopes may not be equal. Since the radius valley is not completely
+empty, the bottom of the radius valley is not clearly defined, and
+there would be challenges locating and fitting the bottom of the
+radius valley. As a result, our observed values may not be fully
+comparable with theoretical model values. Furthermore, the method
+of extracting the radius valley is prone to transit biases, which we
+do not correct for in this work. Rogers et al. (2021) showed that
+even when modelling synthetic transit surveys based on evolving
+planets with theoretical models, the resulting posteriors may not be
+fully consistent with the theoretically predicted slope. Further work,
+such as generating synthetic surveys from both photoevaporation and
+core-powered mass loss models based on conditions similar to that
+of our sample in a method similar to that performed in Rogers et al.
+(2021), and fitting the valley with the same method as in this work, or
+analysing more planets around stars in a larger mass range, is required
+to compare our observations to theoretical models in a homogeneous
+way.
+MNRAS 000, 1–19 (2022)
+
+Sub-Neptunes
+Super-Earths
+Valley at O.1 Gyr
+Valley at 1 Gyr
+10
+Valley at 10 Gyr
+4
+Age (Gyr)
+3
+2
+d
+R
+1
+1
+1.4
+1
+¥3 
+0.3
+1.0
+10
+30
+0.7
+100
+P (days)
+M* (Mo)Deep radius valley with Kepler short cadence
+11
+Figure 11. Radius valley position for planets with host star mass 𝑀★ < 1𝑀⊙ (left), and 𝑀★ ≥ 1𝑀⊙ (right). For 𝑀★ < 1𝑀⊙, 𝑚 = −0.11+0.02
+−0.07 and 𝑐 = 0.35+0.05
+−0.02;
+for 𝑀★ ≥ 1𝑀⊙, 𝑚 = −0.07+0.02
+−0.04 and 𝑐 = 0.35+0.03
+−0.02. The green and blue points show planets above and below the radius valley respectively. The grey shaded
+region represents the ±1𝜎 uncertainty in the radius valley position determined with bootstrapping.
+Table 7. Same as Table 6, but for the radius valley slope on the radius-incident-flux plane.
+Source
+𝑚 = d log 𝑅𝑝/d log 𝑆
+Stellar type
+Observations
+This work
+0.07+0.02
+−0.01
+FGK
+Martinez et al. (2019)
+0.12 ± 0.02
+FGK
+Cloutier & Menou (2020)
+−0.060 ± 0.025
+M
+Petigura et al. (2022)
+0.06 ± 0.01
+FGKM
+Luque & Pallé (2022)
+0.02 ± 0.02
+M
+Table 8. Same as Table 6, but for the radius valley slope on the radius-stellar-mass plane.
+Source
+𝑚 = d log 𝑅𝑝/d log 𝑀★
+Stellar type
+Observations
+This work
+0.23+0.09
+−0.08
+FGK
+Berger et al. (2020)
+0.26+0.21
+−0.16
+FGKM
+Petigura et al. (2022)
+0.18+0.08
+−0.07
+FGKM
+Luque & Pallé (2022)
+0.08+0.12
+−0.12
+M
+Source
+𝑚 = d log 𝑅𝑝/d log 𝑀★
+Model
+Theory
+Gupta & Schlichting (2020)
+0.33
+Core-powered mass loss
+Rogers et al. (2021)
+0.29
+Photoevaporation
+0.32
+Core-powered mass loss
+4.3 Deeper radius valley suggests a homogeneous initial
+planetary core composition
+We now turn to the depth of the radius valley. Using the previously
+defined depth metric (𝐸, equations 9 and 10), we find a valley depth
+of 𝐸avg = 2.98+0.60
+−0.47 (see Section 3.3). We can compare this depth
+to the valley observed by F18. Shifting the planets along the slope
+calculated in Section 3.2, and applying the same metric to their
+filtered sample of 907 planets, we calculate 𝐸SN = 1.99+0.26
+−0.23, 𝐸SE =
+2.28+0.31
+−0.27, giving 𝐸avg = 2.14+0.26
+−0.21 for that sample. For V18, we
+shift the planets according to the slope obtained in their study, i.e.
+𝑚 = −0.09+0.02
+−0.04, and we find 𝐸SN = 7.11+4.70
+−2.49, 𝐸SE = 4.75+3.42
+−1.70,
+giving 𝐸avg = 6.05+3.87
+−2.14. These values imply that compared to F18,
+we observe a deeper radius valley. On the other hand, the radius
+valley appears less deep than observed by V18 for a smaller sample.
+This finding is visualised in Figure 14, which shows the adjusted
+histograms of the sample studied here next to the F18 and the V18
+samples.
+To investigate the reason for observing a deeper valley than F18, we
+compare the 211 planets common in both our sample and the filtered
+sample of F18. To investigate the role of transit fitting, we convert
+MNRAS 000, 1–19 (2022)
+
+M*<1Mo
+4
+TO
+TO
+0
+T
+3
+DT
+.4
+④
+TOI
+R
+2
+R
+五
+中
+中中
+T
+1
+下中
+3
+10
+30
+100
+P (days)M*≥1Mo
+4
+TOITOI
+O
+TOI
+T
+TOI
+Φ
+3
+0
+TO
+D
+中
+(田y)
+TOD.
+2
+R
+!!
+1
+3
+10
+30
+100
+P (days)12
+Ho & Van Eylen
+Figure 12. Posterior distributions of the radius valley location dependence
+with respect to orbital period at constant stellar mass �𝜕 log 𝑅𝑝/𝜕 log 𝑃�
+𝑀★,
+and stellar mass at constant orbital period �𝜕 log 𝑅𝑝/𝜕 log 𝑀★
+�
+𝑃. The dark
+and light-coloured shades represent the 1𝜎 and 2𝜎 uncertainties respectively.
+The theoretical models of photoevaporation and core-powered mass loss are
+taken from Van Eylen et al. (2021).
+Figure 13. Same as Figure 12, but for �𝜕 log 𝑅𝑝/𝜕 log 𝑆�
+𝑀★
+and
+�𝜕 log 𝑅𝑝/𝜕 log 𝑀★
+�
+𝑆.
+all our 𝑅𝑝/𝑅★ into 𝑅𝑝 using 𝑅★ from F18 (even when V18 values
+are available). The results are shown in Figure 15, which compares
+the same planets with the same stellar parameters but different transit
+fitting. We observe that in this case, the 𝑅𝑝 of 56 (27%) and 24 (11%)
+planets change by > 2𝜎 and > 3𝜎 respectively, compared to the
+values reported in F18. We find for this common planetary sample,
+for our planetary parameters, 𝐸SN = 3.63+1.02
+−0.70, 𝐸SE = 2.95+0.86
+−0.63,
+giving 𝐸avg = 3.29+0.91
+−0.59, whereas for parameters from F18, 𝐸SN =
+2.40+0.59
+−0.47, 𝐸SE = 1.93+0.49
+−0.39, giving 𝐸avg = 2.18+0.49
+−0.42. These findings
+suggest that our updated transit fittings are directly responsible for
+deepening (although not fully emptying) the radius valley.
+A deeper radius valley is associated with a more homogeneous
+planet core composition. For example, in photoevaporation models
+the radius valley position is dependent on the mass of the planet core
+(𝑀𝑐), and the density of a 1𝑀⊕ core of a particular core composition
+(𝜌𝑀⊕), as
+𝑅valley ∝ 𝜌−1/3
+𝑀⊕ 𝑀1/4
+𝑐
+.
+(18)
+Hence, if 𝑅valley is known, and the planets’ mean masses are known,
+the planetary core compositions could be deduced (Owen & Wu
+2017).
+Using the above relation, if the planetary cores were icy at forma-
+tion, the radius valley would be located at a higher planetary radius
+than if the cores were rocky/terrestrial at formation. Hence, if the
+planetary cores are of mixed composition, a superposition of the two
+models would be predicted, and we would expect the radius valley
+to be smeared and less distinct, as each type of planet would have
+its own ‘radius valley’ at a different location (Owen & Wu 2017).
+Our deep radius valley found in this work implies the opposite case,
+where the planetary cores are more similar in composition. In this
+scenario, planets inside the valley may have a different (e.g. icy)
+composition.
+Owen & Wu (2017) compared their models to observations, and
+found that the planet compositions are more likely to be Earth-like
+(i.e. rocky), but that the apparent shallowness of the valley suggested
+a wide distribution of iron fractions ( 𝑓Fe) in their cores, as planets
+with a single value iron fraction ( 𝑓Fe = 0.5) produces a deeper
+valley compared to planets with a uniform distribution ( 𝑓Fe ∈ [0, 1]).
+Comparing our finding of a deeper valley to models in Owen & Wu
+(2017) would indicate that the planet compositions are more likely
+to have similar iron fractions with a narrower spread.
+Similarly, in the core-powered mass loss model, the location of the
+radius valley scales as
+𝑅valley ∝ 𝜌−4/9
+𝑐
+(19)
+where 𝜌𝑐 is the planet core density (Gupta & Schlichting 2019).
+The same reasoning as the photoevaporation case then applies: given
+the larger 𝜌𝑐 for icy cores, planets with homogeneous icy cores
+will produce a radius valley at a larger planetary radii compared
+to rocky/terrestrial cores, implying that the radius valley would be
+smeared if planetary cores are of inhomogeneous compositions. Our
+deep radius valley supports the opposite case, i.e., a similar planetary
+core composition.
+Figure 16 shows the stellar parameter distributions for the planet
+host stars in the three planet samples, and the mean and median
+values are listed in Table 9. We notice a similar stellar parameter
+range between this work and F18, however the stars in V18 are
+brighter, have a larger mean radius and mass, and higher effective
+temperature. This is likely due to V18 selecting stars which display
+strong asteroseismic signals, which usually are brighter and larger
+stars. This observation may indicate that the radius valley of such
+stars are emptier, however the details are left for future studies.
+Despite our new results revealing that the radius valley deepens by
+refitting planets with 1-minute short cadence light curves, it is still
+uncertain whether the difference between results from this work and
+F18 is solely due to the cadence in transit data used, as different meth-
+ods are used in the transit fitting process. Mullally et al. (2015) fitted
+planets using the method described in Rowe et al. (2014), which first
+fits a multi-planet transit model to the light curves, with fixed limb
+darkening parameters from Claret & Bloemen (2011), and subse-
+quently fitting for each planet in a system independently by removing
+photometric contributions of other planets based on the parameters
+from the multi-planet fit. In our work, we fit planets in multi-planet
+systems simultaneously, such that each system shares the same stellar
+parameters including limb-darkening parameters and stellar density.
+MNRAS 000, 1–19 (2022)
+
+0.4
+0.3
+0.2
+0.1
+Photoevaporation
+Core-powered mass loss
+0
+This Work
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+(alog Rp/alog P)m,Photoevaporation
+0.2
+Core-powered mass loss
+This Work
+0.1
+0
+-0.1
+-0.2
+0
+0.05
+0.1
+0.15
+0.2
+alog Rp/alog S)mDeep radius valley with Kepler short cadence
+13
+Figure 14. Histogram of planet radii, adjusted to 𝑃 = 10 days for the planetary population in this work (left, identical to Figure 4), F18 (centre), and V18
+(right). Planets in this work and F18 are shifted according to the slope defined in Section 3.2 with the SVM (i.e. −0.11+0.02
+−0.02), whereas planets in V18 are shifted
+according to the slope found in V18 (i.e. 𝑚 = −0.09+0.02
+−0.04). The 𝐸 metrics defining the average peak-to-valley ratio are 2.98+0.60
+−0.47, 2.14+0.26
+−0.21, and 6.05+3.87
+−2.14
+respectively.
+Figure 15. Left: Radius-period plot of the 211 planets common to this work (blue) and F18 (red), using the same stellar radii to calculate planetary radii. Right:
+Same as Figure 14, but the planetary population is filtered to the 211 common planets in both samples. Planets in both samples are adjusted to equivalent radii
+at 𝑃 = 10 days according to the slope calculated in Section 3.2 with the SVM. The red and blue histograms are produced with the parameters obtained from F18
+and this work respectively. The histograms and fitting with the Gaussian Mixture Model show that the observed valley is deeper in this work.
+Mullally et al. (2015) assumed a circular orbit when performing the
+transit fits. On the contrary, we leave orbital eccentricity 𝑒 as a free
+parameter, and place a prior on 𝑒 based on the expected distribution
+from Van Eylen et al. (2019) and the stellar density 𝜌★ from F18.
+However, most of the planets in our sample have near-circular orbits,
+with over 85% of planets having 𝑒 < 0.1. Therefore planetary orbital
+eccentricity is not sufficient to explain the difference between the
+two results. The possible presence of TTVs also do not contribute
+to the discrepancy as planets with known TTVs are excluded in our
+like-for-like planet comparisons. In fact, when fitting transits using
+identical methods, precisions in 𝑅𝑝/𝑅★ obtained from fitting transit
+light curves of shorter cadences has been found to substantially im-
+prove, compared to 30-minute cadence light curves (Camero, Ho &
+Van Eylen, in prep.). We therefore expect the photometry cadence to
+contribute significantly to the difference in the views of the radius val-
+ley. Further work, such as refitting the long cadence data of the same
+planet population with identical transit fitting methods, is needed to
+further investigate the effect of light curve cadence on planet param-
+eter estimates and the radius valley. We leave such considerations for
+future studies.
+4.4 Radius valley relation with stellar age consistent with
+core-powered mass loss model
+In Section 3.5, we present a positive relationship between the radius
+valley location and stellar age. Photoevaporation is predicted to occur
+in the first 100 Myr of the planet’s formation (Owen & Wu 2017), well
+before observations are able to detect the evolution signals, whereas
+MNRAS 000, 1–19 (2022)
+
+70
+This work
+60
+S
+Number of planet
+50
+40
+30
+20
+10
+0
+2
+¥456
+1
+3 F18
+80
+70
+Number of planets
+60
+50
+40
+30
+20
+10
+0
+2
+3
+456
+1
+Rp at P= 10 days (R)V18
+14-
+12
+S
+Number of planets
+10
+8
+6
+4
+2
+0
+1
+2
+3
+456
+Rp at P= 10 days (R)F18
+This work
+4
+ICE
+士
+亞
+HDO
+HOH
+3
+TO
+T
+H
+市
+币
+西
+中
+中
+薄
+H
+生奇
+CDI
+F
+F
+由HO
+由
+2
+10
+中
+Φ
+T0
+西
+中
+1
+T
+0
+TEHO
+中
+HH
+TOIT
+T0
+TTI
+中
+中
+Ho0
+HO由
+HO
+HO
+中
+中
+中中
+He
+I
+1
+3
+10
+30
+L
+100
+P (days)35
+F18
+This work
+30
+S
+Number of planets
+25
+20
+15
+10
+5
+0
+1
+2
+3
+4
+5
+6
+Rp at P= 10 days (R)14
+Ho & Van Eylen
+Figure 16. Distribution of host stars parameters in our work, compared with F18 and V18. The properties shown here, from top left to bottom right, are stellar
+radius 𝑅★, mass 𝑀★, Kepler magnitude Ksmag, and effective temperature 𝑇eff. For systems with multiple transiting planets, stars are counted multiple times.
+Table 9. Average values of stellar properties of the host stars in the planetary sample used in this work, compared with F18 and V18. ¯𝑥 and ˜𝑥 represent the mean
+and median values of the parameter 𝑥 respectively.
+Sample
+¯𝑅★ (𝑅⊙)
+˜𝑅★ (𝑅⊙)
+¯
+𝑀★ (𝑀⊙)
+˜
+𝑀★ (𝑀⊙)
+¯
+Ksmag
+˜
+Ksmag
+¯
+𝑇eff (K)
+˜
+𝑇eff (K)
+This work
+1.08
+0.99
+0.98
+0.95
+12.27
+12.31
+5587
+5630
+F18
+1.23
+1.19
+1.04
+1.03
+11.72
+11.97
+5788
+5860
+V18
+1.50
+1.45
+1.14
+1.14
+11.49
+11.57
+5980
+5952
+core-powered mass loss occurs throughout the main-sequence life-
+time of the stars, on Gyr timescales (Ginzburg et al. 2016; Gupta &
+Schlichting 2019, 2020). Hence, in the photoevaporation case, the
+radius valley is expected to be located at a constant radius. On the
+other hand, in the core-powered mass loss case, the radius valley
+shifts to higher planet radii for older systems, as the atmospheres of
+planets with more massive cores are stripped off later in the evolution
+process than their less massive counterparts (e.g. David et al. 2021;
+Rogers & Owen 2021).
+Our results reveal a weak positive radius valley dependence on
+the stellar age, which is consistent with the core-powered mass loss
+scenario, as is the observed radius valley dependence on stellar mass
+as discussed in Section 4.2. However, a small age dependence does
+not preclude photoevaporation, since even in this scenario a subset of
+planets may still lose their atmospheres and evolve at Gyr timescales
+(David et al. 2021; Rogers et al. 2021), and we are unable to observe
+stars younger than 100 Myr and hence cannot rule out the possibility
+of a dominant photoevaporation effect on planets at the early stages
+of the stars’ lifetime. Also, stellar age measurements are highly un-
+certain; the mean percentage uncertainty in stellar age for our sample
+is 54%, hence there is also a probability that some stars are younger
+than observed.
+Table 10 lists the 50 planets located inside the radius valley in our
+sample. To do so, we here defined the new radius valley region as
+the area bounded by the two lines passing through the supporting
+vectors in the 4D SVM model in Section 3.5, given by equation 15
+with 𝐴 = −0.096, 𝐵 = 0.231, 𝐶 = 0.033, 𝐷lower = 0.272 for the
+lower line, and 𝐷upper = 0.405 for the upper line. These planets
+are potentially interesting for future characterisation study as their
+MNRAS 000, 1–19 (2022)
+
+1.6
+F18
+1.4
+V18
+This work
+1.2
+Density
+1.0
+0.8
+D
+0.6
+0.4
+0.2
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+R(Ro)F18
+3.0
+V18
+This work
+2.5
+Density
+2.0
+1.5
+D
+1.0
+0.5
+0.0
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+M* (Mo)0.7.
+F18
+V18
+0.6
+This work
+0.5
+Density
+0.4
+0.3
+0.2
+0.1
+0.0
+8
+10
+12
+14
+16
+Ksmag1e-4
+16
+F18
+14
+V18
+This work
+12
+Density
+10
+8
+6
+4
+2
+0
+4500
+5000
+5500
+6000
+6500
+7000
+Teff (
+(K)Deep radius valley with Kepler short cadence
+15
+Figure 17. Metallicity and mass of the host stars used in this work. The grey
+dotted line [Fe/H] = 0.06 shows the cut-off between stars of low and high
+metallicity as defined in this work. We note a lack of stars with low metallicity
+and large stellar mass.
+atmospheres and interiors may provide additional insights regarding
+formation and evolution mechanisms.
+4.5 Radius valley depth varies with stellar metallicity
+In Section 3.6, we show a higher average 𝐸 value (i.e. a deeper radius
+valley) for planets around metal-poor stars. This seems to contradict
+the suggestion that the radius valley is deeper for planets around
+metal-rich stars (Owen & Murray-Clay 2018). However, we note
+from Figure 17, that in our sample, the metal-rich host stars span a
+wider range of stellar masses, due to lack of metal-poor stars with
+large radii. As from Section 3.4 we observe that the radius valley
+depends on stellar mass as well, the superposition of the radius
+valley for different stellar masses potentially smears the gap, making
+the radius valley appear shallower.
+The degeneracy between stellar mass and metallicity is not fully
+resolved, hence we are unable to determine the sole effect of stellar
+metallicity on the radius valley in this work. We therefore consider
+the results related to metallicity to be inconclusive and in need of
+further future study.
+5 CONCLUSION
+In summary, we performed transit light curve fitting on 431 plan-
+ets using Kepler 1-minute short cadence data, the vast majority of
+which have not been previously analysed homogeneously using short
+cadence observations. In this paper, we presented their revised plan-
+etary parameters, which in some cases differ substantially from those
+previously reported. These differences are unrelated to stellar param-
+eters but may be related to the details of the transit fitting approach
+or the shorter observing cadence, the effects of which should be
+disentangled in future studies.
+By statistically analysing the small close-in planets in our sam-
+ple, we observed a radius valley which is deeper than that reported
+in several other studies, although not entirely empty. The valley’s
+depth likely implies a homogeneous initial planetary core composi-
+tion where the planets are similar in composition at formation, and
+likely to have similar iron fractions. We provide a table of those plan-
+ets that appear to be inside the valley, as they may warrant further
+study.
+The radius valley has a strong dependence on planetary orbital pe-
+riod and the mass of the host star. It also displays a weak dependence
+on the stellar age. We compared several possible radius valley models
+using support vector machines. We determined that the radius valley
+can best be described in four dimensions using the formula
+𝑅𝑝,valley ∝ 𝑃𝐴𝑀𝐵
+★ (age)𝐶
+(20)
+with 𝐴 = −0.096+0.023
+−0.027, 𝐵 = 0.231+0.053
+−0.064, and 𝐶 = 0.033+0.017
+−0.025.
+Comparing our radius valley dependencies with theoretical mod-
+els, we found that in 𝑅𝑝–𝑆–𝑀★ space, our posterior distributions
+are most consistent with core-powered mass loss, where they agree
+within less than 1𝜎. The models are also consistent with photoevap-
+oration scenarios at ≈ 2𝜎. We did not find a significant dependence
+of the radius valley on stellar metallicity.
+With the Transiting Exoplanet Survey Satellite (TESS, e.g. Ricker
+et al. 2015) now in its extended mission, and the upcoming launch of
+the PLAnetary Transits and Oscillations of stars (PLATO) mission
+(e.g. Rauer et al. 2014), such future planetary studies could drastically
+increase the number of planets with radii measurements and hence
+provide an even more detailed view of the radius valley. This work
+highlights the impact of careful transit fitting using short, 1-minute
+cadence observations to obtain precise planetary radii. This will
+likely be of key importance to derive precise planetary radii using
+transit observations from ongoing and future missions, which will
+ultimately allow us to better understand the formation and evolution
+of small close-in planets.
+ACKNOWLEDGEMENTS
+C.S.K.H. would like to thank the Science and Technology Facilities
+Council (STFC) for funding support through a PhD studentship. We
+would like to thank Erik Petigura, James Rogers, and James Owen
+for insightful discussions. We also thank the anonymous reviewer for
+taking their time to review the paper, and for their valuable sugges-
+tions which have improved the manuscript.
+DATA AVAILABILITY
+The Kepler 1-minute short cadence light curves are available for
+download on the NASA Mikulski Archive for Space Telescopes
+(MAST) database1. The parameter estimates from HMC posteriors
+are provided in the Appendix tables.
+REFERENCES
+Astropy Collaboration et al., 2013, A&A, 558, A33
+Astropy Collaboration et al., 2018, AJ, 156, 123
+Berger T. A., Huber D., Gaidos E., van Saders J. L., Weiss L. M., 2020, AJ,
+160, 108
+Bourque M., et al., 2021, The Exoplanet Characterization Toolkit (ExoCTK),
+doi:10.5281/zenodo.4556063, https://doi.org/10.5281/zenodo.
+4556063
+Claret A., Bloemen S., 2011, A&A, 529, A75
+Cloutier R., Menou K., 2020, AJ, 159, 211
+David T. J., et al., 2021, AJ, 161, 265
+1 https://archive.stsci.edu
+MNRAS 000, 1–19 (2022)
+
+0.4
+0.2
+Fe/H]
+0.0
+-0.2
+-0.4
+-0.6
+0.6
+0.8
+1.0
+1.2
+1.4
+M* (Mo)16
+Ho & Van Eylen
+Table 10. List of planets inside the radius valley as defined by this work. The coordinates (RA and Dec) are taken from Kepler Q1-17 Data Release 25 catalogue
+(Thompson et al. 2018), except for K02533.03, where data is taken from Kepler Q1-16 catalogue (Mullally et al. 2015).
+KOI
+Kepler name
+𝑃 (days)
+𝑡0 (BJD-2454833)
+𝑅𝑝/𝑅★
+𝑅𝑝 (𝑅⊕)
+RA (deg)
+Dec (deg)
+K00049.01
+Kepler-461 b
+8.313784 ± 0.000015
+175.9915 ± 0.0008
+0.0287 ± 0.0010
+4.03+0.17
+−0.17
+292.24902
+46.164822
+K00070.03
+Kepler-20 A d
+77.611598 ± 0.000019
+164.7274 ± 0.0005
+0.0263 ± 0.0003
+2.52+0.07
+−0.07
+287.698
+42.338718
+K00092.01
+KOI-92.01
+65.704594 ± 0.000018
+137.4419 ± 0.0005
+0.0263 ± 0.0006
+3.02+0.11
+−0.11
+283.37482
+43.788219
+K00094.02
+Kepler-89 A c
+10.423684 ± 0.000005
+138.0092 ± 0.0005
+0.0266 ± 0.0006
+3.97+0.13
+−0.13
+297.33307
+41.891121
+K00105.01
+Kepler-463 b
+8.981015 ± 0.000002
+136.6493 ± 0.0005
+0.0313 ± 0.0004
+3.65+0.11
+−0.10
+298.93704
+44.85791
+K00107.01
+Kepler-464 b
+7.256964 ± 0.000011
+134.0232 ± 0.0007
+0.0198 ± 0.0004
+3.46+0.11
+−0.11
+294.83517
+48.982361
+K00108.01
+Kepler-103 b
+15.965333 ± 0.000013
+142.1780 ± 0.0006
+0.0221 ± 0.0002
+3.49+0.04
+−0.04
+288.98456
+40.064529
+K00111.03
+Kepler-104 A d
+51.755294 ± 0.000018
+271.0894 ± 0.0005
+0.0232 ± 0.0003
+2.66+0.07
+−0.07
+287.60461
+42.166779
+K00122.01
+Kepler-95 b
+11.523073 ± 0.000005
+131.9686 ± 0.0004
+0.0205 ± 0.0002
+3.24+0.10
+−0.10
+284.48245
+44.398041
+K00157.02
+Kepler-11 d
+22.687159 ± 0.000014
+148.4549 ± 0.0006
+0.0282 ± 0.0006
+3.34+0.11
+−0.10
+297.11511
+41.909142
+K00174.01
+Kepler-482 b
+56.354185 ± 0.000019
+144.8366 ± 0.0007
+0.0339 ± 0.0013
+2.92+0.14
+−0.14
+296.82291
+48.107552
+K00238.01
+Kepler-123 b
+17.232309 ± 0.000018
+135.0923 ± 0.0008
+0.0223 ± 0.0006
+3.21+0.13
+−0.13
+296.99863
+42.78196
+K00285.01
+Kepler-92 b
+13.748833 ± 0.000015
+179.2788 ± 0.0007
+0.0198 ± 0.0003
+3.71+0.06
+−0.06
+289.08606
+41.562958
+K00317.01
+Kepler-521 b
+22.208119 ± 0.000015
+206.3592 ± 0.0006
+0.0207 ± 0.0003
+3.69+0.11
+−0.11
+298.81638
+43.998039
+K00351.03
+Kepler-90 d
+59.737034 ± 0.000020
+158.9612 ± 0.0009
+0.0217 ± 0.0006
+2.80+0.11
+−0.11
+284.4335
+49.305161
+K00351.04
+Kepler-90 e
+91.940461 ± 0.000020
+134.2987 ± 0.0010
+0.0198 ± 0.0007
+2.56+0.11
+−0.11
+284.4335
+49.305161
+K00386.01
+Kepler-146 b
+31.158789 ± 0.000019
+173.9038 ± 0.0009
+0.0297 ± 0.0007
+3.30+0.12
+−0.12
+294.11075
+38.710232
+K00386.02
+Kepler-146 c
+76.732517 ± 0.000020
+200.6716 ± 0.0010
+0.0258 ± 0.0013
+2.87+0.16
+−0.16
+294.11075
+38.710232
+K00408.01
+Kepler-150 c
+7.381981 ± 0.000007
+173.0729 ± 0.0008
+0.0355 ± 0.0005
+3.34+0.12
+−0.12
+288.2341
+40.520901
+K00416.01
+Kepler-152 b
+18.207957 ± 0.000015
+185.8427 ± 0.0006
+0.0383 ± 0.0006
+3.12+0.10
+−0.09
+286.86548
+41.989079
+K00435.05
+Kepler-154 c
+62.302788 ± 0.000020
+179.0982 ± 0.0010
+0.0266 ± 0.0010
+3.04+0.15
+−0.15
+289.78052
+49.89653
+K00509.02
+Kepler-171 c
+11.463477 ± 0.000013
+137.3859 ± 0.0008
+0.0328 ± 0.0008
+3.30+0.15
+−0.14
+296.77191
+41.755539
+K00510.04
+Kepler-172 e
+35.118523 ± 0.000020
+152.1229 ± 0.0010
+0.0248 ± 0.0021
+2.82+0.26
+−0.26
+283.36841
+41.821861
+K00555.02
+Kepler-598 c
+86.494779 ± 0.000020
+181.8831 ± 0.0009
+0.0272 ± 0.0008
+2.51+0.10
+−0.10
+293.12341
+40.934769
+K00665.01
+Kepler-207 d
+5.868083 ± 0.000009
+170.3244 ± 0.0009
+0.0202 ± 0.0004
+3.69+0.11
+−0.11
+290.03052
+42.16605
+K00707.02
+Kepler-33 f
+41.028059 ± 0.000019
+172.5788 ± 0.0009
+0.0207 ± 0.0004
+3.64+0.13
+−0.13
+289.07755
+46.005219
+K00707.03
+Kepler-33 e
+31.784774 ± 0.000020
+135.8721 ± 0.0009
+0.0188 ± 0.0005
+3.30+0.12
+−0.12
+289.07755
+46.005219
+K00708.01
+Kepler-216 c
+17.406653 ± 0.000017
+171.0063 ± 0.0008
+0.0230 ± 0.0006
+3.88+0.14
+−0.14
+293.72806
+46.12915
+K00711.01
+Kepler-218 c
+44.699505 ± 0.000020
+174.8232 ± 0.0008
+0.0272 ± 0.0007
+3.07+0.11
+−0.11
+295.41281
+46.266472
+K00800.02
+Kepler-234 c
+7.212030 ± 0.000017
+172.8172 ± 0.0009
+0.0283 ± 0.0012
+3.61+0.28
+−0.26
+291.65353
+38.494659
+K00834.02
+Kepler-238 d
+13.233546 ± 0.000019
+140.3211 ± 0.0010
+0.0197 ± 0.0005
+3.37+0.19
+−0.18
+287.89713
+40.637821
+K00834.05
+Kepler-238 f
+50.447315 ± 0.000020
+178.4929 ± 0.0010
+0.0203 ± 0.0009
+3.48+0.23
+−0.22
+287.89713
+40.637821
+K00881.01
+Kepler-712 b
+21.022471 ± 0.000017
+207.6765 ± 0.0007
+0.0391 ± 0.0008
+3.14+0.18
+−0.17
+294.90973
+42.935261
+K00907.04
+Kepler-251 e
+99.640965 ± 0.000021
+198.6899 ± 0.0009
+0.0320 ± 0.0017
+2.82+0.19
+−0.19
+296.56622
+44.105862
+K00921.02
+Kepler-253 d
+18.119898 ± 0.000018
+182.6165 ± 0.0008
+0.0346 ± 0.0011
+3.01+0.16
+−0.15
+291.84198
+44.858089
+K00934.01
+Kepler-254 b
+5.826654 ± 0.000006
+173.0110 ± 0.0009
+0.0368 ± 0.0011
+3.62+0.26
+−0.24
+288.1647
+45.816509
+K00941.01
+Kepler-257 c
+6.581482 ± 0.000009
+174.7876 ± 0.0009
+0.0429 ± 0.0009
+3.75+0.15
+−0.15
+297.31598
+46.023258
+K00954.02
+Kepler-259 c
+36.924954 ± 0.000019
+174.2245 ± 0.0010
+0.0291 ± 0.0017
+2.85+0.19
+−0.19
+288.21194
+46.615002
+K01001.01
+Kepler-264 b
+40.806846 ± 0.000020
+155.7126 ± 0.0009
+0.0159 ± 0.0003
+3.39+0.12
+−0.12
+292.04462
+37.37624
+K01198.01
+Kepler-275 c
+16.088329 ± 0.000018
+139.4928 ± 0.0010
+0.0232 ± 0.0011
+3.55+0.31
+−0.29
+292.47971
+38.514919
+K01215.02
+Kepler-277 c
+33.006310 ± 0.000020
+145.3914 ± 0.0010
+0.0161 ± 0.0008
+3.04+0.17
+−0.17
+286.58316
+39.077202
+K01270.01
+Kepler-57 b
+5.729326 ± 0.000005
+138.5598 ± 0.0009
+0.0356 ± 0.0024
+3.16+0.24
+−0.24
+293.6413
+44.65704
+K01486.02
+Kepler-302 b
+30.183689 ± 0.000018
+146.6480 ± 0.0009
+0.0286 ± 0.0011
+3.27+0.21
+−0.20
+294.31699
+43.629341
+K01563.04
+Kepler-305 d
+16.738655 ± 0.000019
+359.4178 ± 0.0009
+0.0338 ± 0.0022
+2.84+0.22
+−0.22
+299.22433
+40.343182
+K01598.01
+Kepler-310 c
+56.476167 ± 0.000019
+143.8052 ± 0.0008
+0.0301 ± 0.0007
+2.75+0.10
+−0.10
+288.83936
+46.98674
+K02051.01
+Kepler-355 c
+25.762459 ± 0.000020
+147.7050 ± 0.0009
+0.0230 ± 0.0010
+2.95+0.18
+−0.17
+285.79947
+42.811779
+K02390.01
+Kepler-1219 b
+16.104672 ± 0.000020
+135.0156 ± 0.0010
+0.0120 ± 0.0011
+3.51+0.46
+−0.46
+297.21579
+47.378521
+K02414.02
+Kepler-384 c
+45.348527 ± 0.000020
+142.2527 ± 0.0010
+0.0157 ± 0.0012
+2.78+0.22
+−0.22
+286.02612
+44.782871
+K02533.03
+KOI-2533.03
+26.115290 ± 0.000019
+145.5642 ± 0.0010
+0.0122 ± 0.0011
+4.01+0.38
+−0.38
+286.71564
+48.645279
+K02639.01
+KOI-2639.01
+25.108060 ± 0.000020
+146.4407 ± 0.0010
+0.0191 ± 0.0083
+3.69+1.63
+−1.62
+285.36517
+49.201561
+MNRAS 000, 1–19 (2022)
+
+Deep radius valley with Kepler short cadence
+17
+Foreman-Mackey D., et al., 2021, The Journal of Open Source Software, 6,
+3285
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+Fulton B. J., et al., 2017, AJ, 154, 109
+Furlan E., et al., 2017, AJ, 153, 71
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+Ginzburg S., Schlichting H. E., Sari R., 2016, ApJ, 825, 29
+Ginzburg S., Schlichting H. E., Sari R., 2018, MNRAS, 476, 759
+Gupta A., Schlichting H. E., 2019, MNRAS, 487, 24
+Gupta A., Schlichting H. E., 2020, MNRAS, 493, 792
+Holczer T., et al., 2016, ApJS, 225, 9
+Huber D., et al., 2013, ApJ, 767, 127
+Kipping D. M., 2013, MNRAS, 435, 2152
+Lee E. J., Chiang E., 2016, ApJ, 817, 90
+Lee E. J., Chiang E., Ormel C. W., 2014, ApJ, 797, 95
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+time series analysis in Python, Astrophysics Source Code Library
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+Lopez E. D., Rice K., 2018, MNRAS, 479, 5303
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+Learning. The MIT Press
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+APPENDIX A: EXTRA MATERIAL
+MNRAS 000, 1–19 (2022)
+
+18
+Ho & Van Eylen
+Table A1. Host star parameters of planets fitted in this work. 𝑅★, 𝑀★, 𝑇eff, [Fe/H], Ksmag are taken from the following sources: 1: Fulton & Petigura (2018), 2: Van Eylen et al. (2018). Stellar ages are taken from
+Fulton & Petigura (2018). Radius correction factors (RCFs) are taken from Furlan et al. (2017); we take RCF = 1.0000 where there are no measurements in Furlan et al. (2017).
+𝜌★, 𝑢0, 𝑢1 are resulting parameters from the transit fitting. Only the first 10 host stars are shown here; the full table is available online in a machine-readable format.
+KOI
+𝑅★ (𝑅⊙) (1)
+𝑀★ (𝑀⊙) (1)
+𝑇eff (K) (1)
+[Fe/H] (dex) (1)
+Ksmag (1)
+𝑅★ (𝑅⊙) (2)
+𝑀★ (𝑀⊙) (2)
+𝑇eff (K) (2)
+[Fe/H] (dex) (2)
+Ksmag (2)
+Age (Gyr)
+RCF
+𝜌∗ (g cm−3)
+𝑢0
+𝑢1
+41
+1.53+0.03
+−0.03
+1.10+0.07
+−0.03
+5854+60
+−60
+0.10 ± 0.04
+9.768
+1.51+0.01
+−0.01
+1.11+0.02
+−0.02
+5903+53
+−66
+0.10 ± 0.09
+11.197
+6.92+1.91
+−0.80
+1.0083
+0.313 ± 0.015
+0.46 ± 0.08
+0.17 ± 0.11
+46
+1.66+0.05
+−0.04
+1.24+0.03
+−0.09
+5661+60
+−60
+0.39 ± 0.04
+12.011
+N/A
+N/A
+N/A
+N/A
+N/A
+5.25+0.73
+−1.69
+1.0000
+0.279 ± 0.025
+0.52 ± 0.18
+0.27 ± 0.16
+49
+1.29+0.03
+−0.03
+0.95+0.03
+−0.03
+5779+60
+−60
+−0.06 ± 0.04
+11.917
+N/A
+N/A
+N/A
+N/A
+N/A
+10.47+1.21
+−1.21
+1.0000
+0.441 ± 0.021
+0.34 ± 0.12
+0.25 ± 0.13
+69
+0.94+0.02
+−0.02
+0.87+0.03
+−0.03
+5594+60
+−60
+−0.09 ± 0.04
+8.37
+0.91+0.02
+−0.02
+0.89+0.07
+−0.07
+5669+75
+−75
+−0.18 ± 0.10
+9.931
+9.12+2.94
+−2.10
+1.0000
+1.044 ± 0.055
+0.44 ± 0.04
+0.16 ± 0.06
+70
+0.88+0.02
+−0.02
+0.94+0.02
+−0.03
+5508+60
+−60
+0.11 ± 0.04
+10.871
+N/A
+N/A
+N/A
+N/A
+N/A
+3.09+1.99
+−1.57
+1.0054
+1.391 ± 0.052
+0.45 ± 0.04
+0.24 ± 0.07
+72
+1.08+0.03
+−0.03
+0.87+0.02
+−0.01
+5599+60
+−60
+−0.11 ± 0.04
+9.496
+1.07+0.01
+−0.01
+0.92+0.01
+−0.02
+5678+55
+−49
+−0.10 ± 0.06
+10.961
+12.88+1.48
+−0.89
+1.0005
+0.690 ± 0.019
+0.38 ± 0.10
+0.24 ± 0.13
+82
+0.72+0.02
+−0.02
+0.80+0.01
+−0.02
+4909+60
+−60
+0.11 ± 0.04
+9.351
+N/A
+N/A
+N/A
+N/A
+N/A
+1.07+0.89
+−1.58
+1.0000
+2.039 ± 0.055
+0.62 ± 0.06
+0.09 ± 0.07
+85
+1.44+0.03
+−0.03
+1.25+0.01
+−0.03
+6220+60
+−60
+0.13 ± 0.04
+9.806
+1.40+0.01
+−0.01
+1.20+0.03
+−0.03
+6193+43
+−52
+0.15 ± 0.08
+11.018
+2.95+0.34
+−0.41
+1.0001
+0.428 ± 0.019
+0.33 ± 0.06
+0.25 ± 0.07
+92
+1.06+0.02
+−0.02
+1.08+0.03
+−0.05
+5923+60
+−60
+0.09 ± 0.04
+10.3
+1.05+0.03
+−0.03
+1.08+0.11
+−0.11
+5952+119
+−119
+0.02 ± 0.15
+11.667
+2.34+1.35
+−1.40
+1.0000
+0.894 ± 0.049
+0.41 ± 0.12
+0.26 ± 0.13
+94
+1.37+0.03
+−0.03
+1.18+0.03
+−0.02
+6181+60
+−60
+0.07 ± 0.04
+10.926
+N/A
+N/A
+N/A
+N/A
+N/A
+3.47+0.64
+−0.56
+1.0000
+0.462 ± 0.021
+0.33 ± 0.11
+0.37 ± 0.16
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+Table A2. Planetary parameters from transit fits performed in this work. 𝑃, 𝑡0, 𝑅𝑝/𝑅★, 𝑏, 𝑒, 𝜔 are obtained directly from fitting, 𝑎/𝑅★ and 𝑆 are indirectly calculated. The 𝑅𝑝/𝑅★ values from 1: Fulton & Petigura
+(2018) and 2: Van Eylen et al. (2018) are also included for comparison. Only the first 10 planets are shown here; the full table is available online in a machine-readable format.
+KOI
+Kepler name
+𝑃 (days)
+𝑡0 (BJD-2454833)
+𝑅𝑝/𝑅★
+𝑅𝑝/𝑅★ (1)
+𝑅𝑝/𝑅★ (2)
+𝑏
+𝑒
+𝜔 (◦)
+𝑎/𝑅∗
+𝑆 (𝑆⊕)
+K00041.01
+Kepler-100 c
+12.815893 ± 0.000008
+122.9476 ± 0.0005
+0.0138 ± 0.0002
+0.0140+0.0001
+−0.0006
+0.0134+0.0001
+−0.0001
+0.39 ± 0.13
+0.06 ± 0.04
+30 ± 96
+14.36 ± 1.20
+245.37 ± 20.59
+K00041.02
+Kepler-100 b
+6.887062 ± 0.000007
+133.1781 ± 0.0008
+0.0082 ± 0.0001
+0.0081+0.0013
+−0.0002
+0.0079+0.0004
+−0.0002
+0.63 ± 0.05
+0.05 ± 0.04
+3 ± 107
+9.26 ± 0.88
+590.40 ± 56.33
+K00041.03
+Kepler-100 d
+35.333093 ± 0.000019
+153.9835 ± 0.0009
+0.0104 ± 0.0002
+0.0092+0.0023
+−0.0003
+0.0092+0.0004
+−0.0002
+0.81 ± 0.02
+0.05 ± 0.04
+1 ± 104
+27.50 ± 2.51
+66.87 ± 6.13
+K00046.02
+Kepler-101 c
+6.029792 ± 0.000020
+132.4816 ± 0.0010
+0.0073 ± 0.0007
+0.0069+0.0003
+−0.0005
+N/A
+0.41 ± 0.21
+0.05 ± 0.04
+−0 ± 103
+8.13 ± 0.77
+646.80 ± 61.97
+K00049.01
+Kepler-461 b
+8.313784 ± 0.000015
+175.9915 ± 0.0008
+0.0287 ± 0.0010
+0.0259+0.0003
+−0.0003
+N/A
+0.75 ± 0.11
+0.23 ± 0.13
+77 ± 77
+14.75 ± 2.22
+213.71 ± 32.30
+K00069.01
+Kepler-93 b
+4.726739 ± 0.000001
+134.9265 ± 0.0001
+0.0151 ± 0.0001
+0.0159+0.0002
+−0.0008
+0.0149+0.0001
+−0.0001
+0.23 ± 0.13
+0.20 ± 0.09
+105 ± 58
+13.06 ± 1.39
+252.14 ± 27.04
+K00070.01
+Kepler-20 A c
+10.854089 ± 0.000003
+138.6082 ± 0.0002
+0.0290 ± 0.0002
+0.0300+0.0018
+−0.0006
+N/A
+0.18 ± 0.11
+0.09 ± 0.03
+86 ± 39
+22.49 ± 0.80
+75.81 ± 2.80
+K00070.02
+Kepler-20 A b
+3.696115 ± 0.000001
+134.5021 ± 0.0002
+0.0182 ± 0.0002
+0.0209+0.0014
+−0.0023
+N/A
+0.49 ± 0.08
+0.05 ± 0.04
+26 ± 101
+10.24 ± 0.83
+365.46 ± 29.71
+K00070.03
+Kepler-20 A d
+77.611598 ± 0.000019
+164.7274 ± 0.0005
+0.0263 ± 0.0003
+0.0259+0.0005
+−0.0003
+N/A
+0.36 ± 0.13
+0.06 ± 0.04
+33 ± 96
+78.67 ± 6.41
+6.20 ± 0.51
+K00070.05
+Kepler-20 A f
+19.577627 ± 0.000020
+135.2063 ± 0.0009
+0.0091 ± 0.0004
+0.0098+0.0004
+−0.0003
+N/A
+0.65 ± 0.06
+0.05 ± 0.04
+7 ± 104
+30.66 ± 2.84
+40.78 ± 3.81
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+MNRAS 000, 1–19 (2022)
+
+Deep radius valley with Kepler short cadence
+19
+Table A3. Transit jitter and GP parameters from transit fitting of planetary
+systems in this work. Only the first 10 systems are shown here; the full table
+is available online in a machine-readable format.
+KOI
+log 𝜎lc
+log 𝜎gp
+log 𝜌gp
+41
+−8.4379 ± 0.0016
+−9.5822 ± 0.0105
+−4.5121 ± 0.0335
+46
+−7.1119 ± 0.0037
+−9.1794 ± 0.1025
+−4.3449 ± 0.2492
+49
+−7.3268 ± 0.0044
+−8.8595 ± 0.0680
+−1.2568 ± 0.1547
+69
+−8.9861 ± 0.0019
+−10.1248 ± 0.0133
+−4.8968 ± 0.0382
+70
+−7.7689 ± 0.0012
+−9.4665 ± 0.0150
+−3.7110 ± 0.0415
+72
+−8.4175 ± 0.0053
+−9.6473 ± 0.0392
+−4.0440 ± 0.1139
+82
+−8.1671 ± 0.0018
+−8.6193 ± 0.0116
+−3.2278 ± 0.0240
+85
+0.0000 ± 0.0000
+0.0000 ± 0.0000
+0.0000 ± 0.0000
+92
+−8.1505 ± 0.0062
+−9.8036 ± 0.0880
+−3.9919 ± 0.2448
+94
+−7.8819 ± 0.0017
+−9.8028 ± 0.0308
+−2.9898 ± 0.1183
+...
+...
+...
+...
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–19 (2022)
+
diff --git a/SdE2T4oBgHgl3EQfswgq/content/tmp_files/load_file.txt b/SdE2T4oBgHgl3EQfswgq/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf,len=2893
+page_content='MNRAS 000, 1–19 (2022)  Preprint 11 January 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  A deep radius valley revealed by Kepler short cadence observations  Cynthia S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Ho★ and Vincent Van Eylen  Mullard Space Science Laboratory, University College London, Dorking RH5 6NT, UK  Accepted 2022 December 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Received 2022 December 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' in original form 2022 October 17  ABSTRACT  The characteristics of the radius valley, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', an observed lack of planets between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5-2 Earth radii at periods shorter than about  100 days, provide insights into the formation and evolution of close-in planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We present a novel view of the radius valley  by refitting the transits of 431 planets using Kepler 1-minute short cadence observations, the vast majority of which have not  been previously analysed in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' In some cases, the updated planetary parameters differ significantly from previous studies,  resulting in a deeper radius valley than previously observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This suggests that planets are likely to have a more homogeneous  core composition at formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Furthermore, using support-vector machines, we find that the radius valley location strongly  depends on orbital period and stellar mass and weakly depends on stellar age, with 𝜕 log �𝑅𝑝,valley  �/𝜕 log 𝑃 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='096+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='023  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='027,  𝜕 log �𝑅𝑝,valley  �/𝜕 log 𝑀★ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='231+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='053  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='064, and 𝜕 log �𝑅𝑝,valley  �/𝜕 log (age) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='033+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='017  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These findings favour thermally-driven  mass loss models such as photoevaporation and core-powered mass loss, with a slight preference for the latter scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Finally,  this work highlights the value of transit observations with short photometric cadence to precisely determine planet radii, and we  provide an updated list of precisely and homogeneously determined parameters for the planets in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Key words: planets and satellites: composition – planets and satellites: formation – planets and satellites: fundamental parameters  1 INTRODUCTION  The ‘radius valley’, also known as the ‘radius gap’, is the relative  paucity of planets with sizes between about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5 and 2 Earth radii  at orbital periods less than about 100 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This phenomenon has  been predicted theoretically due to the heavy radiation these close-in  planets receive from their host star (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Owen & Wu 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Lopez &  Fortney 2013) and was subsequently seen observationally (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Fulton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Fulton & Petigura 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Several  theories have been suggested to explain the physical origin of the  radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' On one hand, thermally-driven mass loss scenarios have  been proposed, which include photoevaporation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Owen & Wu  2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Lopez & Fortney 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Owen & Wu 2017) and core-powered  mass loss (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Gupta & Schlichting 2019, 2020)  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' In these scenarios, the valley separates planets that have lost  their atmosphere from those that have retained it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Alternatively, late  gas-poor formation, where planets below the valley have formed  atmosphere-free, may also be able to explain the origin of the valley  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Lee & Chiang 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Lopez & Rice 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Cloutier & Menou 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Observed characteristics of the radius valley can therefore reveal  the properties of these close-in planets and their formation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  For example, in photoevaporation models, the location of the radius  valley and its slope as a function of orbital period depend on the  planetary composition and photoevaporation physics (Owen & Wu  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Mordasini 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The valley’s location and relative emptiness  can therefore be used to infer the composition of planets surrounding  it and their relative homogeneity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Planets  ★ E-mail: sze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='ho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='20@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='uk  located inside the radius valley may have a different composition or  could be undergoing the final stages of atmospheric loss by thermally-  driven mechanisms and hence may be important targets for further  studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Owen & Wu 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Gupta & Schlichting 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Petigura  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The valley’s location as a function of orbital period can be  used to distinguish between thermal mass-loss models, which exhibit  a negative slope as a function of orbital period, and late gas-poor  formation models which have the opposite slope (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Van Eylen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Cloutier & Menou 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Within  thermal mass-loss models, photoevaporation and core-powered mass  loss models predict a different dependence of the valley’s location  on stellar mass and age (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Observationally, these valley characteristics have been challeng-  ing to reliably ascertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' A deficit of planets with sizes around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5-2  Earth radii (𝑅⊕) was first observed by Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2017) in a sam-  ple of 2025 planets, with stellar radii determined spectroscopically  as part of the California-Kepler survey (CKS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These planets were  about a factor of two rarer than planets both smaller and larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Independently, Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2018) (V18 hereafter) analysed a  subset of this sample (117 planets), incorporating higher-precision  stellar parameters using asteroseismology and refitting transit light  curves to achieve a median uncertainty on planet sizes of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This  study revealed the valley’s slope as a function of orbital period for  the first time, and suggested the radius valley may be very deep or  even entirely empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='The tension between the valley’s views of Fulton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2017) and V18 was further exacerbated when the precision  of stellar parameters of the former study were further improved by  Fulton & Petigura (2018) (F18 hereafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Despite improving stellar  uncertainties from 11% to 3% by incorporating Gaia parallaxes, the  valley remained partially filled in, with its depth largely unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  © 2022 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04062v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='EP]  10 Jan 2023  2  Ho & Van Eylen  Petigura (2020) investigated the discrepancy in the valley’s depth  between V18 and F18 and concluded it is unlikely to be caused by  differing sample sizes or differing values or uncertainties in stellar  radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The study argued that the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='9% dispersion in planetary radii  is instead primarily caused by a discrepancy in the ratio of planet  to stellar radii (𝑅𝑝/𝑅★)) determined from the transit fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' F18 used  radius ratios from Mullally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2015), which fitted Kepler 30-  minute ’long cadence’ observations, whereas V18 used Kepler 1-  minute ’short cadence’ observations, also used for orbital eccentricity  determination and described in Van Eylen & Albrecht (2015) and Van  Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Here, we seek to refit planet transits for the full subset of F18  for which short cadence observations are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This increases  the sample of planets relevant for the radius valley for which short  cadence transit fits are used from 60 in V18 to 431 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Furthermore,  we will apply the methods to determine the valley’s location and  slope used by V18, notably the use of support vector machines, to  this larger sample, and we expand to other dimensions such as stellar  mass and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  In Section 2, we describe the sample and methodology used to  analyse the radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' In Section 3, we present the results of this  analysis, such as revised planetary sizes, the depth of the valley, and  its dependence on parameters such as the orbital period and stellar  mass and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These findings are compared to other observational  studies and theoretical models in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Finally, we provide con-  clusions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  2 METHODS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='1 Sample selection  We use the sample of planets for which stellar parameters are avail-  able from F18 as a starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' To focus on the radius valley, we  limit the sample to planets with radii 1 ≤ 𝑅𝑝/𝑅⊕ ≤ 4 and orbital  periods 1 ≤ 𝑃/days ≤ 100, resulting in a sample of 1272 planets (for  comparison, applying the same period and radius cuts to the sample  studied by V18 leaves 74 planets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' As Kepler 1-minute short cadence  observations may yield superior precision (Petigura 2020), we fur-  ther limit our sample to those planets for which at least 6 months of  Kepler short cadence data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  To avoid issues with transit fitting related to transit timing varia-  tions (TTVs), we also remove planets with known TTVs based on the  catalogue by Holczer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We further exclude KOI-1576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03,  as we find that the short cadence data suggested an orbital period dif-  ferent to the one recorded in the archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Furthermore, we exclude any  planets that are classified as potential false positives in Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The results in a total sample size of 431 planets, 60 of which  have parameters previously analysed by V18 and 371 which have not  (a further 14 planets in V18 have TTVs and are not reanalysed here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2 Data reduction  The 1-minute Kepler short cadence Pre-search Data Conditioning  SAP (PDCSAP, Stumpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2012) light curves  of these targets are downloaded from the NASA Mikulski Archive  for Space Telescopes (MAST) database using the lightkurve  package (Lightkurve Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2018), which incorporates  astroquery (Ginsburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2019) and astropy (Astropy Collab-  oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2013, 2018) dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We only retain data within  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2 days before the estimated ingress and after the estimated egress of  the transits of the planets of interest, using the transit durations and  mid-times in Mullally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2015) as the expected transit locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  For multi-planet systems, we only retain transits of planets that are  within our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We remove data outliers that lie beyond 6𝜎 from  the median after masking the transits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We then flatten the transits by  dividing the data points with the slope obtained by performing linear  regression on the data points immediately before ingress and after  egress, to remove long-term systematic trends present in the transits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We then again remove data outliers with 𝜎 = 5 to further clean the  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='3 Stellar multiplicity  Around 46% of solar-type stars have at least one stellar companion  (Raghavan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' When a planet orbits a single star, the transit  depth 𝛿 is approximately given by  𝛿 = Δ𝐹  𝐹tot  ≈  𝑅2𝑝  𝑅2  ★  (1)  where 𝐹tot is the total stellar flux, Δ𝐹 is the change in stellar flux, and  𝑅𝑝 and 𝑅★ are the planetary and stellar radius respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' However,  in a multi-stellar system, the total flux is the sum of fluxes of all stars  in the system, but the change in flux during transit is only relative to  the star(s) which the planet transits (Furlan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Therefore  it is important to take into account the effect of nearby stars on the  light curve flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Furlan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2017) compiled a catalogue of Kepler Objects of  Interest (KOI) observations with adaptive optics, speckle interfer-  ometry, lucky imaging, and imaging from space with the Hubble  Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The typical point spread function (PSF) widths  and sensitivities (Δ𝑚) are different for every observation method,  target and bandpass, hence whether stellar companions are detected  is dependent on the above factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' For example, Furlan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2017)  were able to detect a median Δ𝑚 ∼ 8mag with Keck in the K band at  a separation of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5”, but only at ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5” at Lick in the J or H bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  About 30% of KOIs observed in Furlan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2017) have at least one  companion detected within 4” (Furlan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2017), and given a mean  distance of 616pc for the 431 planets in our sample computed from  distances reported in Mathur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2017), corresponds to ∼2464AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Here, we adopt the ‘radius correction factor’ (RCF), given in Furlan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2017) as  RCF =  𝑅𝑝,corr  𝑅𝑝,uncorr  (2)  and multiply the normalised Kepler light curve fluxes by RCF2, and  subtract  �  RCF2 − 1  �  to re-normalise, to obtain the corrected light  curve reflecting the transit of one planet orbiting around one star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  137 of the 431 planets in our sample (32%) have RCF measurements  from Furlan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4 Transit fitting  We use the exoplanet package (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2021) to  generate a transit light curve model with quadratic stellar limb dark-  ening, and then run a Hamiltonian Monte Carlo (HMC) algorithm  implemented in PyMC3 (Salvatier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2016) to perform fitting and  determine orbital parameter posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We also implement a Gaus-  sian Process (GP) model (Rasmussen & Williams 2006) to account  for correlated noise in the light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' However, for Kepler-65 and  Kepler-21 A, we do not fit for a GP model due to convergence  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The parameters fitted for each planet are orbital period  (𝑃), transit mid-time (𝑡0), ratio between planetary and stellar radii  MNRAS 000, 1–19 (2022)  Deep radius valley with Kepler short cadence  3  (𝑅𝑝/𝑅★), impact parameter (𝑏), eccentricity (𝑒), argument of peri-  apsis (𝜔), and stellar density (𝜌★).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' For each light curve, we further  include two quadratic stellar limb darkening parameters (𝑢0 and 𝑢1)  for the host star, with bounds 0 < 𝑢0, 𝑢1 < 1 and implemented with  the Kipping (2013) reparameterisation in exoplanet, the transit jit-  ter (𝜎lc), and two parameters describing the GP contribution (𝜎gp,  𝜌gp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We initialise the HMC chains by using values presented in the  Kepler Q1-16 dataset (Mullally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2015) for 𝑃, 𝑡0, 𝑅𝑝/𝑅★, and 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We set the system to begin with near-circular orbits, with 𝑒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01 and  𝜔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We take initial stellar densities from Fulton & Petigura  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We use the Exoplanet Characterization ToolKit (ExoCTK)  (Bourque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2021) to estimate the initial 𝑢0 and 𝑢1, which takes  the stellar temperature, surface gravity, and metallicity, which we  use values from the Kepler Q1-16 dataset (Mullally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2015), as  inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We apply Gaussian priors to 𝑃, 𝑡0, 𝑢0, 𝑢1, and 𝜌★, using the initial  guesses as the mean, and 𝜎𝑃 = 2 × 10−5 days, 𝜎𝑡0 = 10−3 days,  𝜎𝑢 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2, and the 𝜌★ uncertainty from Fulton & Petigura (2018) if  available, and Mullally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2015) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' A beta distribution  prior according to Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2019) is placed on 𝑒, which is  PDF(𝑒, 𝛼, 𝛽) ∝ 𝑒𝛼(1 − 𝑒)𝛽  (3)  with 𝛼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='58 and 𝛽 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4 for system with only one transiting planet,  and 𝛼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='52 and 𝛽 = 29 for a multi-transiting-planet system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  3 RESULTS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='1 Revised planet parameters  We report the updated orbital periods (𝑃), planetary-to-stellar-radii  ratio (𝑅𝑝/𝑅★), planetary radii (𝑅𝑝), the number of transits in the  fitted light curve (𝑁tr), and their uncertainties of the 431 planets fitted  in this sample in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The full list of parameters are provided in  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We convert our 𝑅𝑝/𝑅★ to 𝑅𝑝 using the updated stellar  parameters available: values used in V18 from asteroseismology (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  taken from Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Silva Aguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Lundkvist  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2016) if the planets are included in the V18 samples, and F18  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Full homogeneity is lost by using stellar radii from two  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' To investigate the consequences of this, we compute the  difference, 𝛿𝑅𝑝, between the planetary radii obtained by converting  𝑅𝑝/𝑅★ to 𝑅𝑝 using 𝑅★ from F18 and V18, and found the mean  𝛿, ¯𝛿 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11, hence 𝛿 = 0 (no difference) is well within  1𝜎, and we conclude that there is no substantial drawbacks of using  multiple sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This sample of 431 planets with updated parameters  is plotted on the radius-orbital period plot as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We present the typical uncertainties of 𝑅𝑝/𝑅★ and 𝑅𝑝 of planets  fitted in this work, compared with F18 and V18 in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' For our  newly fitted results, we find that 𝑅𝑝/𝑅★ has a mean uncertainty of  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='76%, and a median uncertainty of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='44%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This is smaller when  compared to the mean and median uncertainties of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='73% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07%  respectively for the F18 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' For 𝑅𝑝, we find a mean and median  uncertainty of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='70% in our work, again smaller compared  to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='00% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='22% for F18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' However, they are larger than that of  V18, possibly due to V18 analysing brighter stars, hence the light  curves are less noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This can be seen by comparing the mean  photometric flux error of the transit light curves fitted: 1007 parts  per million (ppm) for this work, and 271 ppm for V18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We select the planets in F18 that are in common with the planets in  our sample, and examine the change in 𝑅𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We find that 217 planets  have a larger 𝑅𝑝 after refitting, and smaller for 214 planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Of the  planets whose sizes have increased, the mean change is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='79%, and  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Radius-period plot of all 431 planets refitted in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The  black dotted lines indicate the upper and lower boundaries of the radius  valley defined in V18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='69% for planets with reduced sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Considering all 431 planets,  our new results change 𝑅𝑝 by a mean and median of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='62%  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02% respectively, indicating that our refitting results do not  systematically alter the planet sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We observe that 120 planets  (28%) have a revised 𝑅𝑝 > 2𝜎 away from their corresponding values  from F18, and 62 planets (14%) > 3𝜎 away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2 Radius valley dependence on orbital period  To obtain the most precise planetary sample, we apply the following  conservative cuts to our sample for subsequent analyses in the rest of  this paper:  (i) Precision on 𝑅𝑝: We exclude planets with a planet radius  precision 𝜎𝑅𝑝 > 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  (ii) Radius correction factor (RCF): We exclude planets with an  RCF > 5%, as reported in Furlan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' RCFs may themselves  be uncertain due to observational challenges in detecting nearby stel-  lar companions and measuring their brightness (Furlan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Of the 137 planets with RCF measurements from Furlan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2017),  18 (13%) have RCF > 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The remaining 294 planets do not have  RCF measurements currently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  (iii) Number of transits: We exclude planets with fewer than 3  transits in the Kepler short cadence data to limit the risk that corre-  lated noise during an individual transit strongly affects the resulting  fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  After implementing these filters, 375 planets remain in our sample  which we will use throughout the remainder of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We first use this sample to investigate the location of the radius val-  ley as a function of orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Following the procedure outlined  in Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2018), we calculate the position of the radius  valley by determining the hyperplane of maximum separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We  perform this with a linear support-vector machine (SVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' To ini-  tialise our model, we initially classify our sample into two groups,  ‘above’ and ‘below’ the radius valley on the radius-orbital period  plane, by applying a Gaussian Mixture Model with two components  in the 𝑅𝑝-𝑃 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Since the orders of magnitude of 𝑅𝑝 and 𝑃 are  different, we divide 𝑃 by 5 before applying the above clustering al-  gorithm to allow the model to separate the planetary population into  MNRAS 000, 1–19 (2022)  4  HO  D  0  D  3  TO  DT  D  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  HO  D  TO  O  2  R  D  O  D  勇  DI  D  D  D  3  10  30  100  P (days)4  Ho & Van Eylen  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Table showing estimates of the orbital periods (𝑃), planetary-to-stellar-radii ratio (𝑅𝑝/𝑅★), planetary radii (𝑅𝑝), the number of transits in the fitted  light curve (𝑁tr) of 431 planets refitted in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The source of 𝑅★ to convert 𝑅𝑝/𝑅★ to 𝑅𝑝 is listed in the References column: (1) Fulton & Petigura  (2018), (2) Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='‘Flag’ refers to whether the planet passes the filter checks and is included in the smaller subset for further analyses (1 for  True and 0 for False).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The complete list of parameters are provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Only the first 10 planets are shown here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' the full table is available online in  a machine-readable format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  KOI  Kepler name  𝑃 (days)  𝑅𝑝/𝑅★  𝑅𝑝 (𝑅⊕)  𝑅★ Source  𝑁tr  Flag  K00041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  Kepler-100 c  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='815893 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0138 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='28+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  (2)  93  1  K00041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  Kepler-100 b  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='887062 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0082 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0001  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  (2)  173  1  K00041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  Kepler-100 d  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='333093 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000019  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0104 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='71+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  (2)  35  1  K00046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  Kepler-101 c  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='029792 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0073 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0007  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='32+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='14  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='14  (1)  50  0  K00049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  Kepler-461 b  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='313784 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0287 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0010  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='17  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='17  (1)  34  1  K00069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  Kepler-93 b  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='726739 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0151 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0001  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='04  (2)  275  1  K00070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  Kepler-20 A c  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='854089 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0290 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='07  (1)  116  1  K00070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  Kepler-20 A b  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='696115 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0182 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='05  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  (1)  336  1  K00070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  Kepler-20 A d  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='611598 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000019  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0263 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0003  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='52+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07  (1)  15  1  K00070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  Kepler-20 A f  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='577627 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0091 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='88+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  (1)  62  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Mean and median values radii determined in this work, F18, and  V18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Values are listed as percentages (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Parameter  This work  F18  V18  Sample Size  431  1901  117  𝑅𝑝/𝑅★  Mean  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='76  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='73  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='20  Median  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='44  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='44  𝑅★  Mean  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='23  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='17  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='49  Median  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='78  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='74  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='20  𝑅𝑝  Mean  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='00  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='96  Median  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='70  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='22  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='36  two groups above and below the radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Otherwise, the pop-  ulation would be clustered in a way dominated by the difference in  period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Following Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2018) and David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2021),  we select an SVM penalty parameter of 𝐶 = 10 for the hyperplane,  to minimise misclassification of data points above or below the ra-  dius valley, but still allow the hyperplane location to be determined  by a sufficient number of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' To determine accurate uncer-  tainties on the location of the valley, we then perform a bootstrap  by generating 1000 new sample sets on which we repeat the above  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Each bootstrap sample is generated by generating a new  sample of the same size from the original sample, allowing replace-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Each bootstrapped sample is then categorised into two groups  with a Gaussian Mixture Model, and the SVM procedure is repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Reporting the median value and taking the 16th and 84th percentiles  as the upper and lower uncertainties, we find  log10  �𝑅𝑝/𝑅⊕  � = 𝑚 log10 (𝑃/days) + 𝑐  (4)  with 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02, and 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='37+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The location of the radius  valley is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We also implement the method adopted by Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2022),  which involves computing the planet density in the radius-period  plane with a Gaussian kernel density estimate (KDE), and fitting the  planet radius at the KDE minima between log10 (𝑃/days) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 𝑃 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='16 − 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='6 days) and log10  �𝑅𝑝/𝑅⊕  � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='15 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='35 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  𝑅𝑝 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='41 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='24𝑅⊕) according to equation 4, and performing  bootstrap with 1000 sample sets to find the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The result  is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We use a KDE bandwidth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='467 in  log10 𝑃 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='075 in log10 𝑅𝑝, which is based on the bandwidth used  by Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2022), but scaled up linearly based on the ratios  between the two sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We note that this method fits a narrower  period range compared to the SVM method, which fits for the full  𝑃 = 1 − 100 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' With this method, we obtain 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  and 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='37+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We find this value to be slightly steeper than that  determined with the SVM, however the two values are well within  1𝜎 of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Here, we do not correct for detection completeness,  and we note that this method is highly sensitive to the choice of the  KDE bandwidth, and using a bandwidth five times larger results with  a slope approximately twice as steep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Some studies have opted to study the radius valley location as a  function of incident flux 𝑆, in addition to, or instead of, 𝑃 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Rogers  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We calculate 𝑆 according to the  formula  𝑆 =  𝐿★  4𝜋𝑎2  (5)  where 𝐿★ is the host star’s luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 𝐿★ can itself be calculated  using  𝐿★ = 4𝜋𝑅2  ★𝜎sb𝑇4  eff  (6)  where 𝑅★ is the stellar radius, 𝜎sb is the Stefan-Boltzmann constant,  and 𝑇eff is the effective temperature of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Finally, 𝑎 is the orbital  semi-major axis, which is given by  � 𝑎  𝑅★  �3  = 𝐺𝑃2𝜌★  3𝜋  (1 + 𝑒 sin 𝜔)3  �1 − 𝑒2�1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5  (7)  according to Kepler’s third law of planetary motion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Van Eylen &  Albrecht 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Petigura 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Here, 𝐺 is the gravitational constant,  𝑃 is the orbital period, 𝜌★ is the stellar density, 𝑒 and 𝜔 are the orbital  eccentricity and argument of periapsis respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' As before, we  obtain the stellar properties (𝑅★, 𝜌★, 𝑇eff) from V18 when available,  and from F18 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 𝑃, 𝑒, and 𝜔 are obtained from the transit  fitting results of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Fitting the valley with the SVM, we obtain  MNRAS 000, 1–19 (2022)  Deep radius valley with Kepler short cadence  5  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Dependencies of the radius valley in 𝑛 dimensions (𝑚𝑖), given by the equation log10  �𝑅𝑝/𝑅⊕  � = �𝑛  𝑖=1 𝑚𝑖 𝑥𝑖, with different parameter combinations  𝑥𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The methods used to obtain the equation of the radius valley hyperplanes are also given, where SVM and KDE stand for the support-vector machine and  fitting the minima of the kernel density estimates respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Dimensions  log10 (𝑃/days)  log10  �𝑆/𝑆⊕  �  log10  �𝑀/𝑀⊙  �  log10 (Age/Gyr)  [Fe/H]  Intercept  Method  2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  SVM  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='03  KDE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='04  SVM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='05  SVM  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='03  SVM  4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='018  SVM  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Left: Radius valley location determined with the support-vector machine (SVM), with 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02, and 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03, indicated by the black  solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The dashed lines represent the median boundaries passing through the supporting vectors determining the position of the solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We define the area  between the two lines as the radius valley region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The green and blue points show planets above and below the radius valley respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The grey shaded regions  represent the ±1𝜎 uncertainties of the lines determined using the bootstrap method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The red dotted lines show the plot divided into multiple orbital-period  dependent bins, which is then used for plotting the adjusted histogram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Right: Radius valley position determined by fitting a line through the region  where the kernel density estimate is minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' With this method, we obtain 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05, and 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='37+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03, indicated by the black solid line with ±1𝜎  uncertainty shaded in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  log10 𝑅𝑝/𝑅⊕ = 𝑚 log10 𝑆/𝑆⊕ + 𝑐  (8)  with 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01, and 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The location of the radius  valley in terms of 𝑆 is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='3 Depth of the radius valley  We investigate the depth of the radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' As we find the position  of the radius valley is dependent on orbital period, we divide the  radius valley into multiple tilted bins (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2 left) and  plot an adjusted histogram in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We shift planets  along the slope of the radius valley obtained with the SVM method  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11, and plot a histogram of ‘expected’  planetary radii at an orbital period of 10 days, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We  choose to fit the histogram with a Gaussian Mixture Model of two  clusters, as opposed to a Gaussian kernel density estimate, as the  former is independent of the sizes and locations of the histogram  bins, as well as the Gaussian bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Also, with the Gaussian  Mixture Model, we are able to force the planets to fall into two  groups only, matching the bimodal distribution of small planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  MNRAS 000, 1–19 (2022)  4  3  2  3  10  30  100  P (days)Relative density  YD  2  1  3  TOI  R  2  R  3  10  30  100  P (days)6  Ho & Van Eylen  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Same as Figure 2 left, but as a function of incident flux 𝑆 instead  of orbital period 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Histogram of planet radii, adjusted to equivalent radii at 𝑃 = 10  days, according to the radius valley slope calculated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2 with the  SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Note that completeness corrections are not performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Here, 𝐸avg =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='60  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Here, we propose the metric 𝐸, defined as  𝐸SN = 𝑁sub-Neptune,peak/𝑁valley  (9)  and  𝐸SE = 𝑁super-Earth,peak/𝑁valley  (10)  to compare the number of planets inside the valley and the peak num-  ber outside the valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' A higher 𝐸 indicates a deeper radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  𝑁sub-Neptune,peak and 𝑁super-Earth,peak are the number of planets at  the sub-Neptune and super-Earth Gaussian peaks respectively, and  𝑁valley is the number of planets at the lowest point between the two  Gaussian peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' As 𝑁sub-Neptune, peak, 𝑁super-Earth, peak, and 𝑁valley  are determined directly from the curve resulting from the Gaussian  Mixture Model, this 𝐸 metric is also independent of the histogram  bin sizes or locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' To calculate the uncertainties of 𝐸, we again  perform a bootstrap with 1000 sample sets, where each bootstrap  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Plot of planetary radius against mass of host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The solid line  shows the location of the radius valley determined with the SVM, with slope  𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08, and 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The dashed lines show the boundaries  of the radius valley, given by the same 𝑚, and 𝑐upper = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01, 𝑐lower =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The shaded regions depict the ±1𝜎 uncertainties of the lines  determined with bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  sample is generated by generating a new sample of the same size  from the original, allowing replacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We also replace the ra-  dius of each selected planet by randomly drawing from a normal  distribution with the reported planet radius as the mean, and the  uncertainties on the radius as the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We report the median  of this bootstrap distribution as our results, and the 84th and 16th  percentiles as our ±1𝜎 uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We test this metric on known  planetary samples on F18 and V18, which we know the difference  in the radius valley depth, and find that their corresponding 𝐸 value  differ, hence demonstrating the reliability of such metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The dif-  ference in the depth of the radius valley is further discussed in Sec-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We find that for our new short cadence results, the ratios are  𝐸SN = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='59+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='77  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='62 and 𝐸SE = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='40+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='61  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Averaging the two numbers  gives us 𝐸avg = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='60  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4 Radius valley dependence on stellar mass  We investigate the radius valley location as a function of stellar  mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We first implement a 2-dimensional SVM, using the same  method as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The result is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We find  d log 𝑅𝑝/d log 𝑀★ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08, and the intercept 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2021) suggested the degeneracy between the photo-  evaporation and core-powered mass loss scenarios could be broken  with an analysis of the radius valley in 3 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Hence, we  implement an SVM in 3 dimensions: planet radius 𝑅𝑝, orbital period  𝑃, and mass of the host star 𝑀★, to fit the radius valley in the form  of a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We perform bootstrapping with 1000 sample sets as per  previous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We obtain the relation  log10  �𝑅𝑝/𝑅⊕  � = 𝐴 log10 (𝑃/days) + 𝐵 log10  �𝑀★/𝑀⊙  � + 𝐶  (11)  with 𝐴 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03, 𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='21+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07, 𝐶 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='35+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' An illustration  of the SVM plane is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We also investigate the radius valley location in the 𝑅𝑝–𝑆–𝑀★  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Figure 6 right shows the radius valley location in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  MNRAS 000, 1–19 (2022)  4  甲  TH  3  ®  R  2  R  88  由  申  1000  100  10  1  S (S)70  60  S  f planet  50  40  Number of  30  20  N  10  0  1  2  3  4  5  6  Rp at P= 10 days (R)4  3  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  M* (Mo)Deep radius valley with Kepler short cadence  7  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Plot of planet radius against orbital period and mass of host star (left), and planet radius against incident flux and stellar mass (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The grey plane  shows the radius valley location determined with the SVM in 3 dimensions, with the ±1𝜎 uncertainties shown in pink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The uncertainties on individual planet  parameters are not displayed in the interest of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We find  log10  �𝑅𝑝/𝑅⊕  � = 𝐴 log10  �𝑆/𝑆⊕  � + 𝐵 log10  �𝑀★/𝑀⊙  � + 𝐶  (12)  with 𝐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02, 𝐵 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09, and 𝐶 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5 Radius valley dependence on stellar age  We investigate the location of the radius valley as a function of stellar  age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We obtain the stellar ages from F18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Kepler-174 does not have  stellar age data from this source, hence we omit Kepler-174 b and  Kepler-174 c in our analysis with stellar age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Fitting the valley with the SVM, we obtain  log10  �𝑅𝑝/𝑅⊕  � = 𝑚 log10 (Age/Gyr) + 𝑐  (13)  with 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02, 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01, showing no significant corre-  lation with the radius valley location in 2-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The  result is displayed in Figure 7 (left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We further investigate whether the radius valley depth may be a  function of stellar age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' To do so we generate histograms, similar to  Figure 4, split into different stellar age subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Figure 8 (left  panel) shows that for older stars, the radius valley location shifts to  higher 𝑅𝑝, and the radius valley becomes shallower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The change in  the 𝐸 metric is reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These findings suggest that the  radius valley has a dependence on the age of the host stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We further plot the radius valley in 𝑅𝑝–𝑃–age space, and fit the  valley with the SVM, as shown in Figure 9 (left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We find  log10  �𝑅𝑝/𝑅⊕  � = 𝐴 log10 (𝑃/days) + 𝐵 log10 (Age/Gyr) + 𝐶  (14)  with 𝐴 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02, 𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03, and 𝐶 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We can also combine 𝑅𝑝, 𝑃, 𝑀★, and stellar age, and determine  the radius valley with a 4-dimensional SVM, as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  The resulting equation representing the radius valley is in the form  of a 4-dimensional hyperplane  log10  �𝑅𝑝/𝑅⊕  � = 𝐴 log10 (𝑃/days) + 𝐵 log10  �𝑀★/𝑀⊙  �  + 𝐶 log10 (Age/Gyr) + 𝐷  (15)  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 𝐸 values of the radius valley for different ages of the host stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  𝐸avg = (𝐸SN + 𝐸SE)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  log10 (Age/yr)  Age (Gyr)  𝑁planets  𝐸SN  𝐸SE  𝐸avg  < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='25  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='78  53  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='93  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='64  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='28  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='25 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='78 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='16  95  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='69  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='89  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='75  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='16 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='62  114  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='22  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='32  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='27  > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='75  > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='62  111  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='21  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='40  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='30  with 𝐴 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='096+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='023  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='027, 𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='231+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='053  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='064, 𝐶 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='033+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='017  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='025, 𝐷 =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='339+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='026  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These results imply there is strong evidence the radius  valley location is dependent on 𝑃 and 𝑀★, and weak evidence for its  dependence on stellar age (> 1𝜎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These values are also consistent  within 1𝜎 with their corresponding dependencies in two and three  dimensions (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='6 Radius valley dependence on stellar metallicity  We perform a similar analysis in terms of the stellar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' As  for age, we obtain the stellar metallicity from V18 if available, and  F18 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We find  log10  �𝑅𝑝/𝑅⊕  � = 𝑚[Fe/H] + 𝑐  (16)  with 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08, 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01, again displaying no significant  correlation with the radius valley location in 2-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We divide the planet population into two groups, based on the  median [Fe/H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The adjusted histograms in Figure 8 (right  panel) show that the super-Earth peak is lower for metal-poor stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  The 𝐸 values are reported in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We perform a similar SVM analysis in 𝑅𝑝–𝑃–[Fe/H] space (shown  in Figure 9 right panel), and find  log10  �𝑅𝑝/𝑅⊕  � = 𝐴 log10 (𝑃/days) + 𝐵[Fe/H] + 𝐶  (17)  with 𝐴 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03, 𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04, and 𝐶 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These  MNRAS 000, 1–19 (2022)  Sub-Neptunes  Super-Earths  4  Rp (R)  2  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  1  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  10  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='7  100  P (days)  M* (Mo)Sub-Neptunes  Super-Earths  4  Rp (R)  2  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  1000  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='7  S (S)  1  M* (Mo)8  Ho & Van Eylen  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Same as Figure 5, but for stellar age (left, 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02, 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01), and stellar metallicity [Fe/H] (right, 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08, 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Same as Figure 4, but separated into different stellar ages (left), and metallicities (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Here, 𝑇 = log10 (Age/yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The histograms are normalised  such that the relative density of the sub-Neptune peak equals to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 𝐸 values of the radius valley for different stellar metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 𝐸avg =  (𝐸SN + 𝐸SE)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  [Fe/H]  𝑁planets  𝐸SN  𝐸SE  𝐸avg  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06  181  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='15  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06  194  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='28  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='78  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  values imply that we find no evidence that the radius valley location  depends on stellar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  4 DISCUSSION  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='1 𝑅𝑝-𝑃 relation suggests a thermally-driven mass loss model  As presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2, we observe the radius valley scales  as 𝑚 = d log 𝑅𝑝/d log 𝑃 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This negative period-  dependence is a robust finding which remains roughly similar even  when other parameters are included in the fit (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Different theoretical mechanisms to create the radius valley result  in a different slope as a function of orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' For example,  Lopez & Rice (2018) predicted that if the rocky planets are core  remnants of sub-Neptunes with evaporated atmospheres, the radius  valley location should decrease with increasing orbital period, with  𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09, whereas if those rocky planets were formed after disk  dissipation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', late gas-poor formation), the radius valley location  tends to larger planetary radii at longer orbital periods, with 𝑚 =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Similarly, Owen & Wu (2017) predicted a negative period-  radius valley slope for a photoevaporation model, with −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='25 ≤  𝑚 ≤ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='16 depending on the photoevaporation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' If the  radius valley is thermally driven but powered by the core rather than  photoevaporation, the slope would be similarly negative, with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=',  Gupta & Schlichting (2019) predicting that 𝑚 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Theoretically predicted slopes for different formation mechanisms  are summarised in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Our observed negative slope is consistent with thermally driven  mass-loss models but inconsistent with late gas-poor formation mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We can also compare our observed slope with other observational  MNRAS 000, 1–19 (2022)  4  3  2  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  Fe/H/(dex)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='6  T<9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='25 ≤T<9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5 ≤ T< 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='75  Relative density  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  T≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  3  4  2  5  6  Rp at P=  :10 days (R@)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  [Fe/H] < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06  [Fe/H] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  Relative density  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  2  3  4  5  6  Rp at P= lO days (R)4  3  ④  R  p  R  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  Age (Gyr)Deep radius valley with Kepler short cadence  9  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Same as Figure 6, but in 𝑅𝑝–𝑃–age space (left) and 𝑅𝑝–𝑃–[Fe/H] space (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Slope of the radius valley on the radius-period plane from various sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Source  𝑚 = d log 𝑅𝑝/d log 𝑃  Stellar type  Observations  This work  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  FGK  Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2018)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  FGK  Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2019)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  FGK  MacDonald (2019)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='319+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='088  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='116  FGK  Cloutier & Menou (2020)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='058+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='022  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='022  M  Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2021)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  M  Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2022)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  FGKM  Luque & Pallé (2022)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  M  Source  𝑚 = d log 𝑅𝑝/d log 𝑃  Model  Theory  Owen & Wu (2017)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='25 ≤ 𝑚 ≤ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='16  Photoevaporation  Lopez & Rice (2018)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09  Photoevaporation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11  Gas-poor formation  Gupta & Schlichting (2019)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11  Core-powered mass loss  Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2021)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='16  Photoevaporation  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11  Core-powered mass loss  studies (see again Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The period-radius slope was first observed  by Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2018), who used the SVM approach that we  adopted here and who found 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' A different approach  was followed by Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2019), who divided their planetary  sample into 10 bins with equal number of planets, determined the  minimum radius in each bin, and fitted a linear relationship to obtain  equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These two approaches led to a consistent result, with  𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' MacDonald (2019) adopted machine learning ap-  proaches, and report 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='319+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='088  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The above studies all focus  on samples of FGK stars, where various approaches to model the  valley’s location appear to result in negative slopes with consistent  magnitude, matching thermally-driven atmospheric loss models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  For smaller and cooler (M type) stars, Cloutier & Menou (2020)  found a positive slope (𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='058 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='022) using a method similar  to Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2019), suggesting for these stars the valley may be  the result of gas-poor formation rather than being thermally driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2021) used the SVM approach to measure the  M dwarf valley and found a negative slope instead, of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Luque & Pallé (2022) used the gapfit package (Loyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2020)  and found 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' A recent study by Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2022)  also included M type stars in addition to FGK stars, and they found  𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02 for this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Our sample does not include M  type stars but does span a mass range from about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='6 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4 𝑀★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  To investigate whether the slope of 𝑚 changes with stellar mass  within our sample, we split our planetary sample into two groups:  𝑀★ ≥ 1𝑀⊙, and 𝑀★ < 1𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We determine the 𝑅𝑝 − 𝑃 relation  separately for these two groups with the same methods as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We find for 𝑀★ ≥ 1𝑀⊙, 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04 and 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='35+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' for  MNRAS 000, 1–19 (2022)  Sub-Neptunes  Super-Earths  4  Rp (R)  2  1  1  3  10  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  100  P (days)  Age (Gyr)Sub-Neptunes  Super-Earths  4  Rp (R)  2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5  ¥3  1  0  10  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5  100  P (days)  [Fe/H]10  Ho & Van Eylen  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Radius valley location in terms of orbital period, stellar mass,  and stellar age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The colour bar represents the age of the planetary host stars,  and planes of different colours indicate the radius valley location for different  stellar ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  𝑀★ < 1𝑀⊙, 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07 and 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='35+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These results are  shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The two values are in agreement within 1𝜎,  suggesting that within our sample the radius valley location as a  function of orbital period is inconsistent with the gas-poor formation  scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  We can also look at the slope of the valley as a function of incident  flux (𝑆) rather than orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' By Kepler’s third law (as shown in  equation 7), planets at longer orbital periods are located further away  from the planet, thus the incident flux 𝑆 is lower for planets with  larger star-planet distances as shown in equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Hence, we expect  for a thermally-driven planetary mass loss scenario, the radius valley  location tends to larger planetary radii for higher 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We observe this  positive relationship in this work, in agreement with other previous  observations as shown in Table 7, and consistent with thermally-  driven mass loss models which is also shown in radius-period space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2 𝑅𝑝-𝑀★ relation supports a thermally-driven mass loss  model  As presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4, we find that in two dimensions, 𝑚 =  d log 𝑅𝑝/d log 𝑀★ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  A stellar mass dependence has been predicted by radius valley  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Both thermally driven mass-loss models predict a similar  dependence of the valley on stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' For example, Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  (2021) predicted 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='29 and 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='32 for photoevaporation  (Owen & Wu 2017) and core-powered mass loss models (Gupta &  Schlichting 2019, 2020) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Our results are consistent with  both sets of models within 1𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  A stellar mass dependence was observed by Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2020),  who find 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='21  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='16 by fitting the minima of the 2-dimensional  KDE in 𝑅𝑝 − 𝑀★ space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' A recent study by Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2022),  similarly following a binning approach and incorporating data from  Data Release 2 (DR2) of the California-Kepler Survey (CKS) for  cooler stars, estimated 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='18+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' It is therefore reassuring to  see that despite the different method adopted here, the slope derived  in this work is consistent with both of these studies within 1𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' For  lower mass stars, Luque & Pallé (2022) found 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' this  may be inconsistent with our results at 1𝜎, however the stellar mass  range they studied is significantly lower than that in our sample with  no overlaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The results are summarised in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  When extending our analysis to 3 dimensions as a function of  𝑃 and 𝑀★, we obtain 𝐴 = �𝜕 log 𝑅𝑝/𝜕 log 𝑃�  𝑀★ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03,  𝐵 = �𝜕 log 𝑅𝑝/𝜕 log 𝑀★  �  𝑃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='21+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08 from determining the radius  valley location in the 𝑅𝑝–𝑃–𝑀★ space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Note that this is different to  the total derivative d log 𝑅𝑝/d log 𝑀★ in two dimensions (shown in  Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Based on the models of photoevaporation (Owen & Wu  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Owen & Adams 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Mordasini 2020) and core-powered  mass loss (Gupta & Schlichting 2019, 2020), Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2021)  predicted 𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='19 for a photoevaporation model, and 𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='33  for a core-powered mass-loss model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Our resulting posterior distri-  bution of 𝐴 and 𝐵 determined from the bootstrapping presented in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4, as shown in Figure 12, is consistent with both the pho-  toevaporation and core-powered mass loss cases at 2𝜎, hence we  are unable to distinguish between the two models in this particular  parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2021) proposed an analysis of the radius valley in  𝑅𝑝–𝑆–𝑀★ space that could distinguish between the two different  thermally-driven mass-loss mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Using theoretical models,  they predicted the radius valley scales as a function of 𝑆 and 𝑀★  as equation 11, with 𝐴 = �𝜕 log 𝑅𝑝/𝜕 log 𝑆�  𝑀★ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='12 and 𝐵 =  �𝜕 log 𝑅𝑝/𝜕 log 𝑀★  �  𝑆 ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='17 for a photoevaporation model, and  𝐴 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08 and 𝐵 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='00 for a core-powered mass loss model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Again,  we plot the posterior distributions of 𝐴 and 𝐵 as shown in Figure 13,  and observe that our results are consistent with the core-powered  mass loss case well within 1𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' For the photoevaporation scenario,  our values overlap with the theoretical predictions at the edge of the  2𝜎 confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2021) also measured the planet  density of the California-Kepler Survey (CKS, Fulton & Petigura  2018), and the Gaia-Kepler Survey (GKS, Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2020), in  𝑅𝑝–𝑆–𝑀★ space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' They found for the CKS data, 𝐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05, 𝐵 =  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='21+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='33  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='39, and for the GKS data, 𝐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='10+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02, 𝐵 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='10  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Our results are in agreement with both the CKS and GKS values,  and our measurements have smaller uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  There are some caveats to this comparison between our observation  results and theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Firstly, the thermally-driven mass loss  models predict the slope of the bottom of the valley (Van Eylen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2021), whereas our SVM finds the slope  for the middle of the radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Some studies have suggested  a different planet size dependence with orbital period for super-  Earths and sub-Neptunes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2022), hence these two  slopes may not be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Since the radius valley is not completely  empty, the bottom of the radius valley is not clearly defined, and  there would be challenges locating and fitting the bottom of the  radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' As a result, our observed values may not be fully  comparable with theoretical model values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Furthermore, the method  of extracting the radius valley is prone to transit biases, which we  do not correct for in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2021) showed that  even when modelling synthetic transit surveys based on evolving  planets with theoretical models, the resulting posteriors may not be  fully consistent with the theoretically predicted slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Further work,  such as generating synthetic surveys from both photoevaporation and  core-powered mass loss models based on conditions similar to that  of our sample in a method similar to that performed in Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  (2021), and fitting the valley with the same method as in this work, or  analysing more planets around stars in a larger mass range, is required  to compare our observations to theoretical models in a homogeneous  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  MNRAS 000, 1–19 (2022)  Sub-Neptunes  Super-Earths  Valley at O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='1 Gyr  Valley at 1 Gyr  10  Valley at 10 Gyr  4  Age (Gyr)  3  2  d  R  1  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  1  ¥3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  10  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='7  100  P (days)  M* (Mo)Deep radius valley with Kepler short cadence  11  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Radius valley position for planets with host star mass 𝑀★ < 1𝑀⊙ (left), and 𝑀★ ≥ 1𝑀⊙ (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' For 𝑀★ < 1𝑀⊙, 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07 and 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='35+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  for 𝑀★ ≥ 1𝑀⊙, 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04 and 𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='35+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The green and blue points show planets above and below the radius valley respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The grey shaded  region represents the ±1𝜎 uncertainty in the radius valley position determined with bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Same as Table 6, but for the radius valley slope on the radius-incident-flux plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Source  𝑚 = d log 𝑅𝑝/d log 𝑆  Stellar type  Observations  This work  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  FGK  Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2019)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  FGK  Cloutier & Menou (2020)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='060 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='025  M  Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2022)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  FGKM  Luque & Pallé (2022)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  M  Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Same as Table 6, but for the radius valley slope on the radius-stellar-mass plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Source  𝑚 = d log 𝑅𝑝/d log 𝑀★  Stellar type  Observations  This work  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08  FGK  Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2020)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='21  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='16  FGKM  Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2022)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='18+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07  FGKM  Luque & Pallé (2022)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='12  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='12  M  Source  𝑚 = d log 𝑅𝑝/d log 𝑀★  Model  Theory  Gupta & Schlichting (2020)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='33  Core-powered mass loss  Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2021)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='29  Photoevaporation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='32  Core-powered mass loss  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='3 Deeper radius valley suggests a homogeneous initial  planetary core composition  We now turn to the depth of the radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Using the previously  defined depth metric (𝐸, equations 9 and 10), we find a valley depth  of 𝐸avg = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='60  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='47 (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We can compare this depth  to the valley observed by F18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Shifting the planets along the slope  calculated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2, and applying the same metric to their  filtered sample of 907 planets, we calculate 𝐸SN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='26  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='23, 𝐸SE =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='28+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='31  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='27, giving 𝐸avg = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='14+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='26  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='21 for that sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' For V18, we  shift the planets according to the slope obtained in their study, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04, and we find 𝐸SN = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='70  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='49, 𝐸SE = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='75+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='42  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='70,  giving 𝐸avg = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='87  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These values imply that compared to F18,  we observe a deeper radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' On the other hand, the radius  valley appears less deep than observed by V18 for a smaller sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  This finding is visualised in Figure 14, which shows the adjusted  histograms of the sample studied here next to the F18 and the V18  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  To investigate the reason for observing a deeper valley than F18, we  compare the 211 planets common in both our sample and the filtered  sample of F18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' To investigate the role of transit fitting, we convert  MNRAS 000, 1–19 (2022)  M*<1Mo  4  TO  TO  0  T  3  DT  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  ④  TOI  R  2  R  五  中  中中  T  1  下中  3  10  30  100  P (days)M*≥1Mo  4  TOITOI  O  TOI  T  TOI  Φ  3  0  TO  D  中  (田y)  TOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  2  R  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  1  3  10  30  100  P (days)12  Ho & Van Eylen  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Posterior distributions of the radius valley location dependence  with respect to orbital period at constant stellar mass �𝜕 log 𝑅𝑝/𝜕 log 𝑃�  𝑀★,  and stellar mass at constant orbital period �𝜕 log 𝑅𝑝/𝜕 log 𝑀★  �  𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The dark  and light-coloured shades represent the 1𝜎 and 2𝜎 uncertainties respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  The theoretical models of photoevaporation and core-powered mass loss are  taken from Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Same as Figure 12, but for �𝜕 log 𝑅𝑝/𝜕 log 𝑆�  𝑀★  and  �𝜕 log 𝑅𝑝/𝜕 log 𝑀★  �  𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  all our 𝑅𝑝/𝑅★ into 𝑅𝑝 using 𝑅★ from F18 (even when V18 values  are available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The results are shown in Figure 15, which compares  the same planets with the same stellar parameters but different transit  fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We observe that in this case, the 𝑅𝑝 of 56 (27%) and 24 (11%)  planets change by > 2𝜎 and > 3𝜎 respectively, compared to the  values reported in F18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We find for this common planetary sample,  for our planetary parameters, 𝐸SN = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='63+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='70, 𝐸SE = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='95+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='86  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='63,  giving 𝐸avg = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='29+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='91  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='59, whereas for parameters from F18, 𝐸SN =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='40+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='59  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='47, 𝐸SE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='93+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='49  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='39, giving 𝐸avg = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='18+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='49  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These findings  suggest that our updated transit fittings are directly responsible for  deepening (although not fully emptying) the radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  A deeper radius valley is associated with a more homogeneous  planet core composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' For example, in photoevaporation models  the radius valley position is dependent on the mass of the planet core  (𝑀𝑐), and the density of a 1𝑀⊕ core of a particular core composition  (𝜌𝑀⊕), as  𝑅valley ∝ 𝜌−1/3  𝑀⊕ 𝑀1/4  𝑐  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  (18)  Hence, if 𝑅valley is known, and the planets’ mean masses are known,  the planetary core compositions could be deduced (Owen & Wu  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Using the above relation, if the planetary cores were icy at forma-  tion, the radius valley would be located at a higher planetary radius  than if the cores were rocky/terrestrial at formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Hence, if the  planetary cores are of mixed composition, a superposition of the two  models would be predicted, and we would expect the radius valley  to be smeared and less distinct, as each type of planet would have  its own ‘radius valley’ at a different location (Owen & Wu 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Our deep radius valley found in this work implies the opposite case,  where the planetary cores are more similar in composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' In this  scenario, planets inside the valley may have a different (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' icy)  composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Owen & Wu (2017) compared their models to observations, and  found that the planet compositions are more likely to be Earth-like  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' rocky), but that the apparent shallowness of the valley suggested  a wide distribution of iron fractions ( 𝑓Fe) in their cores, as planets  with a single value iron fraction ( 𝑓Fe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5) produces a deeper  valley compared to planets with a uniform distribution ( 𝑓Fe ∈ [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Comparing our finding of a deeper valley to models in Owen & Wu  (2017) would indicate that the planet compositions are more likely  to have similar iron fractions with a narrower spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Similarly, in the core-powered mass loss model, the location of the  radius valley scales as  𝑅valley ∝ 𝜌−4/9  𝑐  (19)  where 𝜌𝑐 is the planet core density (Gupta & Schlichting 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  The same reasoning as the photoevaporation case then applies: given  the larger 𝜌𝑐 for icy cores, planets with homogeneous icy cores  will produce a radius valley at a larger planetary radii compared  to rocky/terrestrial cores, implying that the radius valley would be  smeared if planetary cores are of inhomogeneous compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Our  deep radius valley supports the opposite case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', a similar planetary  core composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Figure 16 shows the stellar parameter distributions for the planet  host stars in the three planet samples, and the mean and median  values are listed in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We notice a similar stellar parameter  range between this work and F18, however the stars in V18 are  brighter, have a larger mean radius and mass, and higher effective  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This is likely due to V18 selecting stars which display  strong asteroseismic signals, which usually are brighter and larger  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This observation may indicate that the radius valley of such  stars are emptier, however the details are left for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Despite our new results revealing that the radius valley deepens by  refitting planets with 1-minute short cadence light curves, it is still  uncertain whether the difference between results from this work and  F18 is solely due to the cadence in transit data used, as different meth-  ods are used in the transit fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Mullally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2015) fitted  planets using the method described in Rowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2014), which first  fits a multi-planet transit model to the light curves, with fixed limb  darkening parameters from Claret & Bloemen (2011), and subse-  quently fitting for each planet in a system independently by removing  photometric contributions of other planets based on the parameters  from the multi-planet fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' In our work, we fit planets in multi-planet  systems simultaneously, such that each system shares the same stellar  parameters including limb-darkening parameters and stellar density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  MNRAS 000, 1–19 (2022)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='1  Photoevaporation  Core-powered mass loss  0  This Work  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='1  (alog Rp/alog P)m,Photoevaporation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  Core-powered mass loss  This Work  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  alog Rp/alog S)mDeep radius valley with Kepler short cadence  13  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Histogram of planet radii, adjusted to 𝑃 = 10 days for the planetary population in this work (left, identical to Figure 4), F18 (centre), and V18  (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Planets in this work and F18 are shifted according to the slope defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2 with the SVM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02), whereas planets in V18 are shifted  according to the slope found in V18 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 𝑚 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The 𝐸 metrics defining the average peak-to-valley ratio are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='60  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='47, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='14+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='26  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='21, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='87  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='14  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Left: Radius-period plot of the 211 planets common to this work (blue) and F18 (red), using the same stellar radii to calculate planetary radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Right:  Same as Figure 14, but the planetary population is filtered to the 211 common planets in both samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Planets in both samples are adjusted to equivalent radii  at 𝑃 = 10 days according to the slope calculated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2 with the SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The red and blue histograms are produced with the parameters obtained from F18  and this work respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The histograms and fitting with the Gaussian Mixture Model show that the observed valley is deeper in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Mullally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2015) assumed a circular orbit when performing the  transit fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' On the contrary, we leave orbital eccentricity 𝑒 as a free  parameter, and place a prior on 𝑒 based on the expected distribution  from Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2019) and the stellar density 𝜌★ from F18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  However, most of the planets in our sample have near-circular orbits,  with over 85% of planets having 𝑒 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Therefore planetary orbital  eccentricity is not sufficient to explain the difference between the  two results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The possible presence of TTVs also do not contribute  to the discrepancy as planets with known TTVs are excluded in our  like-for-like planet comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' In fact, when fitting transits using  identical methods, precisions in 𝑅𝑝/𝑅★ obtained from fitting transit  light curves of shorter cadences has been found to substantially im-  prove, compared to 30-minute cadence light curves (Camero, Ho &  Van Eylen, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We therefore expect the photometry cadence to  contribute significantly to the difference in the views of the radius val-  ley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Further work, such as refitting the long cadence data of the same  planet population with identical transit fitting methods, is needed to  further investigate the effect of light curve cadence on planet param-  eter estimates and the radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We leave such considerations for  future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4 Radius valley relation with stellar age consistent with  core-powered mass loss model  In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5, we present a positive relationship between the radius  valley location and stellar age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Photoevaporation is predicted to occur  in the first 100 Myr of the planet’s formation (Owen & Wu 2017),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='Rp at P= 10 days (R)14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='Ho & Van Eylen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Distribution of host stars parameters in our work, compared with F18 and V18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The properties shown here, from top left to bottom right, are stellar  radius 𝑅★, mass 𝑀★, Kepler magnitude Ksmag, and effective temperature 𝑇eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' For systems with multiple transiting planets, stars are counted multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Average values of stellar properties of the host stars in the planetary sample used in this work, compared with F18 and V18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' ¯𝑥 and ˜𝑥 represent the mean  and median values of the parameter 𝑥 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Sample  ¯𝑅★ (𝑅⊙)  ˜𝑅★ (𝑅⊙)  ¯  𝑀★ (𝑀⊙)  ˜  𝑀★ (𝑀⊙)  ¯  Ksmag  ˜  Ksmag  ¯  𝑇eff (K)  ˜  𝑇eff (K)  This work  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='95  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='27  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='31  5587  5630  F18  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='72  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='97  5788  5860  V18  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='14  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='49  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='57  5980  5952  core-powered mass loss occurs throughout the main-sequence life-  time of the stars, on Gyr timescales (Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Gupta &  Schlichting 2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Hence, in the photoevaporation case, the  radius valley is expected to be located at a constant radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' On the  other hand, in the core-powered mass loss case, the radius valley  shifts to higher planet radii for older systems, as the atmospheres of  planets with more massive cores are stripped off later in the evolution  process than their less massive counterparts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Rogers & Owen 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Our results reveal a weak positive radius valley dependence on  the stellar age, which is consistent with the core-powered mass loss  scenario, as is the observed radius valley dependence on stellar mass  as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' However, a small age dependence does  not preclude photoevaporation, since even in this scenario a subset of  planets may still lose their atmospheres and evolve at Gyr timescales  (David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2021), and we are unable to observe  stars younger than 100 Myr and hence cannot rule out the possibility  of a dominant photoevaporation effect on planets at the early stages  of the stars’ lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Also, stellar age measurements are highly un-  certain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' the mean percentage uncertainty in stellar age for our sample  is 54%, hence there is also a probability that some stars are younger  than observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Table 10 lists the 50 planets located inside the radius valley in our  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' To do so, we here defined the new radius valley region as  the area bounded by the two lines passing through the supporting  vectors in the 4D SVM model in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5, given by equation 15  with 𝐴 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='096, 𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='231, 𝐶 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='033, 𝐷lower = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='272 for the  lower line, and 𝐷upper = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='405 for the upper line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These planets  are potentially interesting for future characterisation study as their  MNRAS 000, 1–19 (2022)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='6  F18  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  V18  This work  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  Density  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='6  M* (Mo)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  F18  V18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='0  8  10  12  14  16  Ksmag1e-4  16  F18  14  V18  This work  12  Density  10  8  6  4  2  0  4500  5000  5500  6000  6500  7000  Teff (  (K)Deep radius valley with Kepler short cadence  15  Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Metallicity and mass of the host stars used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The grey  dotted line [Fe/H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06 shows the cut-off between stars of low and high  metallicity as defined in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We note a lack of stars with low metallicity  and large stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  atmospheres and interiors may provide additional insights regarding  formation and evolution mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5 Radius valley depth varies with stellar metallicity  In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='6, we show a higher average 𝐸 value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' a deeper radius  valley) for planets around metal-poor stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This seems to contradict  the suggestion that the radius valley is deeper for planets around  metal-rich stars (Owen & Murray-Clay 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' However, we note  from Figure 17, that in our sample, the metal-rich host stars span a  wider range of stellar masses, due to lack of metal-poor stars with  large radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' As from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4 we observe that the radius valley  depends on stellar mass as well, the superposition of the radius  valley for different stellar masses potentially smears the gap, making  the radius valley appear shallower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  The degeneracy between stellar mass and metallicity is not fully  resolved, hence we are unable to determine the sole effect of stellar  metallicity on the radius valley in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We therefore consider  the results related to metallicity to be inconclusive and in need of  further future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  5 CONCLUSION  In summary, we performed transit light curve fitting on 431 plan-  ets using Kepler 1-minute short cadence data, the vast majority of  which have not been previously analysed homogeneously using short  cadence observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' In this paper, we presented their revised plan-  etary parameters, which in some cases differ substantially from those  previously reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' These differences are unrelated to stellar param-  eters but may be related to the details of the transit fitting approach  or the shorter observing cadence, the effects of which should be  disentangled in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  By statistically analysing the small close-in planets in our sam-  ple, we observed a radius valley which is deeper than that reported  in several other studies, although not entirely empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The valley’s  depth likely implies a homogeneous initial planetary core composi-  tion where the planets are similar in composition at formation, and  likely to have similar iron fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We provide a table of those plan-  ets that appear to be inside the valley, as they may warrant further  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  The radius valley has a strong dependence on planetary orbital pe-  riod and the mass of the host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' It also displays a weak dependence  on the stellar age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We compared several possible radius valley models  using support vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We determined that the radius valley  can best be described in four dimensions using the formula  𝑅𝑝,valley ∝ 𝑃𝐴𝑀𝐵  ★ (age)𝐶  (20)  with 𝐴 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='096+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='023  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='027, 𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='231+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='053  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='064, and 𝐶 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='033+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='017  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Comparing our radius valley dependencies with theoretical mod-  els, we found that in 𝑅𝑝–𝑆–𝑀★ space, our posterior distributions  are most consistent with core-powered mass loss, where they agree  within less than 1𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The models are also consistent with photoevap-  oration scenarios at ≈ 2𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We did not find a significant dependence  of the radius valley on stellar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  With the Transiting Exoplanet Survey Satellite (TESS, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Ricker  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2015) now in its extended mission, and the upcoming launch of  the PLAnetary Transits and Oscillations of stars (PLATO) mission  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Rauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2014), such future planetary studies could drastically  increase the number of planets with radii measurements and hence  provide an even more detailed view of the radius valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This work  highlights the impact of careful transit fitting using short, 1-minute  cadence observations to obtain precise planetary radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' This will  likely be of key importance to derive precise planetary radii using  transit observations from ongoing and future missions, which will  ultimately allow us to better understand the formation and evolution  of small close-in planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' would like to thank the Science and Technology Facilities  Council (STFC) for funding support through a PhD studentship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We  would like to thank Erik Petigura, James Rogers, and James Owen  for insightful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' We also thank the anonymous reviewer for  taking their time to review the paper, and for their valuable sugges-  tions which have improved the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  DATA AVAILABILITY  The Kepler 1-minute short cadence light curves are available for  download on the NASA Mikulski Archive for Space Telescopes  (MAST) database1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The parameter estimates from HMC posteriors  are provided in the Appendix tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  REFERENCES  Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content=', 2021, The Exoplanet Characterization Toolkit (ExoCTK),  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4556063, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  4556063  Claret A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', Bloemen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', 2011, A&A, 529, A75  Cloutier R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', Menou K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', 2020, AJ, 159, 211  David T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', 2021, AJ, 161, 265  1 https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='edu  MNRAS 000, 1–19 (2022)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  Fe/H]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4  M* (Mo)16  Ho & Van Eylen  Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' List of planets inside the radius valley as defined by this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The coordinates (RA and Dec) are taken from Kepler Q1-17 Data Release 25 catalogue  (Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2018), except for K02533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03, where data is taken from Kepler Q1-16 catalogue (Mullally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  KOI  Kepler name  𝑃 (days)  𝑡0 (BJD-2454833)  𝑅𝑝/𝑅★  𝑅𝑝 (𝑅⊕)  RA (deg)  Dec (deg)  K00049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  Kepler-461 b  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='313784 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000015  175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='9915 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0287 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0010  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='17  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='17  292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='24902  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='164822  K00070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  Kepler-20 A d  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='611598 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000019  164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='7274 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0263 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0003  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='52+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07  287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', Sari R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content=', Albrecht S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', 2015, ApJ, 808, 126  Van Eylen V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', Agentoft C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', Lundkvist M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', Kjeldsen H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', Owen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', Fulton  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', Petigura E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', Snellen I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', 2018, MNRAS, 479, 4786  Van Eylen V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', 2019, AJ, 157, 61  Van Eylen V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=', 2021, MNRAS, 507, 2154  APPENDIX A: EXTRA MATERIAL  MNRAS 000, 1–19 (2022)  18  Ho & Van Eylen  Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Host star parameters of planets fitted in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 𝑅★, 𝑀★, 𝑇eff, [Fe/H], Ksmag are taken from the following sources: 1: Fulton & Petigura (2018), 2: Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Stellar ages are taken from  Fulton & Petigura (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Radius correction factors (RCFs) are taken from Furlan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' we take RCF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0000 where there are no measurements in Furlan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  𝜌★, 𝑢0, 𝑢1 are resulting parameters from the transit fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Only the first 10 host stars are shown here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' the full table is available online in a machine-readable format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  KOI  𝑅★ (𝑅⊙) (1)  𝑀★ (𝑀⊙) (1)  𝑇eff (K) (1)  [Fe/H] (dex) (1)  Ksmag (1)  𝑅★ (𝑅⊙) (2)  𝑀★ (𝑀⊙) (2)  𝑇eff (K) (2)  [Fe/H] (dex) (2)  Ksmag (2)  Age (Gyr)  RCF  𝜌∗ (g cm−3)  𝑢0  𝑢1  41  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='34+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='35  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='894 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='049  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='13  94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='37+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='18+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  6181+60  −60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='926  N/A  N/A  N/A  N/A  N/A  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='47+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='64  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='56  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='462 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='021  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='16  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  Table A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Planetary parameters from transit fits performed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' 𝑃, 𝑡0, 𝑅𝑝/𝑅★, 𝑏, 𝑒, 𝜔 are obtained directly from fitting, 𝑎/𝑅★ and 𝑆 are indirectly calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' The 𝑅𝑝/𝑅★ values from 1: Fulton & Petigura  (2018) and 2: Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' (2018) are also included for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' Only the first 10 planets are shown here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content=' the full table is available online in a machine-readable format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='  KOI  Kepler name  𝑃 (days)  𝑡0 (BJD-2454833)  𝑅𝑝/𝑅★  𝑅𝑝/𝑅★ (1)  𝑅𝑝/𝑅★ (2)  𝑏  𝑒  𝜔 (◦)  𝑎/𝑅∗  𝑆 (𝑆⊕)  K00041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  Kepler-100 c  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='815893 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000008  122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='9476 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0138 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0140+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0001  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0134+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0001  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  30 ± 96  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='36 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='20  245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='37 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='59  K00041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  Kepler-100 b  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='887062 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000007  133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='1781 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0082 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='0013  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0079+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  3 ± 107  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='88  590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='40 ± 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='33  K00041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  Kepler-100 d  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='333093 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000019  153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='9835 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0009  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0104 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='0023  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0092+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  1 ± 104  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='50 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='51  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='87 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='13  K00046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  Kepler-101 c  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='029792 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000020  132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='4816 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0073 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='0003  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  −0 ± 103  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='77  646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='80 ± 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='97  K00049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  Kepler-461 b  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='313784 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000015  175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='9915 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0287 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0259+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0003  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0003  N/A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='13  77 ± 77  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='75 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='22  213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='71 ± 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='30  K00069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  Kepler-93 b  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='726739 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000001  134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='9265 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0151 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0159+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='0149+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0001  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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+page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09  105 ± 58  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='39  252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='14 ± 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  K00070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='01  Kepler-20 A c  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='854089 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000003  138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='6082 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0290 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0300+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0018  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0006  N/A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  86 ± 39  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='80  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='81 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='80  K00070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='02  Kepler-20 A b  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='696115 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000001  134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='5021 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0182 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0209+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0014  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0023  N/A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  26 ± 101  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='83  365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='46 ± 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='71  K00070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='03  Kepler-20 A d  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='611598 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='000019  164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='7274 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0263 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0259+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0005  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='0003  N/A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='04  33 ± 96  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='67 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
+page_content='41  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfswgq/content/2301.04062v1.pdf'}
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diff --git a/T9E5T4oBgHgl3EQfAg7p/content/tmp_files/2301.05380v1.pdf.txt b/T9E5T4oBgHgl3EQfAg7p/content/tmp_files/2301.05380v1.pdf.txt
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+Prompting Neural Machine Translation with Translation Memories
+Abudurexiti Reheman1, Tao Zhou1, Yingfeng Luo1, Di Yang2, Tong Xiao1,2, Jingbo Zhu1,2*
+1School of Computer Science and Engineering, Northeastern University, Shenyang, China
+2NiuTrans Research, Shenyang, China
+rexiti neu@outlook.com, zhoutao neu@outlook.com, luoyingfengmail@163.com,
+yangdi@niutrans.com, {xiaotong, zhujingbo}@mail.neu.edu.cn
+Abstract
+Improving machine translation (MT) systems with translation
+memories (TMs) is of great interest to practitioners in the MT
+community. However, previous approaches require either a
+significant update of the model architecture and/or additional
+training efforts to make the models well-behaved when TMs
+are taken as additional input. In this paper, we present a sim-
+ple but effective method to introduce TMs into neural ma-
+chine translation (NMT) systems. Specifically, we treat TMs
+as prompts to the NMT model at test time, but leave the train-
+ing process unchanged. The result is a slight update of an ex-
+isting NMT system, which can be implemented in a few hours
+by anyone who is familiar with NMT. Experimental results
+on several datasets demonstrate that our system significantly
+outperforms strong baselines.
+Introduction
+Integrating TM is one of the commonly used techniques to
+improve real-world MT systems. In TM-assisted MT sys-
+tems, it is often assumed that there is a database in which
+high-quality bilingual sentence pairs are stored. When trans-
+lating an input sentence, the most (or top-K) similar sen-
+tence pair, which is retrieved from TM, is used to optimize
+the translation. From the perspective of practical application,
+this approach is particularly useful for MT, especially when
+sentences are highly repetitive, such as in translating tech-
+nical manuals, legal provisions, etc. Previous works show
+that translation quality can be significantly improved when
+a well-matched TM sentence pair is provided both in Sta-
+tistical Machine Translation (SMT) (Ma et al. 2011; Wang,
+Zong, and Su 2013; Li, Way, and Liu 2014) and Neural Ma-
+chine Translation (NMT) (Gu et al. 2018; Khandelwal et al.
+2020).
+However, there are two major problems with this type of
+work in real-world applications. First, it is difficult to find
+such a TM dataset in most cases, especially when users can
+not share their TM data with the public for some reason.
+Second, previous approaches often require model changes,
+including training the model with TM (Bult´e and Tezcan
+2019; Hossain, Ghazvininejad, and Zettlemoyer 2020; Xu,
+*Corresponding author.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+Encoder
+Decoder
+𝑥1
+𝑡𝑚 ··· 𝑥𝑚
+𝑡𝑚 𝑥1
+··· 𝑥𝑛
+<bos> 𝑦1
+𝑡𝑚··· 𝑦𝑘
+𝑡𝑚 𝑦1 ··· 𝑦𝑙
+𝑦1
+𝑡𝑚··· 𝑦𝑘
+𝑡𝑚 𝑦1 ··· 𝑦𝑙 <eos>
+Force decoding
+Figure 1: Structure of the proposed method. The source and
+target sentences of TM are concatenated with the input sen-
+tence and hypothesis in a specific concatenation template,
+respectively. The tokens in the target TM together with the
+concatenation template are generated in a forced manner.
+The lengths of the source and target TM together with the
+concatenation templates are m and k, and the lengths of the
+input sentence and hypothesis are n and l, respectively.
+Crego, and Senellart 2020), changing the NMT model ar-
+chitecture for TM integration (Gu et al. 2018; Bapna and
+Firat 2019; Xia et al. 2019; He et al. 2021), and introducing
+additional modules (Zhang et al. 2018; He et al. 2019; Khan-
+delwal et al. 2020). In this case, it is difficult to incorporate
+TM into the NMT system even if TM data is provided, since
+TM incorporation can not be accomplished on a generic de-
+coder, and a deeply customized decoder is needed.
+Here, we address this problem by using few-shot learning
+(Wang et al. 2021b), which enables the system to quickly
+adapt to a small number of samples. The recent prevalence
+of prompt-based approaches (Brown et al. 2020), which
+transfer the original task into a generation task by design-
+ing an appropriate template without modifying the language
+model, gives us some inspiration that the retrieved TM can
+prompt the translation of the input sentence without modify-
+ing the NMT model.
+Based on this idea, we propose a simple approach to
+quickly adapt the NMT model in the few-shot TM scenario.
+Specifically, we treat TMs as prompts to the NMT model
+during the decoding process, with very small changes to the
+decoder. Our method can cover the advantages of conven-
+tional TM augmented methods and bring some new ideas,
+such as incorporating users’ local historical translation data
+into NMT.
+arXiv:2301.05380v1  [cs.CL]  13 Jan 2023
+
+In order to prompt the translation with the retrieved TM,
+we design several templates to concatenate the source TM
+with the input sentence and feed the concatenated sentence
+into the model encoder. On the decoder side, we generate
+the target TM and the concatenation template in a forced
+way first, then let the model generate the other parts au-
+tomatically. Regarding TM granularity, our method works
+well on sentence-level and fragment-level TM by design-
+ing appropriate templates. Experimental results on several
+datasets show that our method can further improve the trans-
+lation quality on strong NMT models, and with comparable
+performance with the state-of-the-art.
+Background
+NMT Decoding
+Suppose x = {x1, ..., xn} is the source sentence, and NMT
+translates it into the corresponding target sentence y =
+{y1, ..., ym} by using a trained NMT model. In practice, it
+turns the decoding into a searching problem, and a beam
+searcher is adopted to get the target sentence with the high-
+est generation probability. Generally, an NMT model gener-
+ates in an auto-regressive way. Therefore, the generation of
+each token relies on the source sentence and the generated
+prefix of the target sentence. The generation of the whole tar-
+get sentence can be formulated as a conditional probability
+P(y|x) described below:
+P(y|x) =
+m
+�
+i=0
+P(yi|x, y<i)
+(1)
+where y<i = {y1, y2, ..., yi−1} denotes the generated prefix
+tokens of target sentence at time-step i.
+TM
+TM is a database of language pairs that stores seg-
+ments (such as fragments, sentences, paragraphs, or other
+sentence-like units) that have previously been translated by
+human translators for later use. It can provide identical or
+similar segments to help translate the input sentence. In the
+early stage, TM was widely used in Computer Aided Trans-
+lation (CAT) (Dillon and Fraser 2006). When translating an
+input sentence, the target sentence of TM is returned as the
+answer when an identical source sentence is found. In most
+cases, the translation result is obtained by fixing the most
+similar or top-K similar sentences retrieved from TM, since
+it is difficult to find a completely identical sentence. In the
+MT environment, TM is playing the same role as in CAT.
+Many approaches have been proposed to incorporate similar
+TM into MT systems to get a more accurate translation.
+Methodology
+In this section, we introduce our proposed method in detail
+from the perspectives of incorporating TM into NMT de-
+coding, designing templates for concatenation, and retriev-
+ing similar TM.
+Incorporating TMs into NMT Decoding
+We incorporate TMs into the decoding process. For an input
+sentence x and a retrieved TM sentence pair ⟨xtm, ytm⟩, we
+concatenate xtm and x in a specific concatenation template
+and feed it into the model encoder. On the decoder side, we
+first force the model to generate the exact tokens in ytm to-
+gether with the concatenation template. Then the rest of the
+generation, which is returned as the answer, is done automat-
+ically without interfering. In practice, we set the generation
+probability of the tokens in the target TM and the concate-
+nation template as 1 during the force decoding phase.
+Concatenation Templates
+As our method incorporates TM on a pre-trained NMT
+model with no modifications, we must adhere to the follow-
+ing principles when designing the concatenation template:
+tokens in the concatenation template must be recognized by
+the NMT model and each part of the concatenated sentence
+maintain relatively complete semantics. Following this idea,
+we designed several templates for TMs in different gran-
+ularity. In concatenation, the TM comes first on both the
+source and target sides, and a specific template is used on
+both sides.
+Sentence-level TM.
+We concatenate sentence-level TMs
+in five different templates. As the example shown in Table
+1, our designed templates for sentence-level TMs are as fol-
+lows:
+(a) Concatenate directly. We concatenate them directly
+and add a period at the end of TMs if it is not ended up
+with punctuation marks.
+(b) Concatenate with comma. Before concatenation, we
+replace the punctuation mark at the end of TMs with a
+comma or add a comma there if it is not ended up with any
+punctuation mark.
+(c) Concatenate with semicolon. The concatenation is the
+same as the comma concatenation process, using a semi-
+colon instead of the comma.
+(d) Concatenate with conjunctions. In this template, we
+adopt conjunctions that express juxtaposed semantics, such
+as “and” in English and “und” in German. Specifically, we
+add a period at the end of the source and target TM if they are
+not ended up with any punctuation mark, then add a conjunc-
+tion word in the corresponding language and a comma after
+that. Then we concatenate them with the input sentence and
+the hypothesis on the source and target side, respectively.
+(e) Enclose in parentheses. We enclose both source and
+target TMs in parentheses, then perform the concatenation.
+Fragment-level TM.
+Unlike the sentence-level TM, we
+do some preprocessing on fragment-level TMs. First, we ob-
+tain the common fragments between the input sentence and
+source TM. Then, the tokens which are the translation of the
+words in the above common fragments are acquired from
+the target TM, using word alignment tools. After that, we
+construct the fragment-level TM using the tokens from the
+source and target TM above. An example of common frag-
+ments between input sentence and source TM and the word
+
+Input Sentence
+She gave us a full account of the traffic accident .
+Source TM
+She gave the police a full account of the incident .
+Target TM
+Sie gab der Polizei einen voll@@ st¨andigen Bericht ¨uber den Vorfall .
+Sentence Level TMs
+Directly
+En/input
+She gave the police a full account of the incident . She gave us a full account of the traffic accident .
+De/input
+<bos> Sie gab der Polizei einen voll@@ st¨andigen Bericht ¨uber den Vorfall . {Hypothesis}
+Comma
+En/input
+She gave the police a full account of the incident , She gave us a full account of the traffic accident .
+De/input
+<bos> Sie gab der Polizei einen voll@@ st¨andigen Bericht ¨uber den Vorfall , {Hypothesis}
+Semicolon
+En/input
+She gave the police a full account of the incident ; She gave us a full account of the traffic accident .
+De/input
+<bos> Sie gab der Polizei einen voll@@ st¨andigen Bericht ¨uber den Vorfall ; {Hypothesis}
+Conjunction
+En/input
+She gave the police a full account of the incident . And , She gave us a full account of the traffic accident .
+De/input
+<bos> Sie gab der Polizei einen voll@@ st¨andigen Bericht ¨uber den Vorfall . Und , {Hypothesis}
+Parenthesis
+En/input
+( She gave the police a full account of the incident . ) She gave us a full account of the traffic accident .
+De/input
+<bos> ( Sie gab der Polizei einen voll@@ st¨andigen Bericht ¨uber den Vorfall . ) {Hypothesis}
+Fragment Level TMs
+Parenthesis
+En/input
+( She gave ) ( a full account of the ) She gave us a full account of the traffic accident .
+De/input
+<bos> ( Sie gab ) ( einen voll@@ st¨andigen Bericht ¨uber den ) {Hypothesis}
+Table 1: An example of model input in our proposed method. Here, Directly, Comma, Semicolon, Conjunction, and Parenthesis
+denote our designed templates for concatenation, and En/input and De/input denote the input of the encoder and the decoder,
+respectively. The <bos> token denotes the begin-of-sentence tag, and {Hypothesis} denotes the automatically generated part
+of the target sentence. Red tokens denote the concatenation templates.
+Input Sentence:  She  gave   us   a  full  account  of   the   traffic   accident   .
+Source TM:
+She   gave    the  police    a     full     account     of     the     incident   .
+Target TM:
+Sie  gab   der Polizei  einen  voll@@  ständigen  Bericht  über  den  Vorfall   .
+Figure 2: An example of obtaining fragments for fragment-level TMs. For a given bilingual TM, common fragments between
+the input sentence and the source TM are acquired first, then the words in the target TM that align with the words in the common
+fragments are extracted. Common fragments and their corresponding fragments in Target TM are tagged by the same color box,
+and the lines denote word alignments.
+alignment between the source and target TM is given in Fig-
+ure 2, and its corresponding fragment-level TM is given in
+Table 1.
+Specifically, for a given input sentence x and a retrieved
+bilingual TM ⟨xtm, ytm⟩, we acquire the encoder and de-
+coder input for the NMT model in the following steps:
+(a) Perform the longest common subsequence matching
+algorithm to x and xtm, and obtain the longest common sub-
+sequence Ps = {w1, w2, ..., wm}.
+(b) Use word alignment tools to xtm and ytm, and get
+the aligned subsequence Pt = {w′
+1, w′
+2, ..., w′
+n}, which is
+corresponding to Ps, from ytm.
+(c) Group the words, which appear continuously in xtm,
+from Ps in their original order to form source TM fragments
+P ′
+s = {f1, f2, ..., fi}.
+(d) Group the words, which appear continuously in ytm,
+from Pt in the order of the correspondence with P ′
+s to form
+target TM fragments P ′
+t = {f ′
+1, f ′
+2, ..., f ′
+j}.
+(e) Concatenate each fragment in P ′
+s and P ′
+t with a spe-
+cific template to form fragment-level TM, and directly con-
+catenate them with input sentence and hypothesis as is done
+in sentence-level TM.
+As for the concatenation template, we enclose each frag-
+ment in parenthesis to maintain its semantic integrity. In
+practice, we remove the fragments that consist of a single
+stop word and remove the punctuation at two sides of a frag-
+ment.
+Retrieving Similar TMs
+To retrieve the most similar bilingual TM for the input sen-
+tence, we use a word-level fuzzy matching strategy and re-
+move punctuations and numbers from the sentence. Instead
+of retrieving from the whole TM database, we first employ
+the search engine library Apache Lucene (Bialecki, Muir,
+and Ingersoll 2012) to retrieve the top 500 similar bilingual
+sentences from TM. Then we rerank them by adopting Fuzzy
+
+Match Score (FMS) to obtain the most similar TM sentence
+pair. FMS is a length normalized Levenshtein Distance (Li
+and Liu 2007), known as Edit Distance:
+FMS(x, xtm) = 1 −
+LD(x, xtm)
+max(|x|, |xtm|)
+(2)
+where LD(·, ·) denotes the word level Levenshtein Distance,
+and | · | denotes word level length of a sentence.
+Experiments
+In order to verify the validity of our proposed method, we
+conducted several experiments on TM specialized transla-
+tion task and domain adaptation task, respectively. We also
+put our approach into practice on a commercial NMT system
+to assess its usability in the practical setting. In the end, we
+investigated the impact of the NMT model, TM similarity,
+and input sentence length on translation quality.
+Datasets and Models
+For TM specialized translation tasks, we evaluated our
+method on two datasets: 1) DGT-TM, the entire body
+of European legislation in 22 European languages, on
+German-English in both directions (En-De and De-En) and
+2) United Nations Parallel Corpus (UNPC), consisting of
+United Nations General Assembly Resolutions with transla-
+tions in the six official languages, on English-Chinese (En-
+Zh), Russian-Chinese (Ru-Zh) and French-Chinese (Fr-Zh).
+These two datasets are relatively easy to retrieve TM sen-
+tences with a high degree of similarity. For the test set and
+TM database, we cleaned the above corpora first, then ran-
+domly selected 3,000 sentence pairs for the test dataset,
+whereas the remaining corpora were utilized as the TM
+database.
+In addition, we performed an experiment using a home-
+made English-Chinese dataset (denoted as H-m in Table 2)
+of 3401 sentences. Each test sentence has one bilingual TM
+sentence whose source side is similar to the test sentence.
+The statistics of these TM databases and the TM similarity
+ratios of retrieved TMs in FMS metric are shown in Table 2.
+In the domain adaptation task, following Khandelwal
+et al. (2020), we used the multi-domain datasets from Aha-
+roni and Goldberg (2020), which contains German-English
+bilingual datasets in five different domains: Medical, Law,
+IT, Koran, and Subtitles, respectively. We treated the train-
+ing data in each domain as our TM database.
+After cleaning the above corpora and splitting them into a
+test set and TM database, we retrieved the most similar TM
+for each test sentence from the TM database in an offline
+way and applied BPE (Sennrich, Haddow, and Birch 2016)
+to the test sets and TM with the BPE-codes provided by the
+pre-trained NMT models. To obtain the alignment informa-
+tion for the tokens in the TM source and target sentence, we
+trained a word aligner – Mask-Align – as proposed in Chen,
+Sun, and Liu (2021), and constructed corresponding frag-
+ment level TMs. The TM data scale and the TM similarity
+ratios of retrieved TMs in FMS metric are given in Table 3.
+As for the pre-trained NMT model, we applied Face-
+book’s WMT19 De-En, En-De model (Ng et al. 2019) and
+Corpus
+Lang
+TM
+scale
+TM FMS ratio
+[0,
+0.2)
+[0.2,
+0.4)
+[0.4,
+0.6)
+[0.6,
+0.8)
+[0.8,
+1.0)
+DGT
+-TM
+En-De
+3.1M
+3%
+27%
+19%
+21%
+31%
+De-En
+3.1M
+4%
+30%
+18%
+22%
+26%
+UNPC
+En-Zh
+3.9M
+2%
+44%
+22%
+11%
+22%
+Fr-Zh
+11.5M
+2%
+45%
+18%
+11%
+23%
+Ru-Zh
+11.2M
+8%
+46%
+16%
+9%
+20%
+H-m
+En-Zh
+-
+18%
+30%
+35%
+23%
+7%
+Table 2: Sentence numbers in the TM databases and the sim-
+ilarity ratios of the retrieved TM.
+Domain
+TM
+scale
+TM FMS ratio
+[0,
+0.2)
+[0.2,
+0.4)
+[0.4,
+0.6)
+[0.6,
+0.8)
+[0.8,
+1.0)
+Medical
+248K
+7%
+23%
+20%
+17%
+33%
+Law
+467K
+8%
+31%
+18%
+14%
+28%
+IT
+223K
+14%
+18%
+28%
+26%
+14%
+Koran
+18K
+2%
+26%
+33%
+28%
+11%
+Subtitles
+500K
+3%
+27%
+43%
+23%
+4%
+Table 3: Sentence numbers in the TM database in each do-
+main and the similarity ratios of the retrieved TM.
+NiuTrans’ WMT20 En-Zh model (Zhang et al. 2020) as our
+base model. All of these models are very competitive that
+they trained on more than 20 million training data and 10
+million extra back-translated data.
+Main Experiment
+Our main experiment involves the TM specialized transla-
+tion task, the domain adaptation task, and the implementa-
+tion on commercial NMT system.
+TM Specialized translation.
+In this experiment, we de-
+coded the DGT-TM En-De and De-En test sets using face-
+book’s WMT19 En-De and De-En models (Ng et al. 2019),
+respectively. Besides, we decoded the UNPC En-Zh test set
+and the homemade En-Zh test set with NiuTrans’ WMT20
+En-Zh model (Zhang et al. 2020). For the Mask-Align model
+training, we used WMT20 En-Zh training data for the En-Zh
+aligner and DGT-TM’s TM database for En-De and De-En
+aligner. From the experimental results in Table 4, we have
+the following observations.
+First, in sentence-level TM, the BLEU score on DGT-TM
+De-En, En-De, and homemade En-Zh test sets increased sig-
+nificantly, with maximum BLEU score increases of 8.63,
+5.74, and 7.74 points, respectively. Meanwhile, the transla-
+tion improved slightly on the UNPC En-Zh test set in di-
+rectly, semicolon, and parenthesis concatenations. Second,
+in fragment-level TM, the BLEU score increased by about
+1 to 2 points or even decreased, compared to the baseline
+(without TM). The main reason for this is that the NMT
+model is trained on sentence-level training data rather than
+sentence pieces. In addition, it is also affected by the perfor-
+mance of the word aligner, which may provide error align-
+
+ment information.
+Corpus
+DGT-TM
+UNPC
+H-m
+Lang
+De-En
+En-De
+En-Zh
+En-Zh
+W/o TM
+45.40
+39.03
+41.42
+46.43
+Sentence TM
+Directly
+53.74
+44.32
+41.70
+52.97
+Comma
+52.44
+43.03
+41.33
+51.85
+Semico
+53.42
+44.54
+42.31
+52.89
+Conjunc
+53.65
+44.00
+41.15
+54.17
+Parenth
+54.03
+44.77
+41.90
+53.87
+Fragment TM
+47.21
+41.65
+39.85
+47.67
+Table 4: Experimental results on the DGT-TM En-De, De-
+En, UNPC En-Zh, and the homemade En-Zh test sets.
+Domain Adaptation.
+Following the kNN-MT (Khandel-
+wal et al. 2020) and its optimized counterparts, we con-
+ducted the domain adaptation experiment and compared our
+method with kNN-MT. We applied Facebook’s WMT19 De-
+En model (Ng et al. 2019) for decoding. Experimental re-
+sults are given in Table 5.
+From the table, we can find that our method improves the
+translation in all domains except Subtitles, with maximum
+BLEU score improvements of 2.54, 3.05, 4.64 in IT, Law,
+and Medical domains, respectively, whereas in Koran the
+result is only 0.42 BLEU score higher. The fragment-level
+TM method has the same tendency as the above experiment,
+which is slightly higher only in IT and Medical domain than
+the baseline. Besides, the improvement of our method is less
+than kNN-MT in every domain. The original design of our
+approach leads to this result. Our method retrieves the most
+similar single TM and leverages the knowledge it contains
+to improve the translation. How to leverage the knowledge
+provided by TM is fully dependent on the NMT model itself.
+While kNN-MT introduces an extra module to incorporate
+the information explicitly from multiple similar context vec-
+tors.
+The main advantage of our method over kNN-MT is that
+our method performs the retrieval based on string similarity,
+and there is no need to store the context vectors, which saves
+a lot of storage space. At the same time, kNN-MT searches
+the context vectors in each beam in every timestep, which
+is much slower than the vanilla NMT. However, our method
+searches the TM only once and generates two sentences (tar-
+get TM and the hypothesis) in the way of a vanilla NMT.
+This will make our method much faster than kNN-MT.
+Implementation on Commercial NMT System.
+We im-
+plemented our proposed method on a commercial NMT sys-
+tem – NiuTrans Enterprise – to evaluate our method’s ap-
+plicability in the real-world environment. We experimented
+on UNPC En-Zh, Fr-Zh, and Ru-Zh, without modifying the
+NMT model, even not aware of what kind of NMT model
+is used. The word aligners for fragment-level TM on Fr-Zh
+and Ru-Zh are trained on Fr-Zh and Ru-Zh TM databases,
+respectively. Experimental results in Table 6 show that the
+maximum improvement of sentence-level TM on En-Zh, Fr-
+Zh, and Ru-Zh are 2.94, 3.55, 3.06 BLEU points, respec-
+Domains
+IT
+Koran
+Law
+Medical Subtitles
+kNN-MT
+45.82
+19.45
+61.78
+54.35
+31.73
+W/o TM
+38.09
+17.11
+45.92
+41.14
+29.45
+Sentence TM
+Directly
+40.19
+17.20
+48.78
+45.29
+28.18
+Comma
+39.20
+16.46
+47.42
+43.47
+25.09
+Semico
+39.74
+17.09
+48.91
+44.93
+26.44
+Conjunc
+40.13
+17.03
+48.97
+45.13
+27.68
+Parenth
+40.63
+17.53
+48.31
+45.78
+29.03
+Fragment TM
+39.38
+16.49
+45.58
+43.31
+28.24
+Table 5: Experimental results on multi-domain datasets.
+tively, and the fragment-level TM approach still get lower
+BLEU scores than the baseline. The NMT models used
+in this experiment have been trained on much more high-
+quality training data than other models used in the above ex-
+periments. The experimental results demonstrate that even
+strong commercial NMT systems can be further improved
+when similar TMs provided and that the sentence-level TM
+approach can be applied in real-world situations where sim-
+ilar TMs for input sentences are available.
+Lang
+En-Zh
+Fr-Zh
+Ru-Zh
+W/o TM
+41.59
+29.83
+35.62
+Sentence TM
+Directly
+44.18
+33.10
+37.94
+Comma
+43.85
+31.63
+37.36
+Semico
+44.48
+33.38
+37.89
+Conjunc
+44.16
+33.04
+37.97
+Parenth
+44.53
+33.05
+38.68
+Fragment TM
+38.74
+27.78
+32.65
+Table 6: Experimental results on a commercial NMT system.
+NMT Model’s Effect on Translation
+In our proposed method, the generation of each token in the
+target sentence relies on source TM, input sentence, target
+TM and the generated part of target sentence, and the tar-
+get TM is generated in a forced way. Therefore, the trans-
+lation depends on the translation ability of the NMT model,
+“strong” models improve greater, and “weak” models im-
+prove less or even get worse results. In order to investi-
+gate to what extent the results depend on the NMT model’s
+“strength”, we conducted a series of experiments on the
+UNPC En-Zh test set (see Table 2) with different NMT mod-
+els. We measure the translation ability of a model in terms
+of the training data scale and the model architecture. So, we
+trained several NMT models using WMT20 En-Zh training
+data, from the perspectives of training data scale and model
+architecture.
+Training Data Scale.
+The training data of WMT20 En-Zh
+has 20 million bilingual sentences. We uniformly split them
+into four parts after shuffling, then trained four NMT models
+with different data scales, in which the first model is trained
+on the first 5 million datasets, the second model is trained
+
+on the first and second 5 million datasets, and so on. All of
+the models are transformer big models proposed in Vaswani
+et al. (2017). The experimental results are given in Table 7.
+Models
+b5M
+b10M
+b15M
+b20M
+ba20M bb20M
+W/o TM
+41.11
+41.40
+42.53
+43.19
+41.47
+42.53
+Directly
+41.77
+42.45
+44.27
+44.26
+41.87
+43.75
+Comma
+41.29
+41.88
+43.79
+43.99
+41.18
+43.36
+Semico
+42.27
+42.45
+44.64
+45.13
+42.12
+44.28
+Conjunc
+41.48
+42.04
+43.87
+44.33
+41.54
+43.09
+Parenth
+41.68
+42.63
+44.49
+44.53
+41.93
+43.85
+Max ∆
+1.16
+1.23
+2.11
+1.94
+0.65
+1.75
+Table 7: Experimental results on UNPC En-Zh test set with
+different NMT models, including four transformer big mod-
+els trained on 5 million, 10 million, 15 million, and 20 mil-
+lion training data (denoted as b5M, b10M, b15M, b20M,
+respectively), and a transformer base and a bigger model
+trained on 20 million training data (denoted as ba20M and
+bb20M, respectively), max ∆ denotes the maximum im-
+provement comparing to decoding without TM.
+Model Architecture.
+In this experiment, we investigate
+the impact of the model architecture on our proposed
+method. We chose WMT20 En-Zh dataset with 20 million
+bilingual sentences in the above experiment and trained the
+transformer base, big and bigger models, respectively. Their
+attention heads, hidden sizes, and filter sizes are (8, 512,
+2048), (16, 1024, 4096), and (24, 1536, 6144), respectively.
+From the experimental results in Table 7, we have the obser-
+vations below.
+For the same model architecture, with the increase of
+training data scale, the translation ability of the model is get-
+ting stronger, and the BELU improvement of our method is
+also higher compared with the baseline, as the maximum
+BLEU score improvement of the models b15M and b20M
+are higher than that of b5M and b10M. In addition, for the
+models trained on the same training data, the ba20M model
+is “weaker” than the b20M and bb20M models, and the max-
+imum BLEU score improvement is also lower than the latter
+two models. We can find a similar phenomenon if we look
+back to Table 4 and Table 6. The dataset for UNPC En-Zh
+is the same in these two experiments, and the NMT model
+of the commercial NMT system is much “strong” than the
+NMT model used in table 4, and the maximum improvement
+of the former is an 2.94 BLEU points, whereas the latter’s is
+0.89 BLEU points. From these results, we conclude that the
+sentence-level TM approach of our method can further im-
+prove strong baselines, and the “stronger” the NMT model,
+the greater the improvement will be.
+TM Similarity
+As our method obtain useful information from a single TM
+sentence pair, the translation result is influenced by the simi-
+larity of the retrieved TM. TMs with high similarity provide
+more useful information for the translation, while less simi-
+lar ones may introduce noise to decoding. In this experiment,
+[0.1,0.2)
+[0.2,0.3)
+[0.3,0.4)
+[0.4,0.5)
+[0.5,0.6)
+[0.6,0.7)
+[0.7,0.8)
+[0.8,0.9)
+[0.9,1.0]
+Similarity range of TM
+35
+40
+45
+50
+55
+BLEU score
+directly
+comma
+semicolon
+conjunction
+parenthesis
+w/o TM
+Figure 3: BLEU scores on different similarity ranges.
+we explore the similarity threshold that a TM can help or
+not. We decoded the UNPC En-Zh test set using NiuTrans’
+WMT20 En-Zh model (Zhang et al. 2020). Specifically, the
+test set is divided into various portions based on the similar-
+ity score, and each portion of the test set is decoded individ-
+ually both with the sentence-level TM approach and baseline
+(without TM).
+From the experimental results in Figure 3, we can find
+that the BLEU scores of our method are lower than the base-
+line when the similarity score is lower than 0.6; when it is
+between 0.6 and 0.8, some of our concatenation methods
+perform better than the baseline with a marginal advantage;
+when it is higher than 0.8, all of the concatenation meth-
+ods outperform the baseline significantly, with a maximum
+improvement of 7.09 and 6.11 BLEU points, respectively.
+Therefore, the FMS threshold for sentence-level TM of the
+NiuTrans WMT20 En-Zh model is 0.8.
+The threshold is determined by the “strength” of the NMT
+model that “strong” models are more robust to obtaining
+useful information and avoiding noises introduced by less
+similar TMs. Thus, “strong” models have lower FMS thresh-
+olds. In practice, the threshold can be used to decide whether
+to apply TM for decoding.
+Sentence Length
+In this section, we investigate the impact of input sentence
+length on translation. To avoid the TM similarity influencing
+the experimental results, we used the test set itself as the re-
+trieved TM, which means that the TMs are 100% similar to
+the input sentences. We split the test set into groups accord-
+ing to the length of the input sentence, and uniformly chose
+170 sentences from each group as the test set (the minimum
+sentence number of the original groups is 171). The experi-
+mental results are given in Figure 4.
+From the experimental results, we can find a sharp ten-
+dency that the performance of our proposed method de-
+creases and eventually be comparable to the baseline as the
+sentence length increases. The performance of the baseline,
+
+[0,10)
+[10,20)
+[20,30)
+[30,40)
+[40,50)
+[50,60)
+[60,-)
+Input sentence length
+40
+50
+60
+70
+80
+BLEU score
+directly
+comma
+semicolon
+conjunction
+parenthesis
+w/o TM
+Figure 4: input sentence length impact on BLEU.
+however, tends to be steady. This is also determined by the
+initial design of our method. For an input sentence, we em-
+ploy an NMT model that has been trained on a sentence-
+level dataset. As we concatenate the source TM with the
+input sentence before feeding them into the NMT model,
+the length of the input for the model encoder will be dou-
+bled. In this way, the whole sentence length of a lengthy sen-
+tence and its corresponding TM will deviate from the train-
+ing model’s sentence length distribution. This is the main
+cause of our method’s performance in the figure degrading
+on lengthy sentences. Therefore, our method can not han-
+dle long sentences well even though a highly similar sen-
+tence is retrieved. After calculation, we find that the average
+sentence length of the UNPC En-Zh test set and homemade
+En-Zh test set in Table 4 are 29.58 and 10.99. This is why
+the latter can improve the translation more significantly than
+the former even though there are fewer sentences with high
+similarity than there are in the former.
+Related Work
+Many studies have been conducted in recent years to en-
+hance MT quality using TMs. With the emergence of NMT,
+the MT community is seeing an increasing interest in TM
+research. There are mainly two research lines for TM inte-
+gration into NMT: constraining the decoding process with
+TM and using TM to train a more powerful NMT model.
+The main idea of the first research line is to increase
+the generation probability of some target words based on
+the retrieved TM. Zhang et al. (2018) constrained the de-
+coding process by increasing the generation possibility of
+the target words which are in the aligned slices extracted
+from retrieved TM. Following this work, He et al. (2019)
+added positional information for words in the TM slices.
+Unlike the above approaches, Li, Zhang, and Zong (2016)
+and Farajian et al. (2017) embedded the retrieved TM in-
+formation into the NMT model by fine-tuning the NMT
+model with TMs before translating the input sentence. In-
+stead of incorporating sentence level TM, the recent work –
+kNN-MT – retrieved TM from dense vectors (Khandelwal
+et al. 2020). First, they created a key-value datastore from
+the TM database, where the key is the translation context
+vector of each time step, and the value is the true target to-
+ken. In the inference time, kNN-MT interpolates the gen-
+eration probability of the NMT model and retrieved simi-
+lar target distribution from that datastore at each time step.
+Following kNN-MT, several researches optimized kNN-MT
+from different perspectives. Meng et al. (2022) accelerated
+the inference process by narrowing the search range, in-
+stead of searching from the entire data store. By introduc-
+ing a lightweight meta-k network, Zheng et al. (2021) dy-
+namically determines how many neighbors should be intro-
+duced. Wang et al. (2021a) further accelerated kNN-MT in-
+ference by constraining search space when constructing the
+data store. Instead of retrieving a single token, Martins, Mar-
+inho, and Martins (2022) retrieved chunks of tokens from the
+data store to speed up kNN-MT.
+The second research line aims to train the generation
+model to learn how to deal with the retrieved TMs. Bult´e and
+Tezcan (2019) and Xu, Crego, and Senellart (2020) used a
+data augmentation way to concatenate the retrieved TM with
+input sentence during training. While some researches mod-
+ified the NMT model architecture to better integrate TMs.
+Cao and Xiong (2018) and Gu et al. (2018) introduced a
+gating mechanism module to control the signal from the re-
+trieved TM. Cao, Kuang, and Xiong (2020) designed an ad-
+ditional transformer encoder to encode the target sentence
+of retrieved TM, and integrate them through the attention
+mechanism. In Xia et al. (2019), the retrieved multiple TMs
+are compressed into a graph structure for speed up and space
+savings and then are integrated into the model via the at-
+tention mechanism. He et al. (2021) proposed a lightweight
+method to incorporate the target sentence of retrieved TM in
+an extra attention module. Unlike all of the above methods,
+Cai et al. (2021) proposed a method to incorporate mono-
+lingual TM into NMT, and the target sentence retriever and
+NMT model are trained jointly.
+Conclusion and Future Work
+In this paper, we propose a simple but effective method to
+incorporate TM into NMT decoding without modifying the
+pre-trained NMT model. Specifically, we treat the retrieved
+TMs as prompts for the translation of the input sentence by
+concatenating the source TM with the input sentence and
+generating the target token in a forced way. Experiments on
+the TM specialized translation task, domain adaptation task,
+and implementation on commercial MT system verify the
+effectiveness of our method. Our method is easy to imple-
+ment and can be applied to customize a TM-incorporated
+machine translation system for TM data on the user side.
+Our method in this paper suffers from TM sentences with
+low similarity scores and long sentences. In the future, we
+will investigate more effective methods to alleviate the draw-
+backs of our methods in low similarity TM and long sen-
+tence translation situations.
+
+Acknowledgments
+This work was supported by National Key R&D Program
+of China (No. 2020AAA0107904). We are very thankful to
+anonymous reviewers for their comments.
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diff --git a/T9E5T4oBgHgl3EQfAg7p/content/tmp_files/load_file.txt b/T9E5T4oBgHgl3EQfAg7p/content/tmp_files/load_file.txt
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+page_content='Prompting Neural Machine Translation with Translation Memories  Abudurexiti Reheman1, Tao Zhou1, Yingfeng Luo1, Di Yang2, Tong Xiao1,2, Jingbo Zhu1,2*  1School of Computer Science and Engineering, Northeastern University, Shenyang, China  2NiuTrans Research, Shenyang, China  rexiti neu@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='com, zhoutao neu@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='com, luoyingfengmail@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='com,  yangdi@niutrans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='com, {xiaotong, zhujingbo}@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='cn  Abstract  Improving machine translation (MT) systems with translation  memories (TMs) is of great interest to practitioners in the MT  community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' However, previous approaches require either a  significant update of the model architecture and/or additional  training efforts to make the models well-behaved when TMs  are taken as additional input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In this paper, we present a sim-  ple but effective method to introduce TMs into neural ma-  chine translation (NMT) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Specifically, we treat TMs  as prompts to the NMT model at test time, but leave the train-  ing process unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The result is a slight update of an ex-  isting NMT system, which can be implemented in a few hours  by anyone who is familiar with NMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Experimental results  on several datasets demonstrate that our system significantly  outperforms strong baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Introduction  Integrating TM is one of the commonly used techniques to  improve real-world MT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In TM-assisted MT sys-  tems, it is often assumed that there is a database in which  high-quality bilingual sentence pairs are stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' When trans-  lating an input sentence, the most (or top-K) similar sen-  tence pair, which is retrieved from TM, is used to optimize  the translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' From the perspective of practical application,  this approach is particularly useful for MT, especially when  sentences are highly repetitive, such as in translating tech-  nical manuals, legal provisions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Previous works show  that translation quality can be significantly improved when  a well-matched TM sentence pair is provided both in Sta-  tistical Machine Translation (SMT) (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Wang,  Zong, and Su 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Li, Way, and Liu 2014) and Neural Ma-  chine Translation (NMT) (Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  However, there are two major problems with this type of  work in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' First, it is difficult to find  such a TM dataset in most cases, especially when users can  not share their TM data with the public for some reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Second, previous approaches often require model changes,  including training the model with TM (Bult´e and Tezcan  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Hossain, Ghazvininejad, and Zettlemoyer 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Xu,  Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Copyright © 2023, Association for the Advancement of Artificial  Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Encoder  Decoder  𝑥1  𝑡𝑚 ··· 𝑥𝑚  𝑡𝑚 𝑥1  ··· 𝑥𝑛  <bos> 𝑦1  𝑡𝑚··· 𝑦𝑘  𝑡𝑚 𝑦1 ··· 𝑦𝑙  𝑦1  𝑡𝑚··· 𝑦𝑘  𝑡𝑚 𝑦1 ··· 𝑦𝑙 <eos>  Force decoding  Figure 1: Structure of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The source and  target sentences of TM are concatenated with the input sen-  tence and hypothesis in a specific concatenation template,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The tokens in the target TM together with the  concatenation template are generated in a forced manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  The lengths of the source and target TM together with the  concatenation templates are m and k, and the lengths of the  input sentence and hypothesis are n and l, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Crego, and Senellart 2020), changing the NMT model ar-  chitecture for TM integration (Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Bapna and  Firat 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2021), and introducing  additional modules (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Khan-  delwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In this case, it is difficult to incorporate  TM into the NMT system even if TM data is provided, since  TM incorporation can not be accomplished on a generic de-  coder, and a deeply customized decoder is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Here, we address this problem by using few-shot learning  (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2021b), which enables the system to quickly  adapt to a small number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The recent prevalence  of prompt-based approaches (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2020), which  transfer the original task into a generation task by design-  ing an appropriate template without modifying the language  model, gives us some inspiration that the retrieved TM can  prompt the translation of the input sentence without modify-  ing the NMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Based on this idea, we propose a simple approach to  quickly adapt the NMT model in the few-shot TM scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Specifically, we treat TMs as prompts to the NMT model  during the decoding process, with very small changes to the  decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Our method can cover the advantages of conven-  tional TM augmented methods and bring some new ideas,  such as incorporating users’ local historical translation data  into NMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='05380v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='CL]  13 Jan 2023  In order to prompt the translation with the retrieved TM,  we design several templates to concatenate the source TM  with the input sentence and feed the concatenated sentence  into the model encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' On the decoder side, we generate  the target TM and the concatenation template in a forced  way first, then let the model generate the other parts au-  tomatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Regarding TM granularity, our method works  well on sentence-level and fragment-level TM by design-  ing appropriate templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Experimental results on several  datasets show that our method can further improve the trans-  lation quality on strong NMT models, and with comparable  performance with the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Background  NMT Decoding  Suppose x = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=', xn} is the source sentence, and NMT  translates it into the corresponding target sentence y =  {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=', ym} by using a trained NMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In practice, it  turns the decoding into a searching problem, and a beam  searcher is adopted to get the target sentence with the high-  est generation probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Generally, an NMT model gener-  ates in an auto-regressive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Therefore, the generation of  each token relies on the source sentence and the generated  prefix of the target sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The generation of the whole tar-  get sentence can be formulated as a conditional probability  P(y|x) described below:  P(y|x) =  m  �  i=0  P(yi|x, y<i)  (1)  where y<i = {y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=', yi−1} denotes the generated prefix  tokens of target sentence at time-step i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  TM  TM is a database of language pairs that stores seg-  ments (such as fragments, sentences, paragraphs, or other  sentence-like units) that have previously been translated by  human translators for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' It can provide identical or  similar segments to help translate the input sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In the  early stage, TM was widely used in Computer Aided Trans-  lation (CAT) (Dillon and Fraser 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' When translating an  input sentence, the target sentence of TM is returned as the  answer when an identical source sentence is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In most  cases, the translation result is obtained by fixing the most  similar or top-K similar sentences retrieved from TM, since  it is difficult to find a completely identical sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In the  MT environment, TM is playing the same role as in CAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Many approaches have been proposed to incorporate similar  TM into MT systems to get a more accurate translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Methodology  In this section, we introduce our proposed method in detail  from the perspectives of incorporating TM into NMT de-  coding, designing templates for concatenation, and retriev-  ing similar TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Incorporating TMs into NMT Decoding  We incorporate TMs into the decoding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' For an input  sentence x and a retrieved TM sentence pair ⟨xtm, ytm⟩, we  concatenate xtm and x in a specific concatenation template  and feed it into the model encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' On the decoder side, we  first force the model to generate the exact tokens in ytm to-  gether with the concatenation template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Then the rest of the  generation, which is returned as the answer, is done automat-  ically without interfering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In practice, we set the generation  probability of the tokens in the target TM and the concate-  nation template as 1 during the force decoding phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Concatenation Templates  As our method incorporates TM on a pre-trained NMT  model with no modifications, we must adhere to the follow-  ing principles when designing the concatenation template:  tokens in the concatenation template must be recognized by  the NMT model and each part of the concatenated sentence  maintain relatively complete semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Following this idea,  we designed several templates for TMs in different gran-  ularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In concatenation, the TM comes first on both the  source and target sides, and a specific template is used on  both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Sentence-level TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  We concatenate sentence-level TMs  in five different templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' As the example shown in Table  1, our designed templates for sentence-level TMs are as fol-  lows:  (a) Concatenate directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We concatenate them directly  and add a period at the end of TMs if it is not ended up  with punctuation marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  (b) Concatenate with comma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Before concatenation, we  replace the punctuation mark at the end of TMs with a  comma or add a comma there if it is not ended up with any  punctuation mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  (c) Concatenate with semicolon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The concatenation is the  same as the comma concatenation process, using a semi-  colon instead of the comma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  (d) Concatenate with conjunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In this template, we  adopt conjunctions that express juxtaposed semantics, such  as “and” in English and “und” in German.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Specifically, we  add a period at the end of the source and target TM if they are  not ended up with any punctuation mark, then add a conjunc-  tion word in the corresponding language and a comma after  that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Then we concatenate them with the input sentence and  the hypothesis on the source and target side, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  (e) Enclose in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We enclose both source and  target TMs in parentheses, then perform the concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Fragment-level TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Unlike the sentence-level TM, we  do some preprocessing on fragment-level TMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' First, we ob-  tain the common fragments between the input sentence and  source TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Then, the tokens which are the translation of the  words in the above common fragments are acquired from  the target TM, using word alignment tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' After that, we  construct the fragment-level TM using the tokens from the  source and target TM above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' An example of common frag-  ments between input sentence and source TM and the word  Input Sentence  She gave us a full account of the traffic accident .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Source TM  She gave the police a full account of the incident .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Target TM  Sie gab der Polizei einen voll@@ st¨andigen Bericht ¨uber den Vorfall .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Sentence Level TMs  Directly  En/input  She gave the police a full account of the incident .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' She gave us a full account of the traffic accident .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  De/input  <bos> Sie gab der Polizei einen voll@@ st¨andigen Bericht ¨uber den Vorfall .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' {Hypothesis}  Comma  En/input  She gave the police a full account of the incident , She gave us a full account of the traffic accident .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  De/input  <bos> Sie gab der Polizei einen voll@@ st¨andigen Bericht ¨uber den Vorfall , {Hypothesis}  Semicolon  En/input  She gave the police a full account of the incident ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' She gave us a full account of the traffic accident .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  De/input  <bos> Sie gab der Polizei einen voll@@ st¨andigen Bericht ¨uber den Vorfall ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' {Hypothesis}  Conjunction  En/input  She gave the police a full account of the incident .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' And , She gave us a full account of the traffic accident .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  De/input  <bos> Sie gab der Polizei einen voll@@ st¨andigen Bericht ¨uber den Vorfall .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Und , {Hypothesis}  Parenthesis  En/input  ( She gave the police a full account of the incident .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' ) She gave us a full account of the traffic accident .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  De/input  <bos> ( Sie gab der Polizei einen voll@@ st¨andigen Bericht ¨uber den Vorfall .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' ) {Hypothesis}  Fragment Level TMs  Parenthesis  En/input  ( She gave ) ( a full account of the ) She gave us a full account of the traffic accident .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  De/input  <bos> ( Sie gab ) ( einen voll@@ st¨andigen Bericht ¨uber den ) {Hypothesis}  Table 1: An example of model input in our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Here, Directly, Comma, Semicolon, Conjunction, and Parenthesis  denote our designed templates for concatenation, and En/input and De/input denote the input of the encoder and the decoder,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The <bos> token denotes the begin-of-sentence tag, and {Hypothesis} denotes the automatically generated part  of the target sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Red tokens denote the concatenation templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Input Sentence:  She  gave   us   a  full  account  of   the   traffic   accident   .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Source TM:  She   gave    the  police    a     full     account     of     the     incident   .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Target TM:  Sie  gab   der Polizei  einen  voll@@  ständigen  Bericht  über  den  Vorfall   .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Figure 2: An example of obtaining fragments for fragment-level TMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' For a given bilingual TM, common fragments between  the input sentence and the source TM are acquired first, then the words in the target TM that align with the words in the common  fragments are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Common fragments and their corresponding fragments in Target TM are tagged by the same color box,  and the lines denote word alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  alignment between the source and target TM is given in Fig-  ure 2, and its corresponding fragment-level TM is given in  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Specifically, for a given input sentence x and a retrieved  bilingual TM ⟨xtm, ytm⟩, we acquire the encoder and de-  coder input for the NMT model in the following steps:  (a) Perform the longest common subsequence matching  algorithm to x and xtm, and obtain the longest common sub-  sequence Ps = {w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=', wm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  (b) Use word alignment tools to xtm and ytm, and get  the aligned subsequence Pt = {w′  1, w′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=', w′  n}, which is  corresponding to Ps, from ytm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  (c) Group the words, which appear continuously in xtm,  from Ps in their original order to form source TM fragments  P ′  s = {f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=', fi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  (d) Group the words, which appear continuously in ytm,  from Pt in the order of the correspondence with P ′  s to form  target TM fragments P ′  t = {f ′  1, f ′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=', f ′  j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  (e) Concatenate each fragment in P ′  s and P ′  t with a spe-  cific template to form fragment-level TM, and directly con-  catenate them with input sentence and hypothesis as is done  in sentence-level TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  As for the concatenation template, we enclose each frag-  ment in parenthesis to maintain its semantic integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In  practice, we remove the fragments that consist of a single  stop word and remove the punctuation at two sides of a frag-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Retrieving Similar TMs  To retrieve the most similar bilingual TM for the input sen-  tence, we use a word-level fuzzy matching strategy and re-  move punctuations and numbers from the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Instead  of retrieving from the whole TM database, we first employ  the search engine library Apache Lucene (Bialecki, Muir,  and Ingersoll 2012) to retrieve the top 500 similar bilingual  sentences from TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Then we rerank them by adopting Fuzzy  Match Score (FMS) to obtain the most similar TM sentence  pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' FMS is a length normalized Levenshtein Distance (Li  and Liu 2007), known as Edit Distance:  FMS(x, xtm) = 1 −  LD(x, xtm)  max(|x|, |xtm|)  (2)  where LD(·, ·) denotes the word level Levenshtein Distance,  and | · | denotes word level length of a sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Experiments  In order to verify the validity of our proposed method, we  conducted several experiments on TM specialized transla-  tion task and domain adaptation task, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We also  put our approach into practice on a commercial NMT system  to assess its usability in the practical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In the end, we  investigated the impact of the NMT model, TM similarity,  and input sentence length on translation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Datasets and Models  For TM specialized translation tasks, we evaluated our  method on two datasets: 1) DGT-TM, the entire body  of European legislation in 22 European languages, on  German-English in both directions (En-De and De-En) and  2) United Nations Parallel Corpus (UNPC), consisting of  United Nations General Assembly Resolutions with transla-  tions in the six official languages, on English-Chinese (En-  Zh), Russian-Chinese (Ru-Zh) and French-Chinese (Fr-Zh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  These two datasets are relatively easy to retrieve TM sen-  tences with a high degree of similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' For the test set and  TM database, we cleaned the above corpora first, then ran-  domly selected 3,000 sentence pairs for the test dataset,  whereas the remaining corpora were utilized as the TM  database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  In addition, we performed an experiment using a home-  made English-Chinese dataset (denoted as H-m in Table 2)  of 3401 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Each test sentence has one bilingual TM  sentence whose source side is similar to the test sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  The statistics of these TM databases and the TM similarity  ratios of retrieved TMs in FMS metric are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  In the domain adaptation task, following Khandelwal  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' (2020), we used the multi-domain datasets from Aha-  roni and Goldberg (2020), which contains German-English  bilingual datasets in five different domains: Medical, Law,  IT, Koran, and Subtitles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We treated the train-  ing data in each domain as our TM database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  After cleaning the above corpora and splitting them into a  test set and TM database, we retrieved the most similar TM  for each test sentence from the TM database in an offline  way and applied BPE (Sennrich, Haddow, and Birch 2016)  to the test sets and TM with the BPE-codes provided by the  pre-trained NMT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' To obtain the alignment informa-  tion for the tokens in the TM source and target sentence, we  trained a word aligner – Mask-Align – as proposed in Chen,  Sun, and Liu (2021), and constructed corresponding frag-  ment level TMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The TM data scale and the TM similarity  ratios of retrieved TMs in FMS metric are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  As for the pre-trained NMT model, we applied Face-  book’s WMT19 De-En, En-De model (Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2019) and  Corpus  Lang  TM  scale  TM FMS ratio  [0,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='2)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='2,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='4)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='4,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='6)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='6,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='8)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='8,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='0)  DGT  TM  En-De  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='1M  3%  27%  19%  21%  31%  De-En  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='1M  4%  30%  18%  22%  26%  UNPC  En-Zh  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='9M  2%  44%  22%  11%  22%  Fr-Zh  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='5M  2%  45%  18%  11%  23%  Ru-Zh  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='2M  8%  46%  16%  9%  20%  H-m  En-Zh 18%  30%  35%  23%  7%  Table 2: Sentence numbers in the TM databases and the sim-  ilarity ratios of the retrieved TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Domain  TM  scale  TM FMS ratio  [0,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='2)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='2,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='4)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='4,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='6)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='6,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='8)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='8,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='0)  Medical  248K  7%  23%  20%  17%  33%  Law  467K  8%  31%  18%  14%  28%  IT  223K  14%  18%  28%  26%  14%  Koran  18K  2%  26%  33%  28%  11%  Subtitles  500K  3%  27%  43%  23%  4%  Table 3: Sentence numbers in the TM database in each do-  main and the similarity ratios of the retrieved TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  NiuTrans’ WMT20 En-Zh model (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2020) as our  base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' All of these models are very competitive that  they trained on more than 20 million training data and 10  million extra back-translated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Main Experiment  Our main experiment involves the TM specialized transla-  tion task, the domain adaptation task, and the implementa-  tion on commercial NMT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  TM Specialized translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  In this experiment, we de-  coded the DGT-TM En-De and De-En test sets using face-  book’s WMT19 En-De and De-En models (Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2019),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Besides, we decoded the UNPC En-Zh test set  and the homemade En-Zh test set with NiuTrans’ WMT20  En-Zh model (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' For the Mask-Align model  training, we used WMT20 En-Zh training data for the En-Zh  aligner and DGT-TM’s TM database for En-De and De-En  aligner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' From the experimental results in Table 4, we have  the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  First, in sentence-level TM, the BLEU score on DGT-TM  De-En, En-De, and homemade En-Zh test sets increased sig-  nificantly, with maximum BLEU score increases of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='63,  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='74, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='74 points, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Meanwhile, the transla-  tion improved slightly on the UNPC En-Zh test set in di-  rectly, semicolon, and parenthesis concatenations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Second,  in fragment-level TM, the BLEU score increased by about  1 to 2 points or even decreased, compared to the baseline  (without TM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The main reason for this is that the NMT  model is trained on sentence-level training data rather than  sentence pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In addition, it is also affected by the perfor-  mance of the word aligner, which may provide error align-  ment information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Corpus  DGT-TM  UNPC  H-m  Lang  De-En  En-De  En-Zh  En-Zh  W/o TM  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='40  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='03  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='42  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='43  Sentence TM  Directly  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='74  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='32  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='70  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='97  Comma  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='44  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='03  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='33  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='85  Semico  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='42  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='54  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='31  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='89  Conjunc  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='65  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='00  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='15  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='17  Parenth  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='03  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='77  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='90  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='87  Fragment TM  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='21  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='65  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='85  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='67  Table 4: Experimental results on the DGT-TM En-De, De-  En, UNPC En-Zh, and the homemade En-Zh test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Domain Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Following the kNN-MT (Khandel-  wal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2020) and its optimized counterparts, we con-  ducted the domain adaptation experiment and compared our  method with kNN-MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We applied Facebook’s WMT19 De-  En model (Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2019) for decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Experimental re-  sults are given in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  From the table, we can find that our method improves the  translation in all domains except Subtitles, with maximum  BLEU score improvements of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='54, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='05, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='64 in IT, Law,  and Medical domains, respectively, whereas in Koran the  result is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='42 BLEU score higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The fragment-level  TM method has the same tendency as the above experiment,  which is slightly higher only in IT and Medical domain than  the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Besides, the improvement of our method is less  than kNN-MT in every domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The original design of our  approach leads to this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Our method retrieves the most  similar single TM and leverages the knowledge it contains  to improve the translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' How to leverage the knowledge  provided by TM is fully dependent on the NMT model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  While kNN-MT introduces an extra module to incorporate  the information explicitly from multiple similar context vec-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  The main advantage of our method over kNN-MT is that  our method performs the retrieval based on string similarity,  and there is no need to store the context vectors, which saves  a lot of storage space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' At the same time, kNN-MT searches  the context vectors in each beam in every timestep, which  is much slower than the vanilla NMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' However, our method  searches the TM only once and generates two sentences (tar-  get TM and the hypothesis) in the way of a vanilla NMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  This will make our method much faster than kNN-MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Implementation on Commercial NMT System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  We im-  plemented our proposed method on a commercial NMT sys-  tem – NiuTrans Enterprise – to evaluate our method’s ap-  plicability in the real-world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We experimented  on UNPC En-Zh, Fr-Zh, and Ru-Zh, without modifying the  NMT model, even not aware of what kind of NMT model  is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The word aligners for fragment-level TM on Fr-Zh  and Ru-Zh are trained on Fr-Zh and Ru-Zh TM databases,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Experimental results in Table 6 show that the  maximum improvement of sentence-level TM on En-Zh, Fr-  Zh, and Ru-Zh are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='94, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='55, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='06 BLEU points, respec-  Domains  IT  Koran  Law  Medical Subtitles  kNN-MT  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='82  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='45  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='78  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='35  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='73  W/o TM  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='09  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='11  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='92  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='14  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='45  Sentence TM  Directly  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='19  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='20  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='78  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='29  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='18  Comma  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='20  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='46  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='42  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='47  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='09  Semico  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='74  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='09  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='91  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='93  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='44  Conjunc  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='13  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='03  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='97  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='13  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='68  Parenth  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='63  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='53  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='31  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='78  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='03  Fragment TM  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='38  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='49  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='58  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='31  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='24  Table 5: Experimental results on multi-domain datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  tively, and the fragment-level TM approach still get lower  BLEU scores than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The NMT models used  in this experiment have been trained on much more high-  quality training data than other models used in the above ex-  periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The experimental results demonstrate that even  strong commercial NMT systems can be further improved  when similar TMs provided and that the sentence-level TM  approach can be applied in real-world situations where sim-  ilar TMs for input sentences are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Lang  En-Zh  Fr-Zh  Ru-Zh  W/o TM  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='59  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='83  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='62  Sentence TM  Directly  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='18  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='10  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='94  Comma  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='85  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='63  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='36  Semico  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='48  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='38  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='89  Conjunc  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='16  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='04  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='97  Parenth  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='53  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='05  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='68  Fragment TM  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='74  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='78  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='65  Table 6: Experimental results on a commercial NMT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  NMT Model’s Effect on Translation  In our proposed method, the generation of each token in the  target sentence relies on source TM, input sentence, target  TM and the generated part of target sentence, and the tar-  get TM is generated in a forced way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Therefore, the trans-  lation depends on the translation ability of the NMT model,  “strong” models improve greater, and “weak” models im-  prove less or even get worse results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In order to investi-  gate to what extent the results depend on the NMT model’s  “strength”, we conducted a series of experiments on the  UNPC En-Zh test set (see Table 2) with different NMT mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We measure the translation ability of a model in terms  of the training data scale and the model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' So, we  trained several NMT models using WMT20 En-Zh training  data, from the perspectives of training data scale and model  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Training Data Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  The training data of WMT20 En-Zh  has 20 million bilingual sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We uniformly split them  into four parts after shuffling, then trained four NMT models  with different data scales, in which the first model is trained  on the first 5 million datasets, the second model is trained  on the first and second 5 million datasets, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' All of  the models are transformer big models proposed in Vaswani  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The experimental results are given in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Models  b5M  b10M  b15M  b20M  ba20M bb20M  W/o TM  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='11  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='40  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='53  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='19  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='47  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='53  Directly  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='77  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='45  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='27  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='26  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='87  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='75  Comma  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='29  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='88  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='79  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='99  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='18  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='36  Semico  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='27  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='45  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='64  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='13  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='12  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='28  Conjunc  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='48  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='04  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='87  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='33  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='54  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='09  Parenth  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='68  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='63  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='49  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='53  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='93  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='85  Max ∆  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='23  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='75  Table 7: Experimental results on UNPC En-Zh test set with  different NMT models, including four transformer big mod-  els trained on 5 million, 10 million, 15 million, and 20 mil-  lion training data (denoted as b5M, b10M, b15M, b20M,  respectively), and a transformer base and a bigger model  trained on 20 million training data (denoted as ba20M and  bb20M, respectively), max ∆ denotes the maximum im-  provement comparing to decoding without TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Model Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  In this experiment, we investigate  the impact of the model architecture on our proposed  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We chose WMT20 En-Zh dataset with 20 million  bilingual sentences in the above experiment and trained the  transformer base, big and bigger models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Their  attention heads, hidden sizes, and filter sizes are (8, 512,  2048), (16, 1024, 4096), and (24, 1536, 6144), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  From the experimental results in Table 7, we have the obser-  vations below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  For the same model architecture, with the increase of  training data scale, the translation ability of the model is get-  ting stronger, and the BELU improvement of our method is  also higher compared with the baseline, as the maximum  BLEU score improvement of the models b15M and b20M  are higher than that of b5M and b10M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In addition, for the  models trained on the same training data, the ba20M model  is “weaker” than the b20M and bb20M models, and the max-  imum BLEU score improvement is also lower than the latter  two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We can find a similar phenomenon if we look  back to Table 4 and Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The dataset for UNPC En-Zh  is the same in these two experiments, and the NMT model  of the commercial NMT system is much “strong” than the  NMT model used in table 4, and the maximum improvement  of the former is an 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='94 BLEU points, whereas the latter’s is  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='89 BLEU points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' From these results, we conclude that the  sentence-level TM approach of our method can further im-  prove strong baselines, and the “stronger” the NMT model,  the greater the improvement will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  TM Similarity  As our method obtain useful information from a single TM  sentence pair, the translation result is influenced by the simi-  larity of the retrieved TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' TMs with high similarity provide  more useful information for the translation, while less simi-  lar ones may introduce noise to decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In this experiment,  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='2)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
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+page_content='5)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
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+page_content='6)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='6,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='7)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='7,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='8)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='8,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='9)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='9,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='0]  Similarity range of TM  35  40  45  50  55  BLEU score  directly  comma  semicolon  conjunction  parenthesis  w/o TM  Figure 3: BLEU scores on different similarity ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  we explore the similarity threshold that a TM can help or  not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We decoded the UNPC En-Zh test set using NiuTrans’  WMT20 En-Zh model (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Specifically, the  test set is divided into various portions based on the similar-  ity score, and each portion of the test set is decoded individ-  ually both with the sentence-level TM approach and baseline  (without TM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  From the experimental results in Figure 3, we can find  that the BLEU scores of our method are lower than the base-  line when the similarity score is lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' when it is  between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='8, some of our concatenation methods  perform better than the baseline with a marginal advantage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  when it is higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='8, all of the concatenation meth-  ods outperform the baseline significantly, with a maximum  improvement of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='09 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='11 BLEU points, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Therefore, the FMS threshold for sentence-level TM of the  NiuTrans WMT20 En-Zh model is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  The threshold is determined by the “strength” of the NMT  model that “strong” models are more robust to obtaining  useful information and avoiding noises introduced by less  similar TMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Thus, “strong” models have lower FMS thresh-  olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In practice, the threshold can be used to decide whether  to apply TM for decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Sentence Length  In this section, we investigate the impact of input sentence  length on translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' To avoid the TM similarity influencing  the experimental results, we used the test set itself as the re-  trieved TM, which means that the TMs are 100% similar to  the input sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We split the test set into groups accord-  ing to the length of the input sentence, and uniformly chose  170 sentences from each group as the test set (the minimum  sentence number of the original groups is 171).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The experi-  mental results are given in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  From the experimental results, we can find a sharp ten-  dency that the performance of our proposed method de-  creases and eventually be comparable to the baseline as the  sentence length increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' The performance of the baseline,  [0,10)  [10,20)  [20,30)  [30,40)  [40,50)  [50,60)  [60,-)  Input sentence length  40  50  60  70  80  BLEU score  directly  comma  semicolon  conjunction  parenthesis  w/o TM  Figure 4: input sentence length impact on BLEU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  however, tends to be steady.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' This is also determined by the  initial design of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' For an input sentence, we em-  ploy an NMT model that has been trained on a sentence-  level dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' As we concatenate the source TM with the  input sentence before feeding them into the NMT model,  the length of the input for the model encoder will be dou-  bled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In this way, the whole sentence length of a lengthy sen-  tence and its corresponding TM will deviate from the train-  ing model’s sentence length distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' This is the main  cause of our method’s performance in the figure degrading  on lengthy sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Therefore, our method can not han-  dle long sentences well even though a highly similar sen-  tence is retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' After calculation, we find that the average  sentence length of the UNPC En-Zh test set and homemade  En-Zh test set in Table 4 are 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='58 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' This is why  the latter can improve the translation more significantly than  the former even though there are fewer sentences with high  similarity than there are in the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Related Work  Many studies have been conducted in recent years to en-  hance MT quality using TMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' With the emergence of NMT,  the MT community is seeing an increasing interest in TM  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' There are mainly two research lines for TM inte-  gration into NMT: constraining the decoding process with  TM and using TM to train a more powerful NMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  The main idea of the first research line is to increase  the generation probability of some target words based on  the retrieved TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' (2018) constrained the de-  coding process by increasing the generation possibility of  the target words which are in the aligned slices extracted  from retrieved TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Following this work, He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' (2019)  added positional information for words in the TM slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Unlike the above approaches, Li, Zhang, and Zong (2016)  and Farajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' (2017) embedded the retrieved TM in-  formation into the NMT model by fine-tuning the NMT  model with TMs before translating the input sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In-  stead of incorporating sentence level TM, the recent work –  kNN-MT – retrieved TM from dense vectors (Khandelwal  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' First, they created a key-value datastore from  the TM database, where the key is the translation context  vector of each time step, and the value is the true target to-  ken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In the inference time, kNN-MT interpolates the gen-  eration probability of the NMT model and retrieved simi-  lar target distribution from that datastore at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Following kNN-MT, several researches optimized kNN-MT  from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' (2022) accelerated  the inference process by narrowing the search range, in-  stead of searching from the entire data store.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' By introduc-  ing a lightweight meta-k network, Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' (2021) dy-  namically determines how many neighbors should be intro-  duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' (2021a) further accelerated kNN-MT in-  ference by constraining search space when constructing the  data store.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Instead of retrieving a single token, Martins, Mar-  inho, and Martins (2022) retrieved chunks of tokens from the  data store to speed up kNN-MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  The second research line aims to train the generation  model to learn how to deal with the retrieved TMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Bult´e and  Tezcan (2019) and Xu, Crego, and Senellart (2020) used a  data augmentation way to concatenate the retrieved TM with  input sentence during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' While some researches mod-  ified the NMT model architecture to better integrate TMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Cao and Xiong (2018) and Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' (2018) introduced a  gating mechanism module to control the signal from the re-  trieved TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Cao, Kuang, and Xiong (2020) designed an ad-  ditional transformer encoder to encode the target sentence  of retrieved TM, and integrate them through the attention  mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' (2019), the retrieved multiple TMs  are compressed into a graph structure for speed up and space  savings and then are integrated into the model via the at-  tention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' (2021) proposed a lightweight  method to incorporate the target sentence of retrieved TM in  an extra attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Unlike all of the above methods,  Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' (2021) proposed a method to incorporate mono-  lingual TM into NMT, and the target sentence retriever and  NMT model are trained jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Conclusion and Future Work  In this paper, we propose a simple but effective method to  incorporate TM into NMT decoding without modifying the  pre-trained NMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Specifically, we treat the retrieved  TMs as prompts for the translation of the input sentence by  concatenating the source TM with the input sentence and  generating the target token in a forced way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Experiments on  the TM specialized translation task, domain adaptation task,  and implementation on commercial MT system verify the  effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Our method is easy to imple-  ment and can be applied to customize a TM-incorporated  machine translation system for TM data on the user side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Our method in this paper suffers from TM sentences with  low similarity scores and long sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' In the future, we  will investigate more effective methods to alleviate the draw-  backs of our methods in low similarity TM and long sen-  tence translation situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  Acknowledgments  This work was supported by National Key R&D Program  of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2020AAA0107904).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' We are very thankful to  anonymous reviewers for their comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content='  References  Aharoni, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' and Goldberg, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
+page_content=' Unsupervised Domain  Clusters in Pretrained Language Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
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+page_content='  Bapna, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
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+page_content=' In  Proceedings of the 57th Conference of the Association for  Computational Linguistics, ACL 2019, Florence, Italy, July  28- August 2, 2019, Volume 1: Long Papers, 1800–1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
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+page_content=' In Proceedings of the 59th Annual Meeting of the As-  sociation for Computational Linguistics and the 11th Inter-  national Joint Conference on Natural Language Process-  ing, ACL/IJCNLP 2021, (Volume 1: Long Papers), Virtual  Event, August 1-6, 2021, 7307–7318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
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+page_content=' In ECAI 2020 - 24th European Conference on Arti-  ficial Intelligence, 29 August-8 September 2020, Santiago de  Compostela, Spain, August 29 - September 8, 2020 - Includ-  ing 10th Conference on Prestigious Applications of Artificial  Intelligence (PAIS 2020), volume 325 of Frontiers in Artifi-  cial Intelligence and Applications, 1982–1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
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+page_content=' In Proceedings of the Fifth Conference on Ma-  chine Translation, WMT@EMNLP 2020, Online, November  19-20, 2020, 338–345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E5T4oBgHgl3EQfAg7p/content/2301.05380v1.pdf'}
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+Recovering Dantzig-Wolfe Bounds by Cutting Planes
+Rui Chen1, Oktay G¨unl¨uk2, Andrea Lodi1
+1 Cornell Tech, Cornell University ({rui.chen,andrea.lodi}@cornell.edu)
+2 School of ORIE, Cornell University (ong5@cornell.edu)
+January 31, 2023
+Abstract
+Dantzig-Wolfe (DW) decomposition is a well-known technique in mixed-integer programming
+for decomposing and convexifying constraints to obtain potentially strong dual bounds. We
+investigate Fenchel cuts that can be derived using the DW decomposition algorithm and show
+that these cuts can provide the same dual bounds as DW decomposition. We show that these
+cuts, in essence, decompose the objective function cut one can simply write using the DW bound.
+Compared to the objective function cut, these Fenchel cuts lead to a formulation with lower
+dual degeneracy, and consequently a better computational performance under the standard
+branch-and-cut framework in the original space. We also discuss how to strengthen these cuts
+to improve the computational performance further.
+We test our approach on the Multiple
+Knapsack Assignment Problem and show that the proposed cuts are helpful in accelerating the
+solution time without the need to implement branch and price.
+1
+Introduction
+In this paper, we present a computationally effective approach for generating cutting planes from
+Dantzig-Wolfe (DW) decomposition [1] to enhance branch and cut in the space of original variables.
+We focus on mixed-integer (linear) programs (MIPs) with the following structure:
+z∗ := min c⊤x
+s.t. xI(j) ∈ P j,
+j ∈ J := {1, . . . , q},
+Ax ≥ b,
+x ∈ X,
+(1)
+where the index set I(j) ⊆ {1, . . . , n} contains the indices of the variables in “block” j ∈ J, and
+we use the notation xI to denote the subvector of x with indices in I. The set P j = {y ∈ R|I(j)| :
+Gjy ≥ gj} is a polyhedral set for j ∈ J; and X ⊆ Rn represents integrality constraints on some of
+the variables. We do not assume the index sets I(j) for j ∈ J to be disjoint, see Figure 1.
+DW decomposition was originally developed for solving large-scale linear programs with loosely
+coupled blocks and later extended to MIPs with similar block structures to obtain strong dual
+bounds. Typically, these so-called DW bounds are stronger than the linear programming (LP)
+relaxation bounds as they exploit the block structure through convexification of the block con-
+straints using integrality information. DW decomposition has been found to be effective in various
+applications, such as transportation [2], traffic scheduling [3], energy [4] and multi-stage stochastic
+planning [5].
+1
+arXiv:2301.13149v1  [math.OC]  30 Jan 2023
+
+Figure 1: Constraint Matrices of MIPs With Different Types of Block Structures (Left: Loosely
+Coupled, Middle: Two-Stage, Right: Overlapping)
+G1
+G2
+G3
+G4
+A
+G1
+G2
+G3
+G4
+G1
+G2
+G3
+G4
+Computing the DW bound requires reformulating the MIP using the extreme points and ex-
+treme rays of the mixed-integer hulls of the block problems. Consequently, while the DW refor-
+mulation approach often leads to good dual bounds, using this to solve the MIP exactly requires
+specialized techniques that are not readily present in off-the-shelf solvers. In other words, to ex-
+ploit DW decomposition one needs to solve the continuous relaxation of the reformulated MIP by
+column generation followed by an ad-hoc branching step [6], after which one has to again resort
+to column generation, leading to an algorithmic framework called branch and price [7]. While this
+approach has been successfully implemented in some special cases, most notably for vehicle routing
+problems [8], generic branch-and-price solvers, including ABACUS [9], G12 [10], GCG [11], DIP
+[12], BaPCod [13], to name a few, are still in theirs infancy with respect to solving general MIPs
+with block structure. However, the most up-to-date (and always improving) MIP solvers are based
+on the branch-and-cut (or cut-and-branch) scheme [14].
+In this paper, we aim to make a step towards bridging this gap by developing a new scheme
+to incorporate the dual bounds produced by DW reformulations into the standard cut-and-branch
+framework.
+More precisely, we first compute the DW reformulation bound zD of the MIP (1)
+at the root node and then generate cutting planes to incorporate this bound into the original
+formulation to solve the problem to optimality in the space of the original variables using standard
+MIP technology. Our approach, therefore, requires column generation only at the root node and
+not throughout the enumeration tree. Apart from our proposed approach, the only general and
+trivial method known so far to enforce the DW bound zD in the original MIP is to augment the
+formulation by adding the objective function cut, c⊤x ≥ zD, which is known (folklore) to be not only
+computationally ineffective but also numerically unstable. To the best of our knowledge, however,
+no detailed theoretical or computational investigation has been conducted on the objective function
+cut. Our approach, on the other hand, appears to be practical and computationally effective.
+Paper Contributions.
+Our first contribution is to confirm the instability of the approach using
+the objective cut by mostly attributing it to dual degeneracy. Then, we show how to overcome
+the limitation of the objective constraint by proposing a family of Fenchel cutting planes [15]
+that essentially decomposes the objective function cut leading to a formulation with lower dual
+degeneracy, and consequently a better computational performance under the standard branch-
+and-cut framework in the original space. Moreover, we propose two distinct ways of strengthening
+these Fenchel cuts for an additional improvement of their computational performance. As a case
+study, we test our approach on the Multiple Knapsack Assignment Problem and show that the
+proposed cuts are helpful in accelerating the solution time without the need to implement branch
+and price. Finally, we show how to put all these pieces together in an algorithm that, exploiting
+2
+
+a standard multi-thread computational environment, (i) runs a standard branch and cut on some
+threads, (ii) computes in parallel the DW dual bound in a separate thread and (iii) on the fly
+enhances the original MIP formulation by the Fenchel cuts associated with the DW bound if they
+are predicted to improve the overall solution time for solving the instance. The framework is simple
+and general, holding the promise of being implementable in a relatively easy way in general-purpose
+MIP solvers.
+Paper Organization.
+The remainder of the paper is organized as follows. In Section 2, we
+provide some algorithmic preliminaries on DW decomposition, while in Section 3, we propose
+our new approach for recovering the DW bound in the original formulation by Fenchel cuts and
+compare it with the objective function cut. In Section 4, we discuss Lagrangian relaxation as an
+alternative approach for computing the DW bound and generating cutting planes. Some techniques
+for strengthening the proposed cuts are presented in Section 5. Section 6 reports our computational
+investigation on the Multiple Knapsack Assignment Problem, while Section 7 proposes and tests
+the multi-thread hybrid algorithm. Finally, some short conclusions are drawn in Section 8.
+2
+Preliminaries
+We assume MIP (1) is feasible. Throughout the paper, we call formulation (1) the original for-
+mulation of the MIP and call constraints Ax ≥ b (potentially empty) linking constraints. In the
+context of Dantzig-Wolfe decomposition, (1) is sometimes called the compact formulation. MIPs
+of this form with disjoint index sets I(j) are called loosely coupled MIPs [16]. Note that we do
+not assume the original MIP (1) is loosely coupled and as such the supports of different blocks
+can have overlaps (e.g., in MIPs with two-stage [17] or overlapping [18] block structures). We call
+the LP relaxation of (1), obtained by dropping the integrality constraints x ∈ X, the natural LP
+relaxation, and denote its optimal objective value by zL. Throughout, we assume that all data is
+rational.
+2.1
+Dantzig-Wolfe Decomposition
+We next reformulate (1) by replacing the constraints xI(j) ∈ P j with xI(j) ∈ conv(Qj), where
+Qj = {y ∈ R|I(j)| : Gjy ≥ gj, y ∈ Xj},
+and Xj has the integrality constraints inherited from X for the variables xI(j). Replacing the
+constraints xI(j) ∈ P j with xI(j) ∈ conv(Qj) in (1), one obtains the following Dantzig-Wolfe
+reformulation of problem (1):
+z∗ := min c⊤x
+s.t. xI(j) ∈ conv(Qj),
+j ∈ J,
+Ax ≥ b,
+x ∈ X.
+(2)
+We call the continuous relaxation of (2), obtained by relaxing the integrality constraints x ∈ X, the
+DW relaxation of (1), and the bound zD obtained by solving the DW relaxation the DW bound
+of (1). The DW bound zD is potentially stronger than the natural LP relaxation bound zL as
+P j ⊇ conv(Qj) for all j ∈ {1, . . . , q}. Consequently, we have
+z∗ ≥ zD ≥ zL.
+3
+
+In practice, computing zD is not as straightforward as solving the continuous relaxation of (2)
+since polyhedra
+�
+conv(Qj)
+�q
+j=1 are not given explicitly. One often has to apply the so-called DW
+decomposition to compute zD.
+By the Minkowski-Weyl theorem and Meyer’s theorem [19] for
+rational polyhedra, each conv(Qj) can be represented by its extreme points and extreme rays. For
+j ∈ J, let V j and Rj denote the set of extreme points and the set of extreme rays of conv(Qj),
+respectively. The DW reformulation (2) can be further reformulated using (V j)j∈J and (Rj)j∈J,
+leading to the following extended reformulation of (2):
+z∗ = min c⊤x
+(3a)
+s.t. xI(j) =
+�
+v∈V j
+λvv +
+�
+r∈Rj
+µrr,
+j ∈ J,
+(3b)
+�
+v∈V j
+λv = 1,
+j ∈ J,
+(3c)
+λ ≥ 0, µ ≥ 0
+(3d)
+Ax ≥ b,
+x ∈ X.
+(3e)
+The LP relaxation of (3) (obtained by dropping the integrality constraints x ∈ X, and equivalent
+to DW relaxation) can be solved iteratively via column generation. Specifically, at each iteration,
+a restricted LP is solved by replacing V j and Rj with a small collection of extreme points ˆV j ⊆ V j
+and extreme rays ˆRj ⊆ Rj. A new extreme point or a new extreme ray is generated by solving a
+pricing subproblem
+Dj(πj) := min
+�
+(πj)⊤v : v ∈ conv(Qj)
+�
+= min
+�
+(πj)⊤v : v ∈ Qj�
+(4)
+for each block j ∈ J, where πj above is the dual solution associated with constraints (3b) in the
+restricted LP relaxation of (3). By convention, we define Dj(πj) = −∞ if the pricing subproblem
+(4) is unbounded, and when this happens one can obtain an extreme ray r ∈ Rj by finding an
+unbounded ray of min{(πj)⊤v : v ∈ P j}, which is added to ˆRj. If, on the other hand, Dj(πj) is
+finite, one obtains a solution of (4) as an extreme point v ∈ V j. This point is added to ˆV j provided
+that it has a negative reduced cost that is computed by subtracting the dual variable associated
+with the j-th constraint of (3c) from Dj(πj). The restricted LP, with augmented vertices and rays,
+is then solved again and this process is repeated until no such points or rays are generated.
+The restricted LP at each iteration gives an upper bound for zD, and these upper bounds
+converge to zD in a finite number of iterations as |V j| and |Rj| are finite for all j ∈ J. The algorithm
+iterates between the restricted LP master problem and pricing subproblems until the solutions of
+the pricing subproblems all have nonnegative reduced costs, in which case it can be shown that the
+optimal objective value of the restricted LP takes value zD. Such an inner approximation approach
+for solving the LP relaxation of (3) is called DW decomposition [6]. Algorithm 1 in Appendix A
+presents a pseudocode describing the standard DW decomposition. When the algorithm terminates,
+in addition to the lower bound zD, an optimal solution to the LP relaxation of (2) is obtained.
+This solution does not necessarily satisfy the integrality constraints x ∈ X in (3). If this solution
+is fractional, branching is necessary to obtain an optimal solution of the original problem (3). The
+subproblems in the branch-and-bound tree can again be solved using column generation, and this
+procedure for finding the exact solution of (3) is called branch and price [7]. A notable extension
+of branch and price is the so-called branch-and-cut-and-price algorithm, where cutting planes that
+are valid for the original formulation are added to LP relaxations at branch-and-price nodes to
+further strengthen the formulation (see, e.g., [20, 21]).
+4
+
+Figure 2: Solutions of DW Pricing Subproblems Build Both Inner (Blue, Dashed) and Outer (Red,
+Solid) Approximations
+2.2
+Inner and Outer Approximations
+In DW decomposition, the restricted LP at each iteration corresponds to an inner approximation of
+the DW reformulation (2). The inner approximation is iteratively augmented by extreme points and
+extreme rays generated from the pricing subproblems. At the same time, the pricing subproblems
+can also be used to derive an outer approximation of (2). Specifically, by definition of the pricing
+subproblems, for each j ∈ {1, . . . , q} and πj ∈ RI(j), inequality (πj)⊤y ≥ Dj(πj) is valid for
+conv(Qj). Therefore, given the particular dual multipliers (πj(τ))q
+j=1 obtained at iteration τ of
+DW decomposition, the following inequalities are valid for the DW relaxation of (2):
+(πj(τ))⊤xI(j) ≥ Dj(πj(τ)),
+j ∈ J.
+(5)
+Then, for each iteration t of DW decomposition, the following inequality holds:
+zD ≥ min c⊤x
+s.t.
+�
+πj(τ)
+�⊤xI(j) ≥ Dj
+�
+πj(τ)
+�
+,
+j ∈ J, τ = 1, . . . , t,
+xI(j) ∈ P j,
+j ∈ J,
+Ax ≥ b.
+The above inner approximation can be potentially stronger than the natural LP relaxation of (1)
+as (5) may cut off solutions that are feasible to the natural LP relaxation. The overall inner and
+outer process is depicted in Figure 2.
+3
+Incorporating the DW Bound Into the Formulation
+As anticipated, it has been shown in numerous applications that the DW bound can be much
+stronger than the natural LP relaxation bound [22, 23, 24]. However, directly using formulation
+(3) is not practical since formulation (3) has potentially exponentially many variables and requires
+enumerating all vertices and extreme rays of
+�
+conv(Qj)
+�q
+j=1.
+Even when the LP relaxation of
+formulation (3) is solved by column generation and the DW bound zD is effectively computed,
+enforcing integrality of x ∈ X requires specialized branching and DW decompostion must be
+applied at every node, leading to the branch-and-price algorithm. A straightforward approach for
+enforcing the DW bound zD in the original space is to add to the LP relaxation of (1) a cut of the
+form
+c⊤x ≥ zD,
+5
+
+which we call the objective function cut. However, it is well known that adding such an objective
+function cut often slows down the branch-and-cut solution process in practice.
+We first observe a basic property of the optimal face of the LP relaxation after adding the
+objective function cut.
+Proposition 1. Let P be a polyhedron in Rn. If neither c⊤x ≤ v nor c⊤x ≥ v is valid for P, then
+dim(P ∩ {x : c⊤x = v}) = dim(P) − 1.
+Proof. Define P + := P ∩ {x : c⊤x ≤ v}. Note that c⊤x ≤ v is an irredundant inequality for P +
+because c⊤x ≤ v is not valid for P. By [25, Lemma 3.26], dim(P ∩ {x : c⊤x = v}) = dim(P +) − 1.
+We next show that the affine hull of P + is equal to the affine hull of P, and thus dim(P +) = dim(P).
+By [25, Theorem 3.17], we only need to show the following two statements:
+1. Equality c⊤x = v is not valid for P +;
+2. If a⊤x ≤ a0 is valid for P but a⊤x = a0 is not valid for P, then a⊤x = a0 is not valid for P +.
+Note that c⊤x ≥ v is not valid for P. Therefore, there exists ˆx ∈ P such that c⊤ˆx < v. The
+first statement then follows from the fact that ˆx ∈ P +. Similarly, there exists ¯x ∈ P such that
+a⊤¯x < a0. For λ ∈ (0, 1), define xλ := (1 − λ)ˆx + λ¯x. Because a⊤¯x ≤ a0, we have xλ ∈ P + but
+a⊤xλ < a0 for a small enough λ. The second statement then follows.
+When zD > zL, adding the objective function cut into the original formulation would create an
+optimal face almost as high-dimensional as the original LP relaxation polyhedron under the mild
+assumption that the LP relaxation of (1) contains a point x′ with c⊤x′ > zD. Consequently, adding
+the objective function cut would often yield a formulation with a large optimal face. This, in turn,
+can cause not only performance variability [26], but also serious computational issues especially in
+early stages of the branch and cut in terms of branching [27], as well as cutting plane generation.
+3.1
+An alternative approach
+In Section 2.2, we observed that valid inequalities for the DW relaxation can be generated while
+solving the pricing subproblems, and adding these valid inequalities into the natural LP relaxation
+can at most yield DW bound zD. We next show that relatively few of these cutting planes can
+readily recover zD. Specifically, assume at iteration t of DW decomposition, the restricted LP we
+solve is of the form
+zt
+D = min c⊤x
+s.t. xI(j) =
+�
+v∈ ˆV j
+vλj
+v +
+�
+r∈ ˆRj
+rµj
+r,
+j ∈ J,
+(πj)
+Ax ≥ b,
+(β)
+�
+v∈ ˆV j
+λj
+v = 1,
+j ∈ J,
+(θj)
+λj ≥ 0, µj ≥ 0,
+j ∈ J.
+(6)
+Let (π1,t, . . . , πq,t, βt, θt
+1, . . . , θt
+q) denote the optimal values of dual variables (π1, . . . , πq, β, θ1, . . . , θq)
+for the restricted LP at iteration t. We then have the following result for the valid inequalities
+derived at the last iteration of DW decomposition.
+6
+
+Theorem 2. Assume DW decomposition terminates in T iterations. Then,
+zD = min c⊤x
+(7a)
+s.t. (πj,T )⊤xI(j) ≥ Dj(πj,T ),
+j ∈ J,
+(7b)
+Ax ≥ b.
+(7c)
+Proof. The “≥” direction is implied by the definition of zD as inequality (7b) is valid for conv(Qj)
+for j ∈ J. We next show the “≤” direction. Based on LP duality of (6) at iteration T and the
+termination condition of DW decomposition, the following equalities hold:
+1. zD = b⊤βT + �q
+j=1 θT
+j ;
+2. ci = A⊤
+i βT + �
+j:i∈I(j) πj,T
+i
+, i = 1, . . . , n.
+Note that at the last iteration T, the DW pricing subproblems are bounded. Let (vj,T )q
+j=1 denote
+the solutions of the DW pricing subproblems at iteration T. Note that the reduced costs associated
+with points (vj,T )q
+j=1 are nonnegative at iteration T of DW decomposition, i.e., (πj,T )⊤vj,T − θT
+j =
+Dj(πj,T ) − θT
+j ≥ 0 for j ∈ J. Therefore, for each solution x satisfying (7b) and (7c), we have the
+following inequality:
+c⊤x =
+n
+�
+i=1
+cixi =
+n
+�
+i=1
+�
+xiA⊤
+i βT +
+�
+j:i∈I(j)
+xiπj,T
+i
+�
+= (βT )
+����
+≥0
+⊤ Ax
+����
+≥b
++
+q
+�
+j=1
+(πj,T )⊤xI(j)
+�
+��
+�
+≥Dj(πj,T )
+≥ b⊤βT +
+q
+�
+j=1
+Dj(πj,T ) ≥ b⊤βT +
+q
+�
+j=1
+θT
+j = zD.
+(8)
+We call inequalities (7b) last-iteration DW cuts. Theorem 2 shows that q last-iteration DW
+cuts together with linking constraints recover the DW bound zD. We remark that the last-iteration
+DW cuts are not necessarily all nontrivial. It is possible that πj,T = 0 for some j ∈ J, which implies
+that the convexification of the j-th block has no impact on improving the dual bound.
+It is worth emphasizing that (7) is not a valid formulation for the MIP (1) even if we add
+integrality constraints x ∈ X to it. One should use last-iteration DW cuts as cutting planes and
+add them to the original formulation (1) to obtain a valid formulation whose LP relaxation bound
+is precisely zD. Also note that Theorem 2 does not imply that last-iteration DW cuts dominate
+other cutting planes of the form (5) that can be generated at intermediate iterations τ < T, in the
+sense that intermediate-iteration DW cuts may still cut off fractional points that do not violate
+any of the last-iteration DW cuts.
+3.2
+Dual Degeneracy and LP Optimal Face
+When comparing the strength of different collections of cutting planes or different formulations,
+very often the LP relaxation bound is used as the sole criterion. However, the effectiveness of two
+formulations in branch and cut may differ significantly even if they have the same or very similar
+LP relaxation bounds. A particular measure that should also be taken into account is the dual
+degeneracy level of the LP relaxation of the formulation [27]. A dual basic solution of an LP is
+called dual degenerate if at least one of the dual basic variable is set to 0 in that solution. Next, we
+formally define the degeneracy level of a dual basic solution of an LP (given in inequality form).
+7
+
+Definition 1. Consider an LP with n variables and m inequality constraints, and let α be a basic
+feasible dual solution. We define the degeneracy level of α to be n − ∥α∥0.
+A highly dual degenerate LP relaxation is associated with many alternative LP basic primal
+optimal solutions, which usually corresponds to a large optimal face. The following result shows
+how the size of the optimal face, in particular, its dimension, is related to the degeneracy level of
+a dual basic optimal solution.
+Proposition 3. Assume α∗ ∈ Rm
++ is a dual basic optimal solution of an LP with n variables and
+m inequality constraints. Then, the optimal face of the LP has dimension at most n − ∥α∗∥0. Fur-
+thermore, if α∗ is the unique dual optimal solution, then the optimal face of the LP has dimension
+exactly n − ∥α∗∥0.
+Proof. Assume the LP is of the form min{c⊤x : Gx ≥ h}. Let F denote the optimal face of the
+LP, i.e.,
+F = {x : Gx ≥ h, c⊤x ≤ h⊤α∗}.
+(9)
+Let (gk)⊤ denote the k-th row of G.
+By complementary slackness of LP, (gk)⊤x = hk for all
+x ∈ F for all k with α∗
+k > 0. Since α∗ is a dual basic optimal solution, {(gk, hk)}k:α∗
+k>0 are linearly
+independent. Otherwise, there exists β ∈ Rm \ {0} such that βk = 0 for all k with α∗
+k = 0 and
+�
+k:α∗
+k>0 βk(gk, hk) = 0. Note that α∗ + ϵβ and α∗ − ϵβ are then both dual optimal solutions of the
+LP for small enough positive ϵ, which contradicts the fact that α∗ is a dual basic optimal solution.
+Therefore, dim(F) ≤ n − rank({gk}k:α∗
+k>0) = n − rank({(gk, hk)}k:α∗
+k>0) = n − ∥α∗∥0. Here, the
+first inequality follows from [25, Theorem 3.17], the second equality follows from the consistency
+of the linear system {(gk)⊤x = hk}k:α∗
+k>0, and the third equality follows from linear independence
+of {(gk, hk)}k:α∗
+k>0.
+If α∗ is the unique dual optimal solution, by strict complementary slackness of LP [28], there
+exists an optimal solution x∗ ∈ F of the LP, such that (gk)⊤x∗ > hk for all k with α∗
+k = 0. It implies
+that {(gk)⊤x = hk}k:α∗
+k>0 and c⊤x = h⊤α∗ are exactly all the implicit equalities that hold in the
+inequality description (9) of F. By LP duality, c⊤x = h⊤α∗ is implied by {(gk)⊤x = hk}k:α∗
+k>0.
+Therefore, by [25, Theorem 3.17], dim(F) = n − rank({gk}k:α∗
+k>0) = n − ∥α∗∥0.
+We remark that Proposition 3 does not extend to dual nonbasic optimal solutions, moreover,
+the ℓ0-norm of dual nonbasic optimal solutions can be greater than n. Note that the dual optimal
+solution is unique when the primal solution is nondegenerate. For the primal degenerate case,
+even if there exists a unique dual basic optimal solution, it is possible that the dimension of the
+optimal face of the LP is strictly less than the degeneracy level of that dual basic optimal solution.
+See Example 2 in Appendix C for an example where the unique dual basic optimal solution has a
+strictly positive dual degeneracy level but the primal optimal solution is still unique.
+Under some mild assumptions, Proposition 3 implies the dimension of the optimal face of the
+LP (7).
+Proposition 4. Assume the LP (7) has a unique dual optimal solution. Then, the optimal face
+of (7) has dimension n − q − ∥βT ∥0.
+Proof. Note that the proof of Theorem 2 implies that (1, . . . , 1, βT ) is a dual optimal solution of
+(7). The result then follows from Proposition 3.
+Proposition 4 also implies that if the dual optimal solution is unique, then none of the DW
+last-iteration cuts can be redundant. If this is not the case, the dimension of the optimal face
+8
+
+after adding the last iteration cuts depends on the number of cuts that are active. Note that when
+applying the last-iteration DW cuts in practice, we would add them to the original formulation
+(1), resulting in an LP optimal face whose size can be even smaller due to constraints xI(j) ∈ P j,
+j ∈ J.
+4
+Bound Computation and Cut Generation via Lagrangian Re-
+laxation
+Lagrangian relaxation [29] is an alternative approach for computing the DW bound zD. In La-
+grangian relaxation, separate auxiliary variables are created for each block and these auxiliary
+variables are related to the original variables using additional (copying) constraints. The copying
+constraints together with the linking constraints Ax ≥ b are then dualized into the objective to
+obtain a Lagrangian relaxation of (2). More precisely, one writes
+zD = min c⊤x
+(10a)
+s.t. yj ∈ conv(Qj),
+j ∈ J,
+(10b)
+yj = xI(j),
+j ∈ J,
+(πj)
+(10c)
+Ax ≥ b.
+(β)
+(10d)
+After dualizing constraints (10c) and (10d) using variables π and β ≥ 0, one obtains the following
+Lagrangian relaxation:
+z(π, β) = min c⊤x +
+q
+�
+j=1
+(πj)⊤(yj − xI(j)) + β⊤(b − Ax),
+s.t. yj ∈ Qj,
+j ∈ J.
+(11)
+Note that when the index sets {I(j)}q
+j=1 are nonoverlapping, (10) and (11) can be simplified by
+properly removing the copying constraints and the associated dual variables π.
+In general, it follows from Lagrangian duality [30] that the largest Lagrangian relaxation bound
+matches zD, i.e.,
+zD = max
+β≥0,π z(π, β).
+(12)
+Note that x is unconstrained in (11) and therefore z(π, β) = −∞ unless the coefficients of the x
+variables in the objective function are zero, i.e.,
+ci −
+�
+j:i∈I(j)
+πj
+i − β⊤Ai = 0
+for all i ∈ {1, . . . , n},
+where Ai denotes the i-th column of A. Consequently, (12) can also be written as
+zD = max z(π, β)
+(13a)
+s.t.
+�
+j:i∈I(j)
+πj
+i + β⊤Ai = ci,
+i = 1, . . . , n,
+(13b)
+β ≥ 0.
+(13c)
+Problem (13) is called the Lagrangian dual problem. For (π, β) satisfying (13b) and (13c), it holds
+that
+z(π, β) =
+q
+�
+j=1
+Dj(πj) + b⊤β,
+9
+
+Figure 3: Comparison of the Cutting Plane Method (Left) and the Level Method (Right) on Solving
+the Lagrangian Dual Problem
+0
+200
+400
+600
+800
+1000
+No. of Iterations
+1000
+800
+600
+400
+200
+0
+Relative Gap (%)
+cutting plane method UB
+cutting plane method LB
+0
+200
+400
+600
+800
+1000
+No. of Iterations
+1000
+800
+600
+400
+200
+0
+Relative Gap (%)
+level method UB
+level method LB
+where Dj : R|I(j)| → R ∪ {−∞} is a piecewise linear concave function of the form
+Dj(πj) = min{(πj)⊤v : v ∈ Qj}.
+(14)
+Therefore, (13) is a nonsmooth convex optimization problem with a separable objective function. It
+is worth emphasizing that the pricing problem (4) in DW decomposition has exactly the same form
+as (14). The function values and supergradients of the concave function Dj(·) can be evaluated by
+solving (14) [29] (an optimal solution of (14) is a supergradient of Dj(·) at πj). This alternative
+way of viewing DW bound zD as the optimal value of the Lagrangian dual problem allows us
+to use various convex optimization methods for computing DW bound zD.
+For example, DW
+decomposition is equivalent to applying the classical cutting plane method [31] to solve (13). Since
+the description of Qj involves integer variables in general, functions Dj(·) are often piecewise
+linear concave with exponentially many pieces. In that case, convex optimization methods with
+some stabilization techniques (e.g., the level method [32]) often outperform the cutting plane
+method, and the difference can be significant. Figure 3 is a representative example of the difference
+in performance between the cutting plane method and the level method for the DW bound zD
+(averaged over a set of multiple knapsack assignment problem instances that are used in Section
+6). Details about our implementation of the level method are presented in Appendix B.
+4.1
+Dantzig-Wolfe Fenchel Cuts
+The cutting planes we derived in Section 2.2 belong to a more general class of cutting planes
+called Fenchel cuts [15]. Fenchel cuts are cutting planes that can be derived from a subsystem of
+constraints (including integrality constraints) of a given MIP formulation. Let the feasible region
+corresponding to such a subsystem is given by
+Q = {x ∈ Rn : Gx ≥ g, x ∈ X}.
+Given a direction µ ∈ Rn, a Fenchel cut (associated with Q in direction µ) is given by µ⊤x ≥ f(µ),
+where f(µ) = minx{µ⊤x : x ∈ Q}. In other words, an inequality is a Fenchel cut (associated with
+Q) if and only if it is valid and have the tightest possible right-hand side for Q.
+In particular, the j-th inequality in (5) is a Fenchel cut associated with the subsystem defined
+by Q = {x ∈ Rn : GjxI(j) ≥ gj, xI(j) ∈ Xj}. We call such a Fenchel cut associated with a
+particular block of the DW reformulation a Dantzig-Wolfe Fenchel cut.
+10
+
+Definition 2. We call an inequality a Dantzig-Wolfe Fenchel (DWF) cut for (1) if it is of the form
+(πj)⊤xI(j) ≥ Dj(πj)
+(15)
+for some j ∈ {1, . . . , q}, where Dj(·) is defined in (4).
+We have shown in Section 3 that DWF cuts associated with the last iteration of DW decompo-
+sition can recover DW bound zD and applying these DWF cuts rather than the objective function
+cut for enforcing DW bound yields root node relaxations with lower dual degeneracy. Another
+advantage of using DW cuts is regarding the dimension of the cuts, which is often used as a mea-
+sure of the strength of the cutting plane. A higher dimension of the associated face means that
+the cutting plane is closer to being facet-defining and irredundant, and therefore stronger in some
+sense. Let S denote the feasible region of the original problem (1). We next present a relationship
+between the dimension of a DWF cut in conv(S) and its restriction in conv(Qj).
+Definition 3. We say that MIP (1) has block relative feasibility if for each j ∈ {1, . . . , q} and
+y ∈ Qj there exists x ∈ S such that xI(j) = y, i.e., projxI(j)(S) = Qj for j ∈ J.
+The block relative feasibility assumption holds for a broad class of MIP problems. For example,
+for two-stage stochastic integer programs, relatively complete recourse [33] implies block relative
+feasibility if each block is defined by first-stage and second-stage constraints associated with a
+particular scenario.
+Proposition 5. Assume problem (1) has block relative feasibility. If (πj)⊤y ≥ Dj(πj) defines a
+d-dimensional face of conv(Qj) for some j ∈ {1, . . . , q}, then (15) defines a face of conv(S) of
+dimension at least d.
+Proof. Since (πj)⊤y ≥ Dj(πj) defines a d-dimensional face of conv(Qj), there exists d + 1 affinely
+independent points {yk}d+1
+k=1 ⊆ Qj. By the block relative feasibility assumption, there exist points
+{xk}d+1
+k=1 ⊆ S such that xk
+I(j) = yk. Points {xk}d+1
+k=1 are affinely independent from each other by
+affine independence of {yk}d+1
+k=1. The conclusion then follows from the fact that {xk}d+1
+k=1 are all on
+the face associated with (15) in conv(S).
+Proposition 5 indicates that DWF cuts whose restrictions in the space of xI(j) correspond
+to high-dimensional faces of conv(Qj) are likely to define high-dimensional faces in conv(S). It
+motivates the idea of strengthening some DWF cuts to obtain higher-dimensional DWF cuts, which
+will be discussed in Section 5. In contrast to DWF cuts that potentially define high-dimensional
+faces of conv(S), the objective function cut c⊤x ≥ zD usually corresponds to an empty face of
+conv(S) unless zD = z∗. Even if zD = z∗, the face associated with the objective function cut may
+still be low-dimensional unless the problem has many alternative optimal primal solutions.
+4.2
+Cut Generation From the Lagrangian Dual
+As discussed earlier, solving the Lagrangian dual problem can be computationally more efficient
+than the standard DW decomposition. During the solution of the Lagrangian dual problem, DWF
+cuts similar to (5) can also be generated every time we evaluate the function values of Dj(·).
+The following result demonstrates the strength of DWF cuts generated from the evaluation of the
+Lagrangian dual function at any point (π, β) with z(π, β) > −∞.
+11
+
+Proposition 6. Let (π, β) be Lagrangian dual multipliers for (11) satisfying constraints (13b) and
+(13c). Then,
+z(π, β) ≤ min c⊤x
+s.t. (πj)⊤xI(j) ≥ Dj(πj),
+j ∈ J,
+Ax ≥ b.
+(16)
+Proof. Consider any x ∈ Rn feasible to the right-hand side LP of (16). Since ci = �
+j:i∈I(j) πj
+i +
+β⊤Ai for i = 1, . . . , n by (13b), we have
+c⊤x =
+n
+�
+i=1
+�
+j:i∈I(j)
+πj
+i xi + β⊤Ax =
+q
+�
+j=1
+(πj)⊤xI(j)
+�
+��
+�
+≥Dj
+�
+πj�
++ β
+����
+≥0
+⊤ Ax
+����
+≥b
+≥
+p
+�
+j=1
+Dj(πj) + b⊤β(τ) = z(π, β).
+Note that, unlike Theorem 2, Proposition 6 does not depend how the dual multiplier is obtained.
+If one can solve the Lagrangian dual problem to optimality, then Proposition 6 implies that one
+can recover DW bound using DWF cuts associated with an optimal solution of the Lagrangian
+dual problem.
+Corollary 7. Let (¯π, ¯β) be an optimal solution of (13). Then,
+zD = min c⊤x
+s.t. (¯πj)⊤xI(j) ≥ Dj(¯πj),
+j ∈ J,
+Ax ≥ b.
+(17)
+Results similiar to Propostion 4 can be derived for the optimal face of the LP (17) that utilizes
+DWF cuts associated with an optimal Lagrangian dual solution.
+Even if the Lagrangian dual
+problem is not solved to optimality, Proposition 6 still guarantees that DWF cuts can provide
+strength at least as strong as the best Lagrangian dual bound by generating DWF cuts associated
+with the dual solution that provides the strongest bound obtained so far. Besides, there may be
+values for adding DWF cuts obtained at different dual multipliers. See Example 1 in Appendix C
+showing that DWF cuts can potentially provide stronger dual bounds than the best Lagrangian
+dual bound.
+5
+Generating a Stronger Relaxation
+In this section we describe how to strengthen the DWF cuts to obtain a stronger relaxation. The
+strengthened cutting planes are still DWF cuts valid for blocks Qj, and therefore do not lead to
+better bounds than zD. However, these strengthened cutting planes may potentially make more
+constraints active in the LP relaxation, and therefore can help reduce the dual degeneracy level of
+the formulation and improve the branch-and-cut performance. Moreover, the strengthened cutting
+planes are more likely to define high-dimensional faces of the original problem, as they define
+higher-dimensional faces of the block polyhedra conv(Qj).
+12
+
+Figure 4: Disjunctive Coefficient Strengthening for Three Cases
+xi
+0
+1
+xi
+0
+1
+r
+xi
+0
+1
+x0
+x1
+5.1
+Disjunctive Coefficient Strengthening
+We first describe a disjunctive coefficient strengthening technique for binary variables [34] that
+strengthens the coefficients of a valid inequality one at a time. Given a valid inequality a⊤x ≥ f
+for the set Q, let Q= := {x ∈ Q : a⊤x = f}. For a binary variable xi, its coefficient ai in the cut
+can be strengthened if one of the following two cases hold: (i) xi = 0 for all x ∈ Q=, or, (ii) xi = 1
+for all x ∈ Q=. If there are points x0, x1 ∈ Q= such that x0
+i = 0 and x1
+i = 1, then this approach
+does not improve (i.e., decrease) the coefficient of the variable. Figure 4 shows an example of all
+three cases.
+Note that if we solve
+¯f = min{a⊤x : x ∈ Q, xi = 1}
+(18)
+and observe that ¯f > f, then we can strengthen the original inequality a⊤x ≥ f to be a⊤x ≥
+f + ( ¯f − f)xi using the disjunction
+Q = {x ∈ Q : xi = 0} ∪ {x ∈ Q : xi = 1}.
+Similarly, if we solve
+¯f = min{a⊤x : x ∈ Q, xi = 0},
+(19)
+and observe that ¯f > f, then we can strengthen the original inequality a⊤x ≥ f to be a⊤x ≥
+f + ( ¯f − f)(1 − xi). If either problem (18) or (19) is infeasible, then one can simply fix variable xi
+to 0 or 1, respectively. It is easy to verify that the original inequality is implied by the strengthened
+inequality together with the bound constraint xi ≥ 0 or xi ≤ 1. Therefore, if the original inequality
+is active in the LP relaxation, then one round of coefficient strengthening (if applicable) would
+potentially make a bound constraint active in the LP relaxation and reduce the dual degeneracy
+level. Note that this approach does not increase the size of the formulation.
+For strengthening a DWF cut obtained from a block j, we set Q = Qj and apply coefficient
+strengthening sequentially to all coefficients of binary variables. In practice, we keep a set L of
+points that are known to be elements of Q=, generated from previous solutions of (14), (18) and
+(19). If there are x0, x1 ∈ L such that x0
+i = 0 and x1
+i = 1, then without solving (18) or (19) we
+conclude that disjunctive strengthening cannot be applied to the i-th coefficient.
+5.2
+Strengthening via Tilting
+We next describe a tilting technique introduced in [35, 36] that starts with a valid inequality and
+iteratively tilts it to obtain a facet-defining inequality. Let a⊤x ≥ f be a valid inequality for Q
+and assume that it is not facet-defining. Also assume that conv(Q) is full dimensional and there
+is a set Q′ ⊆ Q such that all points x ∈ Q′ satisfy a⊤x = f. The algorithm first generates a point
+13
+
+¯x ∈ Q \ Q′ that satisfies a⊤¯x > f, and a vector (v, w) such that v⊤x = w for all x ∈ Q′ ∪ {¯x}. The
+algorithm then does the following:
+1. If v⊤x ≥ w is valid for Q, then the algorithm outputs ¯x. Otherwise the algorithm computes
+the largest λ+ ∈ R+ such that (a + λ+v)⊤x ≥ f + λ+w is valid for Q and outputs this
+inequality together with a point ¯x+ ∈ Q \ Q′ satisfying (a + λ+v)⊤¯x+ = f + λ+w;
+2. If v⊤x ≤ w is valid for Q, then the algorithm outputs ¯x. Otherwise the algorithm computes
+the largest λ− ∈ R+ such that (a − λ−v)⊤x ≥ f − λ−w is valid for Q and outputs this
+inequality together with a point ¯x− ∈ Q \ Q′ satisfying (a − λ−v)⊤¯x− ≥ f − λ−w.
+Note that when conv(Q) is full dimensional, both v⊤x ≥ w and v⊤x ≤ w cannot be valid and
+consequently we obtain two (possibly identical) valid inequalities whose conic combination implies
+the original inequality a⊤x ≥ f. Moreover, each one of the these inequalities has one more known
+feasible point on its associated face than a⊤x ≥ f. In [36], the authors show that recursively tilting
+one of the two obtained valid inequalities leads to a facet-defining inequality for conv(Q).
+In our context, we apply the tilting idea with the following modifications. For a DWF cut
+associated with block j, we set Q to be Qj, and instead of picking one of the two tilted inequalities
+for the subsequent tilting iteration, we apply tilting to both. By doing so, we create a binary tree
+where the root node corresponds to the original DWF cut and the remaining nodes correspond to
+DWF cuts obtained by tilting the inequalities associated with their parent nodes. For any such
+tree, cuts associated with the leaf nodes imply the original DWF cut associated with the root node.
+We call the collection of inequalities associated with the leaf nodes of a depth-d tree depth-d tilted
+DWF cuts. Using a depth-d tree means replacing one DWF cut with up to 2d DWF cuts, which
+can be computationally expensive if d is large. Therefore, it is often beneficial to choose a relatively
+small value for d. Note that this process does not improve the DW bound but can potentially help
+reduce the dimension of the LP optimal face as more constraints are likely to become active at the
+optimal face.
+To improve computational performance, we keep track of the feasible points for block j ∈ J
+that we encounter during the overall algorithm and store them in a set ˆQj ⊆ Qj. Using this set of
+points reduces the number of oracle calls needed in the following two ways. First, we can check if
+any point in ˆQj satisfies the condition for ¯x and use it for choosing (v, w), thus avoiding an extra
+call to the optimization oracle. In addition, when computing λ+, we use ˆQj to obtain the following
+upper bound on it:
+λ+ :=
+min
+x∈Qj:v⊤x<w
+a⊤x − f
+w − v⊤x
+≤
+min
+x∈ ˆQj:v⊤x<w
+a⊤x − f
+w − v⊤x.
+This upper bound can be very close to the final λ+ value and, in practice, we have observed a
+significantly reduction in the number of oracle calls needed to compute λ+. This in turn reduces
+the time spent on tilting significantly. We use the same idea when computing λ− ∈ R+.
+6
+Multiple Knapsack Assignment Problem: A Case Study
+We test our approaches on the Multiple Knapsack Assignment Problem (MKAP) [37] that is an
+extension of the well-studied multiple knapsack and assignment problems. A detailed description of
+the MKAP problem, its DW reformulation as well as the test instances are presented in Appendix
+D. To our knowledge, solving MKAP as an integer program by a branch-and-cut MIP solver is the
+only exact solution approach that has been tried.
+We compare the performance of the following formulations:
+14
+
+Figure 5: Tilting One Valid Inequality Into Two Stronger Valid Inequalities
+1. MIP: The original formulation (1);
+2. OBJ: Objective cut c⊤x ≥ zD added into the original formulation;
+3. DWF: Last-iteration DWF cuts added into the original formulation;
+4. STR: Last-iteration DWF cuts with disjunctive coefficient strengthening added into the orig-
+inal formulation;
+5. DdT: Last-iteration DWF cuts with disjunctive coefficient strengthening and depth-d tilting
+added into the original formulation.
+All experiments are run on a computer with a Intel(R) Xeon(R) Gold 6258R processor running
+at 2.7GHz, with up to 4 threads used. All optimization problems are solved using the optimization
+solver Gurobi [38] version 9.5.0.
+The parameters K, M, and N define the size of these instances. In all the tables presented in
+this section, statistics presented in a row correspond to averages over 30 instances for a particular
+combination of (K, M, N).
+6.1
+Comparing Relaxations
+We first present some results regarding lower bounds. We compare the DW bound zD with the LP
+relaxation bound zL and the Gurobi root node bound (denoted by zR) obtained by adding solver
+cuts. With some abuse of notation, we use zD to denote the best Lagrangian dual bound obtained
+by the level method. In theory, zD should be equal to the DW bound. However, numerical issues
+happen occasionally when solving the quadratic program in the level method, especially when the
+DW bound is close to the LP relaxation bound, in which case zD can be strictly less than zL.
+In Table 1, we present the relative gaps rL := (zD −zL)/|zD| and rR := (zD −zR)/|zD| between
+zD and natural LP relaxation bound zL as well as the Gurobi root bound zR, respectively. The
+running time tR spent at the root node and running time tD spent on solving the Lagrangian dual
+problem by the level method are also reported.
+Depending on the instance size, the relative difference between the DW bound and the natrual
+LP relaxation bound varies. When the gap is small, Gurobi can often generate a root node relax-
+ation that is almost as strong as or even slightly stronger than the DW relaxation. When the gap
+is large, Gurobi can close some gap at the root node but the bound is often significantly weaker
+than DW bound.
+In our preliminary experiments, we observe that DWF cuts are more likely to be effective
+when DW bound is significantly stronger than the natrual LP relaxation bound. Therefore, we
+focus on instance classes (defined by the size parameters |K|, |M| and |N|) whose average natrual
+15
+
+Table 1: Comparison of Different Bounds and Their Computing Time
+|K|
+|M|
+|N|
+rL (%)
+rR (%)
+tR (s)
+tD (s)
+2
+10
+50
+0.16
+-0.04
+0.1
+0.8
+100
+0.00
+-0.03
+0.3
+2.0
+200
+0.00
+-0.01
+0.5
+9.9
+300
+0.00
+-0.01
+0.5
+22.0
+20
+50
+1.98
+0.08
+0.4
+0.7
+100
+0.02
+0.00
+0.3
+2.8
+200
+0.00
+0.00
+0.6
+12.7
+300
+-0.01
+-0.01
+0.8
+37.0
+30
+50
+12.05
+-0.06
+0.1
+0.6
+100
+0.18
+0.10
+0.4
+2.8
+200
+0.00
+0.00
+0.8
+15.4
+300
+-0.05
+-0.05
+1.1
+49.7
+40
+50
+39.11
+0.00
+0.0
+0.7
+100
+0.68
+0.18
+1.9
+2.7
+200
+0.00
+0.00
+0.9
+17.3
+300
+-0.34
+-0.34
+1.4
+55.3
+5
+10
+50
+1.27
+0.44
+0.3
+0.6
+100
+0.11
+0.07
+0.2
+2.4
+200
+0.01
+0.00
+0.4
+12.6
+300
+0.00
+0.00
+0.6
+43.0
+20
+50
+2.97
+0.00
+0.3
+0.8
+100
+0.10
+0.06
+0.3
+3.0
+200
+-0.44
+-0.44
+0.5
+12.9
+300
+-0.02
+-0.02
+0.8
+50.5
+30
+50
+12.27
+0.08
+0.1
+1.0
+100
+0.41
+0.15
+1.5
+3.3
+200
+0.01
+0.00
+0.8
+17.7
+300
+0.00
+0.00
+1.2
+55.8
+40
+50
+39.11
+0.00
+0.0
+1.2
+100
+1.00
+0.14
+1.5
+3.4
+200
+0.01
+0.00
+1.1
+23.1
+300
+-0.05
+-0.05
+1.5
+47.0
+|K|
+|M|
+|N|
+rL (%)
+rR (%)
+tR (s)
+tD (s)
+10
+10
+50
+7.03
+0.86
+0.2
+0.5
+100
+5.65
+4.13
+0.2
+1.2
+200
+3.53
+2.64
+0.3
+4.5
+300
+3.64
+2.79
+0.5
+14.0
+20
+50
+4.76
+-0.08
+0.2
+1.1
+100
+0.74
+0.37
+1.4
+2.7
+200
+0.02
+0.02
+0.6
+17.5
+300
+0.00
+0.00
+0.9
+45.5
+30
+50
+12.82
+0.01
+0.1
+1.4
+100
+0.88
+0.15
+1.2
+3.5
+200
+0.02
+0.01
+0.9
+22.8
+300
+0.00
+0.00
+1.3
+26.1
+40
+50
+39.11
+0.00
+0.0
+1.8
+100
+1.51
+0.12
+1.0
+4.0
+200
+0.04
+0.02
+1.2
+26.6
+300
+0.00
+0.00
+1.8
+25.8
+25
+10
+50
+43.05
+0.21
+0.1
+0.9
+100
+54.17
+1.76
+0.3
+1.2
+200
+58.14
+13.64
+0.8
+1.6
+300
+66.02
+16.63
+1.5
+2.1
+20
+50
+11.08
+-0.01
+0.1
+2.0
+100
+9.40
+0.45
+0.7
+2.7
+200
+8.74
+6.83
+0.9
+5.1
+300
+10.18
+8.78
+0.9
+9.4
+30
+50
+14.35
+0.00
+0.0
+2.8
+100
+3.80
+0.13
+1.0
+4.2
+200
+1.79
+0.85
+7.0
+12.6
+300
+1.30
+1.29
+1.4
+28.3
+40
+50
+39.21
+0.01
+0.0
+3.5
+100
+3.13
+0.00
+1.1
+5.5
+200
+0.75
+0.25
+11.5
+22.7
+300
+0.25
+0.24
+2.0
+24.8
+LP relaxation bound is more than 1% weaker than average DW bound, especially the ones with
+Gurobi root bound more than 1% worse than DW bound. We denote these two sets of instances
+by I1 and I2(⊆ I1), respectively. In Table 1, I1 instances correspond to rows with bold rL and
+I2 instances correspond to rows with bold rR. Set I1 consists of 870 instances and set I2 consists
+of 270 instances. Over the I1 instances, we observe that zD is very close to the DW bound with
+relative gaps always less than 0.01% except on 3 instances (whose relative gaps are 0.03%, 0.16%
+and 0.34%, respectively).
+We next investigate the dual degeneracy of different formulations. In Table 2, we report the
+relative normalized degeneracy level of the optimal dual solution computed by Gurobi for LP
+relaxations of different formulations. The relative degeneracy level is calculated as the degeneracy
+level n−∥α∗∥0 divided by the dimension n, i.e., 1−∥α∗∥0/n. Bold rows correspond to I2 instances.
+Due to numerical tolerances, the optimal values α∗
+k of some dual variables can sometimes be very
+close to but not equal to zero, especially the ones associated with bound constraints. We treat α∗
+k
+as 0 if |α∗
+k| ≤ 10−8 for all entries α∗
+k of α∗ when computing ∥α∗∥0. Although the strengthening
+techniques in Section 5 do not guarantee a monotonic decrease in degeneracy level, it is clear from
+Table 2 that formulations with stronger strengthening tend to have less degenerate optimal dual
+solutions. We observe that the degeneracy level of DWF is already comparable to MIP on I2
+instances but is higher than MIP on I1 \ I2 instances. With coefficient strengthening, STR has
+lower average dual degeneracy than MIP in almost all cases except on the instance class (5,40,50).
+The dual degeneracy continues to decrease with further tilting. Finally, it is worth noting that, as
+expected, the relative degeneracy level of OBJ is extremely high, close to 100%. Time spent on
+generating these different formulations is presented in Appendix E.
+16
+
+Table 2: Relative Degeneracy Levels of LP Optimal Dual Solutions for Different Formulations
+(|K|, |M|, |N|)
+(1 − ∥α∗∥0/n) × 100%
+MIP
+OBJ
+DWF
+STR
+D1T
+D2T
+D3T
+D4T
+D5T
+D6T
+( 2,20, 50)
+56.27%
+99.90%
+78.30%
+41.85%
+36.37%
+30.94%
+25.77%
+23.87%
+21.86%
+21.88%
+( 2,30, 50)
+57.75%
+99.94%
+70.21%
+52.25%
+46.32%
+41.81%
+39.24%
+39.52%
+39.89%
+39.26%
+( 2,40, 50)
+58.06%
+99.95%
+71.04%
+55.17%
+51.67%
+52.29%
+51.82%
+50.46%
+50.46%
+50.46%
+( 5,10, 50)
+50.84%
+99.82%
+77.44%
+42.29%
+32.33%
+22.32%
+14.84%
+11.09%
+11.25%
+11.04%
+( 5,20, 50)
+53.20%
+99.91%
+68.12%
+36.42%
+28.41%
+22.50%
+20.10%
+19.95%
+19.04%
+19.42%
+( 5,30, 50)
+54.60%
+99.94%
+65.54%
+45.31%
+34.81%
+34.89%
+35.45%
+36.05%
+36.11%
+36.11%
+( 5,40, 50)
+54.90%
+99.95%
+70.72%
+55.61%
+51.21%
+50.62%
+50.61%
+50.61%
+50.61%
+50.61%
+(10,10, 50)
+46.60%
+99.83%
+57.63%
+26.99%
+11.24%
+3.01%
+3.35%
+3.46%
+3.46%
+3.46%
+(10,10,100)
+50.65%
+99.91%
+56.98%
+35.05%
+22.84%
+7.77%
+1.36%
+1.14%
+1.49%
+1.54%
+(10,10,200)
+53.30%
+99.95%
+53.77%
+38.02%
+26.06%
+17.45%
+7.19%
+1.05%
+0.26%
+0.28%
+(10,10,300)
+54.23%
+99.97%
+53.50%
+39.77%
+26.75%
+20.75%
+17.99%
+7.47%
+1.43%
+0.61%
+(10,20, 50)
+48.77%
+99.92%
+57.55%
+24.16%
+16.96%
+11.23%
+11.46%
+11.36%
+11.36%
+11.36%
+(10,30, 50)
+50.05%
+99.94%
+57.02%
+38.97%
+33.06%
+31.63%
+31.42%
+31.42%
+31.42%
+31.42%
+(10,40, 50)
+50.32%
+99.96%
+62.98%
+45.03%
+44.05%
+43.36%
+43.36%
+43.36%
+43.36%
+43.36%
+(10,40,100)
+54.98%
+99.98%
+80.55%
+45.61%
+35.69%
+27.61%
+25.77%
+24.92%
+25.41%
+25.50%
+(25,10, 50)
+37.28%
+99.87%
+26.68%
+11.18%
+5.86%
+5.86%
+5.86%
+5.86%
+5.86%
+5.86%
+(25,10,100)
+44.57%
+99.92%
+40.47%
+30.85%
+14.51%
+4.40%
+3.70%
+3.70%
+3.70%
+3.70%
+(25,10,200)
+49.75%
+99.96%
+47.23%
+37.99%
+29.72%
+13.19%
+3.71%
+1.59%
+1.36%
+1.68%
+(25,10,300)
+51.73%
+99.97%
+56.84%
+48.89%
+34.16%
+24.02%
+8.05%
+1.77%
+1.03%
+1.17%
+(25,20, 50)
+39.01%
+99.93%
+34.14%
+11.26%
+6.44%
+6.44%
+6.44%
+6.44%
+6.44%
+6.44%
+(25,20,100)
+47.02%
+99.96%
+52.18%
+26.21%
+10.38%
+3.34%
+3.57%
+3.57%
+3.57%
+3.57%
+(25,20,200)
+52.44%
+99.98%
+62.38%
+38.30%
+24.36%
+7.41%
+0.93%
+0.91%
+1.06%
+1.25%
+(25,20,300)
+54.69%
+99.98%
+62.72%
+42.19%
+30.72%
+22.26%
+4.78%
+0.56%
+0.53%
+0.62%
+(25,30, 50)
+40.04%
+99.96%
+41.02%
+23.72%
+22.94%
+22.94%
+22.94%
+22.94%
+22.94%
+22.94%
+(25,30,100)
+47.84%
+99.97%
+64.22%
+32.04%
+16.22%
+7.41%
+7.57%
+7.57%
+7.57%
+7.57%
+(25,30,200)
+53.39%
+99.99%
+73.83%
+43.35%
+30.70%
+15.59%
+6.42%
+5.51%
+5.67%
+5.89%
+(25,30,300)
+55.68%
+99.99%
+78.99%
+48.56%
+41.44%
+26.72%
+8.59%
+4.23%
+3.58%
+3.27%
+(25,40, 50)
+40.26%
+99.97%
+50.78%
+39.69%
+39.52%
+39.52%
+39.52%
+39.52%
+39.52%
+39.52%
+(25,40,100)
+48.39%
+99.98%
+61.63%
+29.76%
+20.09%
+17.55%
+17.17%
+17.17%
+17.17%
+17.17%
+Figure 6: Relative Gaps Between Bounds and Optimal Value by MIP (Left) and OBJ (Right)
+0
+20000
+40000
+60000
+80000
+100000
+No. of Nodes Explored
+8
+6
+4
+2
+0
+2
+4
+Relative Gap (%)
+MIP UB
+MIP LB
+0
+20000
+40000
+60000
+80000
+100000
+No. of Nodes Explored
+8
+6
+4
+2
+0
+2
+4
+Relative Gap (%)
+OBJ UB
+OBJ LB
+6.2
+Objective Function Cut
+We next emprically investigate how the objective function cut (formulation OBJ) may hurt the
+branch-and-cut solution comparing to simply using the original formulation MIP.
+We consider MKAP instances with (|K|, |M|, |N|) = (25, 20, 300), for which DW bound is
+much stronger than the natrual LP relaxation bound (with average gap 10.18%). Out of these
+30 instances, Gurobi can solve 22 using MIP and 2 using OBJ. We plot in Figure 6 the average
+relative upper and lower bound gaps (with respect to the optimal value) obtained by Gurobi using
+formulations MIP and OBJ on (25, 20, 300) instances as the number of branching nodes explored
+increases. We observe that although DW bound is almost equal to the optimal value for these
+instances (with a 0.01% average relative gap), Gurobi can hardly produce good feasible solutions
+to improve the primal bound when using the OBJ formulation. Instead, Gurobi can find better
+17
+
+feasible solutions much more efficiently when using MIP, where all optimal solutions are found
+after exploring a bit more than 20,000 nodes, leaving a few instances unsolved because Gurobi
+cannot prove optimality within the timelimit. This shows that a single objective function cut can
+already significantly influence the performance of Gurobi’s primal heuristics, which is plausibly due
+to worse branching decisions and cut generation for OBJ. A table reporting the average number of
+different cutting planes generated when using different formulations is presented in Appendix F.
+6.3
+Branch-and-Cut Performance
+Table 3: Comparison of Branch-and-Cut Performance on I2 Instances
+(|K|, |M|, |N|)
+Number of Instances Solved
+Average B&C Time (s)
+MIP
+OBJ
+DWF
+STR
+D3T
+D6T
+MIP
+OBJ
+DWF
+STR
+D3T
+D6T
+(10,10,100)
+28/30
+18/30
+30/30
+30/30
+30/30
+30/30
+≥ 65
+≥298
+2
+0
+0
+0
+(10,10,200)
+11/30
+10/30
+30/30
+30/30
+30/30
+30/30
+≥424
+≥457
+6
+1
+2
+2
+(10,10,300)
+7/30
+11/30
+30/30
+30/30
+30/30
+30/30
+≥489
+≥454
+41
+2
+12
+7
+(25,10,100)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+0
+1
+0
+0
+0
+0
+(25,10,200)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+2
+25
+0
+0
+0
+0
+(25,10,300)
+30/30
+29/30
+30/30
+30/30
+30/30
+30/30
+6
+≥ 40
+0
+0
+1
+3
+(25,20,200)
+29/30
+9/30
+30/30
+30/30
+30/30
+30/30
+≥ 44
+≥499
+3
+2
+1
+1
+(25,20,300)
+22/30
+2/30
+29/30
+30/30
+30/30
+30/30
+≥224
+≥590
+≥ 48
+5
+3
+7
+(25,30,300)
+1/30
+0/30
+0/30
+2/30
+1/30
+3/30
+≥595
+≥600
+≥600
+≥590
+≥591
+≥575
+Table 4: Comparison of Branch-and-Cut Performance on I2 Instances (Cont’d)
+(|K|, |M|, |N|)
+Average Optimality Gap (%)
+Average Number of Nodes
+MIP
+OBJ
+DWF
+STR
+D3T
+D6T
+MIP
+OBJ
+DWF
+STR
+D3T
+D6T
+(10,10,100)
+0.04
+0.03
+0.00
+0.00
+0.00
+0.00
+≥
+22761
+≥1698156
+4481
+220
+2
+1
+(10,10,200)
+0.27
+0.09
+0.00
+0.00
+0.00
+0.00
+≥2778880
+≥3898948
+6828
+407
+33
+3
+(10,10,300)
+0.21
+0.09
+0.00
+0.00
+0.00
+0.00
+≥2616971
+≥2781174
+131139
+1208
+631
+6
+(25,10,100)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+93
+1123
+1
+1
+1
+1
+(25,10,200)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+3853
+20272
+1
+1
+1
+1
+(25,10,300)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+8488
+≥
+25864
+1
+1
+1
+1
+(25,20,200)
+0.02
+0.08
+0.00
+0.00
+0.00
+0.00
+≥
+27987
+≥ 438310
+2255
+426
+3
+2
+(25,20,300)
+0.24
+0.17
+0.00
+0.00
+0.00
+0.00
+≥ 113353
+≥ 410728
+≥ 13861
+929
+1
+1
+(25,30,300)
+0.84
+0.53
+0.40
+0.33
+0.29
+0.22
+≥
+81545
+≥ 379999
+≥105115
+≥35197
+≥20036
+≥6568
+Finally, we compare the branch-and-cut performance of formulations MIP, OBJ, STR, D3T
+and D6T on the selected MKAP instances.
+We set a 10-minute branch-and-cut timelimit for
+each formulation. For I2 instances, we summarize in Tables 3 and 4 some statistics regarding the
+number of instances solved by different formulations, average branch-and-cut time (excluding time
+for the Lagrangian dual problem and cut generation/strengthening), average ending optimality
+gap and average number of branching nodes needed to solve the instances.
+We observe that
+branch and cut on OBJ solves fewer instances than MIP, which is consistent with the folklore that
+simply adding an objective function cut to improve dual bound is mostly ineffective. The ending
+optimality gap of OBJ might be better than that of MIP because OBJ has a much stronger lower
+bound, whereas finding good feasible solutions can be hard for OBJ. Without any strengthening,
+formulation DWF already significantly outperforms both MIP and OBJ in branch and cut. For
+most instances, D6T requires the smallest number of branching nodes to solve the problem, while
+STR has the best solution time. This can be explained by the fact that D6T has the strongest
+LP relaxation while STR has a more compact formulation.
+Compared to D3T or D6T, the
+processing of each branching node takes less time for STR. For the hardest (25, 30, 300) instances,
+a stronger polyhedral relaxation becomes more important for closing the optimality gap, where
+D6T has the best performance.
+In Figure 7 we plot gap closed as time or number of nodes
+explored increases for different methods on the hardest (25, 30, 300) instances where the relative
+18
+
+gap is computed by comparing with the best feasible solution found by all methods. We observe
+that although all methods except MIP start from the same root bound (DW bound zD), with
+coefficient strengthening and tilting the solver can find good primal solutions more efficiently. We
+also note that each branching node takes longer time to explore when coefficient strengthening and
+tilting are applied. Additional tables comparing branch-and-cut performance on I1 \ I2 instances
+are presented in Appendix G.
+Figure 7: Gap Closed as Time (Left) and Number of Nodes Explored (Right) Increases
+0
+100
+200
+300
+400
+500
+600
+Solution Time (s)
+2
+1
+0
+1
+2
+3
+4
+5
+Relative Gap (%)
+MIP UB
+OBJ UB
+DWF UB
+STR UB
+D6T UB
+MIP LB
+OBJ LB
+DWF LB
+STR LB
+D6T LB
+0
+20000
+40000
+60000
+80000
+100000
+No. of Nodes Explored
+2
+1
+0
+1
+2
+3
+4
+5
+Relative Gap (%)
+MIP UB
+OBJ UB
+DWF UB
+STR UB
+D6T UB
+MIP LB
+OBJ LB
+DWF LB
+STR LB
+D6T LB
+7
+Hybrid Implementation in a Multi-Thread Context
+Computationally, using the last iteration DWF cuts (and their strengthened versions) does not
+always lead to a performance improvement as it (i) requires additional computational effort to
+generate the cuts and (ii) increases the size of the formulation. Similar to other MIP techniques,
+one needs to decide whether or not to use this approach for a given problem instance. Machine
+learning (ML) techniques have been widely applied to make algorithmic decisions inside MIP
+solvers. For example, FICO Xpress implements a random forest model to determine whether or
+not local cuts should be generated in nodes other than the root of the branch-and-bound tree
+[39], while CPLEX uses ML to decide whether to linearize the objective function of mixed-integer
+quadratic programming problems [40].
+In the remainder of this section, we present some preliminary experiment results with a simple
+ML model to predict whether or not branch and cut may benefit from DWF cuts. We consider
+a computational environment with 4 threads where one starts with running branch and cut on
+MIP (using 3 threads) and solves the Lagrangian dual problem (using 1 thread) in parallel until
+the Lagrangian dual problem is solved. After this first phase, one decides to either allocate all 4
+threads to the original formulation MIP or to the strengthened formulation STR.
+To collect labels for the training phase, we use 20% of I1 instances as the training set. We first
+run standard branch and cut (using 3 threads) and Lagrangian dual (using 1 thread) in parallel
+until the Lagrangian dual problem is solved, and collect some simple features for each instance.
+We then separately try allocating all 4 threads to (i) solving the original formulation MIP, and
+(ii) solving the strengthened formulation STR. We label an instance as promising if one of the
+following holds when both methods are run for 10 minutes:
+1. The instance can be solved with STR but not with MIP, or,
+19
+
+Table 5: Comparing MIP, STR and HYB on Test Instances
+MIP
+STR
+HYB
+Number of Instances Solved
+167/242
+206/242
+208/242
+Average Optimality Gap (%)
+0.18%
+0.05%
+0.05%
+Average Solution Time (s)
+≥ 223
+≥ 117
+≥ 110
+2. Neither method can solve the instance, and STR has a smaller optimality gap (at least 0.01%
+smaller), or,
+3. Both methods can solve the instance and STR reduces the solution time by 10% or more.
+Using this labeled data, we train a simple decision stump (depth-1 decision tree) to determine
+whether or not we should restart the MIP using the formulation STR. The features we use are
+composites of zL, zD, zR, zLB and zUB, where zLB and zUB denote the lower and upper bounds
+obtained by the solver from solving the original formulation until the Lagrangian dual problem is
+solved. In our experiments, zUB is always finite as the solver can easily find out that the all-zero
+solution is feasible for MKAP. The resulting decision stump obtained after training is as follows:
+“If (zD − zLB)/zUB > 0.05%, then switch to the strengthened formulation STR.”
+Instances that can be solved using the original MIP formulation before we finish computing zD
+are dropped from the training set and the test set as there is no algorithmic decision of interest
+to make in that situation. This leaves us a training set with 62 instances and a test set with 242
+instances.
+Next we compare a hybrid implementation with default Gurobi. To further simplify the imple-
+mentation, we replace the parallelization of branch and bound and Lagrangian dual by a simulated
+parallelization. Such a simulated hybrid implementation is described below:
+Step 1.
+(a) Using 1 thread, solve the Lagrangian dual problem using the level method. Store the
+solution time tD.
+(b) Using 3 threads, run a standard MIP solver using the original MIP formulation with
+timelimit tD.
+Step 2. If the MIP is solved to optimality in Step 1, then stop.
+Else if (zD − zLB)/zUB > 0.05%, then restart the MIP solver using all 4 thread with the
+strengthened formulation STR and the upper bound (primal solution) obtained from the
+MIP solver in Step 1(b).
+Else, allocate all 4 threads to the MIP solver for continuing with the original formulation.
+In Table 5 we summarize results obtained by directly solving the original formulation (‘MIP’),
+directly generating and solving the strengthened formulation (‘STR’), and applying our simulated
+hybrid implementation (‘HYB’). For each method, we report the number of instances solved within
+the timelimit of 10 minutes, the ending optimality gap and average time spent on solving the
+problem including cut generating time for STR and HYB in seconds. Note that STR is likely to
+outperform MIP on the test instances because the DW bound is already significantly stronger than
+the LP relaxation bound on those instances. We observe that both STR and HYB outperform
+MIP on the test instances while HYB slightly outperforms MIP in terms of solution time. This is
+20
+
+because HYB avoids solving the more expensive STR formulation in some cases when the bound
+provided by STR is not significantly stronger than MIP. Another interesting observation is that
+HYB solves exactly all instances that can be solved by either MIP or STR, including two instances
+that cannot be solved by STR but can be solved by MIP, demonstrating the value of having the
+flexibility of switching between different formulations. We believe this hybrid version would be
+relatively easy to implement by modern MIP solvers.
+8
+Conclusions
+We investigated methods for generating and strengthening cutting planes that are derivable from
+DW reformulation to accelerate branch and cut for MIPs with block structures. Numerical exper-
+iments have shown that adding these cutting planes is effective in cases when the DW bound is
+strong. We also proposed ways to incorporate our methodology into a MIP solver when multiple
+threads are available.
+Empirically, we observed that if many of the cutting planes are added into the formulation,
+then the LP solution at each branching node can be significantly slowed down. A potential fix is to
+consider cut selection within our cut generation approach. One may also consider some adaptive
+tilting schemes rather than tilting each inequality to the same depth. We also observed that the
+Lagrangian dual problem often has nonunique optimal solutions.
+It is therefore interesting to
+investigate whether using alternative dual solutions can lead to stronger cutting planes.
+References
+[1] G. B. Dantzig and P. Wolfe, “Decomposition principle for linear programs,” Operations Re-
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+railroads,” Transportation Science, vol. 37, no. 2, pp. 183–197, 2003.
+[3] J. Rios and K. Ross, “Massively parallel Dantzig-Wolfe decomposition applied to traffic flow
+scheduling,” Journal of Aerospace Computing, Information, and Communication, vol. 7, no. 1,
+pp. 32–45, 2010.
+[4] M. F. Anjos, A. Lodi, and M. Tanneau, “A decentralized framework for the optimal coordi-
+nation of distributed energy resources,” IEEE Transactions on Power Systems, vol. 34, no. 1,
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+24
+
+A
+Dantzig-Wolfe Decomposition
+Algorithm 1 Standard Dantzig-Wolfe Decomposition
+1: Initialize:
+ˆV j ← a subset of extreme points of conv(Qj), j = 1, 2, . . . , p
+ˆRj ← a subset of extreme rays of conv(Qj), j = 1, 2, . . . , p
+t ← 0
+2: t ← t + 1, solve
+zt
+D = min c⊤x
+s.t. xI(j) =
+�
+v∈ ˆ
+V j
+vλj
+v +
+�
+r∈ ˆ
+Rj
+rµj
+r,
+j ∈ J,
+(πj)
+Ax ≥ b,
+(β)
+�
+v∈ ˆV j
+λj
+v = 1,
+j ∈ J,
+(θj)
+λj ≥ 0, µj ≥ 0,
+j ∈ J
+(20)
+(assume { ˆV j}q
+j=1 and { ˆRj}q
+j=1 are initialized such that (20) is always feasible)
+3: let (π1, . . . , πq, θ1, . . . , θq) denote the values of optimal dual variables for (20)
+4: for j = 1, 2, . . . , q do
+5:
+solve the following pricing problem:
+Dj(πj) := min{(πj)⊤v : v ∈ conv(Qj)} = min{(πj)⊤v : v ∈ Qj}.
+(21)
+6:
+if the pricing problem (21) is bounded then
+7:
+let vj denote an optimal solution
+8:
+ζj ← (πj)⊤vj − θj
+9:
+if ζj < 0 then
+10:
+ˆV j ← ˆV j ∪ {vj}
+11:
+end if
+12:
+else
+13:
+ζj ← −∞
+14:
+let rj denote an extreme ray of conv(Qj) with (πj)⊤rj < 0
+15:
+ˆRj ← ˆRj ∪ {rj}
+16:
+end if
+17:
+if ζj ≥ 0 for all j ∈ J then
+18:
+return zD = zt
+D
+19:
+else
+20:
+go to step 2
+21:
+end if
+22: end for
+B
+The Level Method for the Lagrangian Dual Problem
+The level method adds on top of the cutting plane method a regularization step that requires
+solving a convex quadratic program to find a promising candidate multiplier (π, β) = (¯π, ¯β) to
+explore while staying close to the previous multiplier that is already explored. A pseudocode of
+the level method is given in Algorithm 2. For generating the upper bound ¯z, we give the original
+formulation to the solver. We pick the second feasible solution ¯x found by the solver (often much
+better than the first feasible solution) and set ¯z = c⊤¯x. Constraint (22d) is important in early
+iterations of the algorithm to ensure that problem (13) is bounded, but becomes redundant when
+UB< ¯z in later iterations of the algorithm. Within the level method, LB and UB store the best
+lower and upper bounds of zD found by the algorithm. If we set the termination condition to be
+25
+
+Algorithm 2 The Level Method for Solving (13)
+1: Initialize:
+ˆV j ← ∅, ˆRj ← ∅, j = 1, 2, . . . , p
+¯z ← an upper bound of zD
+LB ← −∞, UB ← ∞, t ← 0
+2: Main Loop: t ← t + 1, solve:
+UB ← max
+q
+�
+j=1
+θj + b⊤β
+(22a)
+s.t. θj ≤ v⊤πj,
+v ∈ ˆV j, j ∈ J,
+(22b)
+r⊤πj ≥ 0,
+r ∈ ˆRj, j ∈ J,
+(22c)
+q
+�
+j=1
+θj + b⊤β ≤ ¯z,
+(22d)
+�
+j:i∈I(j)
+πj
+i + β⊤Ai = ci,
+i = 1, . . . , n,
+(22e)
+β ≥ 0.
+(22f)
+(22g)
+3: if LB = −∞ then
+4:
+(¯π, ¯β) ← optimal value of (π, β) in (22)
+5: else
+6:
+solve:
+min
+���
+π − ¯π, β − ¯β
+���2
+2
+s.t.
+q
+�
+j=1
+θj + b⊤β ≥ 0.7 · UB + 0.3 · LB
+(22b) − (22f)
+(23)
+7:
+(¯π, ¯β) ← optimal value of (π, β) in (23)
+8: end if
+9: for j = 1, 2, . . . , q do
+10:
+solve (14) for πj = ¯πj
+11:
+if (14) is bounded then
+12:
+let vj denote an optimal solution
+13:
+ˆV j ← ˆV j ∪ {vj}
+14:
+else
+15:
+let rj denote an extreme ray of conv(Qj) with (πj)⊤rj < 0
+16:
+ˆRj ← ˆRj ∪ {rj}
+17:
+end if
+18: end for
+19: LB ← max
+�
+LB, �q
+j=1 Dj(¯πj) + b⊤ ¯β
+�
+20: if UB-LB is small enough then
+21:
+return LB
+22: else
+23:
+go to step 2
+24: end if
+UB-LB=0, then the algorithm returns the exact DW bound zD. To avoid some numerical issues,
+when implementing the level method we terminate the algorithm if difference between LB and UB
+is within 10−6 relative tolerance, or if the solver fails to solve the quadratic master problem (23).
+We observe that the second case rarely happens but can sometimes lead to a Lagrangian dual
+bound weaker than the natrual LP relaxation bound since we do not solve the Lagranigan dual
+problem to optimality.
+26
+
+C
+Examples
+Example 1. Consider the following MIP:
+z∗ = min x1 + x2 + 2x3 + 2x4
+s.t. x2 + x4 ≥ 3,
+3x1 + x2 + 3x3 + x4 ≥ −12,
+0.5 ≤ xi ≤ 2.5, xi ∈ Z, i = 1, 2, 3, 4.
+Define I1 = {1, 2}, I2 = {3, 4}, Qj = {y ∈ Z2 : 0.5 ≤ y1 ≤ 2.5, 0.5 ≤ y2 ≤ 2.5} for j ∈ {1, 2} and
+the linking constraints Ax ≥ b to be
+� 0
+1
+3
+1
+� � x1
+x2
+�
++
+� 0
+1
+3
+1
+� � x3
+x4
+�
+≥
+� 3
+12
+�
+.
+In this example the MIP has symmetric blocks but asymmetric objective coefficients. The LP relax-
+ation of the problem gives the lower bound 7. Let π(1) = (1, 1/4, 2, 5/4)⊤, β(1) = (3/4, 0)⊤, π(2) =
+(2/11, 8/11, 13/11, 19/11)⊤, β(2) = (0, 3/11)⊤. Note that the dual multipliers
+�
+π(τ), β(τ)
+�
+τ∈{1,2}
+satisfy the assumptions in Proposition 6. The best dual bound is given by
+max
+τ∈{1,2} z
+�
+π(τ), β(τ)
+�
+= max{27/4, 78/11} = 78/11.
+Adding DWF cuts associated with π(1) and adding DWF cuts associated with π(2) into the LP
+relaxation of the original formulation yield bounds 125/16 and 149/19, respectively. These bounds
+are stronger than the corresponding Lagrangian relaxation bounds 27/4 and 78/11. On the other
+hand, the LP
+min c⊤x
+s.t.
+�
+πj(τ)
+�⊤xI(j) ≥ Dj
+�
+πj(τ)
+�
+,
+j ∈ J, τ = 1, 2,
+Ax ≥ b
+has the optimal objective value equal to 8, which happens to be z∗ in this example.
+Example 2. Consider the following primal and dual LP pair:
+(Primal) : min x1 − x2
+s.t. x1 + x2 ≥ 1,
+x2 ≥ 1,
+−x2 ≥ −1;
+(Dual) : max α1 + α2 − α3
+s.t. α1
+= 1,
+α1 + α2 − α3 = −1,
+α ≥ 0.
+In this example, the primal LP has a unique optimal solution x = (0, 1)⊤. Therefore, the dimension
+of the optimal face is 0. Also, the problem has a unique dual basic solution α = (1, 0, 2)⊤, whose
+0-norm is 2, which is strictly less than n minus the dimension of the optimal face. However, note
+that all dual solutions of the form α = (1, λ, 2 + λ)⊤ with λ ≥ 0 are optimal for the dual LP, i.e.,
+the problem has nonunique dual optimal solutions.
+27
+
+D
+MKAP and Test Instances
+In MKAP, there are a finite number of items, knapsacks and item classes. Each item belongs to
+exactly one item class. The decision maker has to pack a subset of items into knapsacks, subject
+to constraints that only items belonging to the same item class with a total weight no larger than
+the capacity of the knapsack can be packed into the same knapsack, with the aim of maximizing
+the total profit of the packed items. Specifically, let N, M and K denote the index sets of items,
+knapsacks and item classes, respectively. For j ∈ N, let pj denote the profit of item j and wj
+denote the weight of item j. For i ∈ M, let Ci denote the capacity of knapsack i. For k ∈ K,
+let Sk denote the set of items that belong to item class k, i.e., {Sk}k∈K is a partition of N. We
+use variables x ∈ {0, 1}M+N to represent the packing decisions, where xij = 1 if and only if item
+j is packed into knapsack i. Variables y ∈ {0, 1}M+K represent the assignment decisions, where
+yik = 1 if and only if knapsack i is used to pack items from item class k. MKAP can then be
+formulated as the following integer program (we flip the sign of the objective function to formulate
+it as a minimization problem):
+min −
+�
+i∈M
+�
+j∈N
+pjxij
+(24a)
+s.t.
+�
+j∈Sk
+wjxij ≤ Ciyik,
+i ∈ M, k ∈ K,
+(24b)
+�
+i∈M
+xij ≤ 1,
+j ∈ N,
+(24c)
+�
+k∈K
+yik ≤ 1,
+i ∈ M,
+(24d)
+x ∈ {0, 1}M×N, y ∈ {0, 1}M×K.
+(24e)
+It has been observed in [41] that applying Lagrangian relaxation to (24) with constraints (24c)
+and (24d) dualized can lead to a dual bound potentially much stronger than the LP relaxation
+bound. By equivalence between DW decomposition and Lagrangian relaxation, this observation
+also applies to DW decomposition. Specifically, we define |M| × |K| blocks in our DW decom-
+position, where for each i ∈ M and k ∈ K block Qi,k is defined by a knapsack constraint
+�
+j∈Sk wjxij ≤ Ciyik together with constraints forcing (xij)j∈Sk and yik to be binary.
+We generate the instances following the scheme of [37] but using different instance sizes. We
+consider instances with |K| ∈ {2, 5, 10, 25}, |M| ∈ {10, 20, 30, 40} and |N| ∈ {50, 100, 200, 300}.
+For each combination (|K|, |M|, |N|) of the parameters, we generate three types (uncorrelated,
+weakly correlated, strongly correlated) of MKAP instances with 10 instances generated using 10
+different random seeds for each type. We refer readers to [37] for a more detailed description of
+the instance generation procedure. It is worth mentioning that the item classes {Sk}k∈K all have
+equal sizes |N|/|K| for our test instances.
+E
+Time Spent on Obtaining Different Formulations
+Since both OBJ and DWF can be obtained directly after solving the Lagrangian dual problem,
+we only report in Table 6 the total time spent on applying strengthening and different levels of
+tilting for STR and DdT with d ∈ {1, . . . , 6}. Bold rows correspond to I2 instances and other rows
+correspond to I1 \ I2 instances.
+28
+
+Table 6: Comparison of Time Spent on Obtaining Different Formulations
+(|K|, |M|, |N|)
+Generation Time Excluding Lagrangian Dual Time (s)
+STR
+D1T
+D2T
+D3T
+D4T
+D5T
+D6T
+( 2,20, 50)
+0.1
+0.3
+0.5
+1.1
+2.1
+3.7
+6.0
+( 2,30, 50)
+0.1
+0.2
+0.3
+0.6
+0.9
+1.0
+1.2
+( 2,40, 50)
+0.1
+0.2
+0.1
+0.2
+0.2
+0.2
+0.2
+( 5,10, 50)
+0.0
+0.1
+0.4
+0.8
+1.6
+2.6
+3.6
+( 5,20, 50)
+0.1
+0.2
+0.5
+0.9
+1.3
+1.5
+1.6
+( 5,30, 50)
+0.1
+0.2
+0.3
+0.5
+0.5
+0.5
+0.5
+( 5,40, 50)
+0.1
+0.1
+0.1
+0.1
+0.1
+0.1
+0.1
+(10,10, 50)
+0.0
+0.2
+0.4
+0.8
+1.1
+1.2
+1.2
+(10,10,100)
+0.1
+0.3
+1.0
+2.2
+4.5
+8.7
+15.7
+(10,10,200)
+0.3
+0.9
+2.2
+5.0
+10.3
+20.9
+42.4
+(10,10,300)
+0.9
+2.9
+7.0
+15.2
+30.5
+60.0
+118.7
+(10,20, 50)
+0.1
+0.2
+0.4
+0.6
+0.7
+0.7
+0.7
+(10,30, 50)
+0.1
+0.2
+0.3
+0.3
+0.3
+0.3
+0.3
+(10,40, 50)
+0.1
+0.1
+0.1
+0.1
+0.1
+0.1
+0.1
+(10,40,100)
+0.3
+0.9
+2.0
+3.8
+5.5
+6.6
+6.9
+(25,10, 50)
+0.0
+0.2
+0.3
+0.3
+0.3
+0.3
+0.3
+(25,10,100)
+0.0
+0.4
+1.1
+2.1
+2.8
+2.8
+2.8
+(25,10,200)
+0.1
+0.7
+2.1
+4.9
+10.3
+20.2
+38.7
+(25,10,300)
+0.1
+0.9
+3.0
+7.2
+15.5
+31.7
+63.5
+(25,20, 50)
+0.1
+0.2
+0.2
+0.2
+0.2
+0.2
+0.2
+(25,20,100)
+0.1
+0.6
+1.6
+2.6
+3.1
+3.1
+3.1
+(25,20,200)
+0.2
+1.3
+4.0
+9.0
+17.7
+32.1
+53.9
+(25,20,300)
+0.4
+2.0
+5.8
+13.5
+28.1
+56.1
+109.5
+(25,30, 50)
+0.1
+0.2
+0.2
+0.2
+0.2
+0.2
+0.2
+(25,30,100)
+0.2
+0.8
+1.7
+2.4
+2.5
+2.5
+2.5
+(25,30,200)
+0.4
+1.9
+5.4
+11.8
+21.9
+36.1
+51.1
+(25,30,300)
+0.8
+3.1
+8.3
+19.3
+39.7
+77.9
+148.1
+(25,40, 50)
+0.1
+0.1
+0.0
+0.0
+0.0
+0.0
+0.0
+(25,40,100)
+0.2
+0.8
+1.5
+1.9
+2.0
+2.0
+2.0
+At each row, the averages of 30 instances are reported.
+F
+Number of Cutting Planes Generated Using Different Formu-
+lations
+In Table 7, we report the number of cutting planes generated by Gurobi within 10 minutes for
+formulations MIP, OBJ, DWF, STR, D3T and D6T averaged over the (25, 20, 300) MKAP
+instances.
+G
+Branch-and-Cut Results for I1 \ I2 instances
+In Tables 8 and 9 we report branch-and-cut results for I1\I2 instances. We observe that the original
+MIP formulation is very competitive on these instances, whose performance is often better than
+DWF and even STR. This can be explained by the fact that although the natrual LP relaxation
+is weak (rL is large), Gurobi can close much of the gap by presolve and cutting planes at the root
+node (rR is small).
+29
+
+Table 7: Average Number of Cutting Planes Generated by Gurobi
+MIP
+OBJ
+DWF
+STR
+D3T
+D6T
+Gomory
+132.1
+41.5
+10.4
+17.9
+0.4
+0.0
+Lift-and-project
+5.3
+1.7
+0.6
+0.6
+0.0
+0.0
+Cover
+616.1
+176.5
+121.5
+173.3
+16.3
+0.0
+Clique
+362.3
+248.5
+148.3
+266.4
+108.3
+0.0
+MIR
+90.2
+24.6
+9.7
+13.2
+3.1
+0.0
+StrongCG
+94.3
+49.8
+12.7
+21.9
+2.4
+0.0
+Flow cover
+228.8
+61.6
+18.5
+3.1
+0.0
+0.0
+Zero half
+34.9
+13.9
+2.0
+0.4
+0.0
+0.0
+RLT
+19.7
+0.8
+1.4
+0.7
+0.0
+0.0
+Relax-and-lift
+10.0
+0.0
+0.1
+0.0
+0.0
+0.0
+Table 8: Comparison of Branch-and-Cut Performance on I1 \ I2 Instances
+(|K|, |M|, |N|)
+Number of Solved
+Average B&C Time (s)
+MIP
+OBJ
+DWF
+STR
+D3T
+D6T
+MIP
+OBJ
+DWF
+STR
+D3T
+D6T
+( 2,20, 50)
+30/30
+23/30
+30/30
+30/30
+30/30
+30/30
+2
+≥202
+7
+5
+4
+4
+( 2,30, 50)
+30/30
+29/30
+30/30
+30/30
+30/30
+30/30
+0
+≥ 25
+0
+0
+0
+0
+( 2,40, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+0
+0
+0
+0
+0
+0
+( 5,10, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+4
+62
+7
+4
+2
+1
+( 5,20, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+1
+42
+4
+2
+1
+0
+( 5,30, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+0
+19
+0
+0
+0
+0
+( 5,40, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+0
+0
+0
+0
+0
+0
+(10,10, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+0
+1
+0
+0
+0
+0
+(10,20, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+0
+16
+0
+0
+0
+0
+(10,30, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+0
+1
+0
+0
+0
+0
+(10,40, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+0
+0
+0
+0
+0
+0
+(10,40,100)
+30/30
+1/30
+29/30
+30/30
+30/30
+30/30
+40
+≥600
+≥103
+72
+15
+12
+(25,10, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+0
+0
+0
+0
+0
+0
+(25,20, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+0
+0
+0
+0
+0
+0
+(25,20,100)
+30/30
+20/30
+30/30
+30/30
+30/30
+30/30
+1
+≥204
+0
+0
+0
+0
+(25,30, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+0
+0
+0
+0
+0
+0
+(25,30,100)
+30/30
+10/30
+30/30
+30/30
+30/30
+30/30
+5
+≥432
+1
+1
+0
+0
+(25,30,200)
+16/30
+1/30
+10/30
+15/30
+23/30
+26/30
+≥377
+≥600
+≥475
+≥402
+≥255
+≥143
+(25,40, 50)
+30/30
+30/30
+30/30
+30/30
+30/30
+30/30
+0
+0
+0
+0
+0
+0
+(25,40,100)
+30/30
+9/30
+30/30
+30/30
+30/30
+30/30
+2
+≥464
+1
+1
+0
+0
+At each row, the averages of 30 instances are reported.
+Table 9: Comparison of Branch-and-Cut Performance on I1 \ I2 Instances (Cont’d)
+(|K|, |M|, |N|)
+Average Optimality Gap (%)
+Average Number of Nodes
+MIP
+OBJ
+DWF
+STR
+D3T
+D6T
+MIP
+OBJ
+DWF
+STR
+D3T
+D6T
+( 2,20, 50)
+0.00
+0.06
+0.00
+0.00
+0.00
+0.00
+3439
+≥755510
+28880
+16808
+16826
+7963
+( 2,30, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+89
+≥177782
+341
+796
+188
+131
+( 2,40, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+1
+1
+1
+1
+1
+1
+( 5,10, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+12677
+669382
+62160
+23721
+5498
+1509
+( 5,20, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+1865
+334795
+20703
+9992
+929
+614
+( 5,30, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+275
+151202
+161
+61
+17
+18
+( 5,40, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+1
+1
+1
+1
+1
+1
+(10,10, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+1077
+3227
+39
+51
+1
+1
+(10,20, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+344
+46614
+803
+379
+14
+14
+(10,30, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+131
+3045
+372
+129
+1
+1
+(10,40, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+1
+1
+1
+1
+1
+1
+(10,40,100)
+0.00
+0.14
+0.01
+0.01
+0.00
+0.00
+23261
+≥302450
+≥137206
+82878
+11574
+9120
+(25,10, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+2
+229
+1
+1
+1
+1
+(25,20, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+1
+212
+1
+1
+1
+1
+(25,20,100)
+0.00
+0.03
+0.00
+0.00
+0.00
+0.00
+279
+≥202331
+1
+1
+1
+1
+(25,30, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+1
+2
+1
+1
+1
+1
+(25,30,100)
+0.00
+0.15
+0.00
+0.00
+0.00
+0.00
+9866
+≥185765
+1903
+1121
+1
+1
+(25,30,200)
+0.12
+0.44
+0.22
+0.15
+0.07
+0.04
+≥45681
+≥341784
+≥ 67369
+≥48540
+≥20550
+≥14553
+(25,40, 50)
+0.00
+0.00
+0.00
+0.00
+0.00
+0.00
+1
+28
+1
+1
+1
+1
+(25,40,100)
+0.00
+0.08
+0.00
+0.00
+0.00
+0.00
+450
+≥257189
+1520
+2483
+47
+47
+At each row, the averages of 30 instances are reported.
+30
+
diff --git a/XtFPT4oBgHgl3EQfsjWt/content/tmp_files/load_file.txt b/XtFPT4oBgHgl3EQfsjWt/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8ce42a21cf59e7e7b14335b867e2712a816d5278
--- /dev/null
+++ b/XtFPT4oBgHgl3EQfsjWt/content/tmp_files/load_file.txt
@@ -0,0 +1,1999 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf,len=1998
+page_content='Recovering Dantzig-Wolfe Bounds by Cutting Planes  Rui Chen1, Oktay G¨unl¨uk2, Andrea Lodi1  1 Cornell Tech, Cornell University ({rui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='chen,andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='lodi}@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='edu)  2 School of ORIE, Cornell University (ong5@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='edu)  January 31, 2023  Abstract  Dantzig-Wolfe (DW) decomposition is a well-known technique in mixed-integer programming  for decomposing and convexifying constraints to obtain potentially strong dual bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We  investigate Fenchel cuts that can be derived using the DW decomposition algorithm and show  that these cuts can provide the same dual bounds as DW decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We show that these  cuts, in essence, decompose the objective function cut one can simply write using the DW bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Compared to the objective function cut, these Fenchel cuts lead to a formulation with lower  dual degeneracy, and consequently a better computational performance under the standard  branch-and-cut framework in the original space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We also discuss how to strengthen these cuts  to improve the computational performance further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We test our approach on the Multiple  Knapsack Assignment Problem and show that the proposed cuts are helpful in accelerating the  solution time without the need to implement branch and price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  1  Introduction  In this paper, we present a computationally effective approach for generating cutting planes from  Dantzig-Wolfe (DW) decomposition [1] to enhance branch and cut in the space of original variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We focus on mixed-integer (linear) programs (MIPs) with the following structure:  z∗ := min c⊤x  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' xI(j) ∈ P j,  j ∈ J := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , q},  Ax ≥ b,  x ∈ X,  (1)  where the index set I(j) ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , n} contains the indices of the variables in “block” j ∈ J, and  we use the notation xI to denote the subvector of x with indices in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The set P j = {y ∈ R|I(j)| :  Gjy ≥ gj} is a polyhedral set for j ∈ J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' and X ⊆ Rn represents integrality constraints on some of  the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We do not assume the index sets I(j) for j ∈ J to be disjoint, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  DW decomposition was originally developed for solving large-scale linear programs with loosely  coupled blocks and later extended to MIPs with similar block structures to obtain strong dual  bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Typically, these so-called DW bounds are stronger than the linear programming (LP)  relaxation bounds as they exploit the block structure through convexification of the block con-  straints using integrality information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' DW decomposition has been found to be effective in various  applications, such as transportation [2], traffic scheduling [3], energy [4] and multi-stage stochastic  planning [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='13149v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='OC]  30 Jan 2023  Figure 1: Constraint Matrices of MIPs With Different Types of Block Structures (Left: Loosely  Coupled, Middle: Two-Stage, Right: Overlapping)  G1  G2  G3  G4  A  G1  G2  G3  G4  G1  G2  G3  G4  Computing the DW bound requires reformulating the MIP using the extreme points and ex-  treme rays of the mixed-integer hulls of the block problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Consequently, while the DW refor-  mulation approach often leads to good dual bounds, using this to solve the MIP exactly requires  specialized techniques that are not readily present in off-the-shelf solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In other words, to ex-  ploit DW decomposition one needs to solve the continuous relaxation of the reformulated MIP by  column generation followed by an ad-hoc branching step [6], after which one has to again resort  to column generation, leading to an algorithmic framework called branch and price [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' While this  approach has been successfully implemented in some special cases, most notably for vehicle routing  problems [8], generic branch-and-price solvers, including ABACUS [9], G12 [10], GCG [11], DIP  [12], BaPCod [13], to name a few, are still in theirs infancy with respect to solving general MIPs  with block structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' However, the most up-to-date (and always improving) MIP solvers are based  on the branch-and-cut (or cut-and-branch) scheme [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  In this paper, we aim to make a step towards bridging this gap by developing a new scheme  to incorporate the dual bounds produced by DW reformulations into the standard cut-and-branch  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  More precisely, we first compute the DW reformulation bound zD of the MIP (1)  at the root node and then generate cutting planes to incorporate this bound into the original  formulation to solve the problem to optimality in the space of the original variables using standard  MIP technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Our approach, therefore, requires column generation only at the root node and  not throughout the enumeration tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Apart from our proposed approach, the only general and  trivial method known so far to enforce the DW bound zD in the original MIP is to augment the  formulation by adding the objective function cut, c⊤x ≥ zD, which is known (folklore) to be not only  computationally ineffective but also numerically unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' To the best of our knowledge, however,  no detailed theoretical or computational investigation has been conducted on the objective function  cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Our approach, on the other hand, appears to be practical and computationally effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Paper Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Our first contribution is to confirm the instability of the approach using  the objective cut by mostly attributing it to dual degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Then, we show how to overcome  the limitation of the objective constraint by proposing a family of Fenchel cutting planes [15]  that essentially decomposes the objective function cut leading to a formulation with lower dual  degeneracy, and consequently a better computational performance under the standard branch-  and-cut framework in the original space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Moreover, we propose two distinct ways of strengthening  these Fenchel cuts for an additional improvement of their computational performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' As a case  study, we test our approach on the Multiple Knapsack Assignment Problem and show that the  proposed cuts are helpful in accelerating the solution time without the need to implement branch  and price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Finally, we show how to put all these pieces together in an algorithm that, exploiting  2  a standard multi-thread computational environment, (i) runs a standard branch and cut on some  threads, (ii) computes in parallel the DW dual bound in a separate thread and (iii) on the fly  enhances the original MIP formulation by the Fenchel cuts associated with the DW bound if they  are predicted to improve the overall solution time for solving the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The framework is simple  and general, holding the promise of being implementable in a relatively easy way in general-purpose  MIP solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Paper Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In Section 2, we  provide some algorithmic preliminaries on DW decomposition, while in Section 3, we propose  our new approach for recovering the DW bound in the original formulation by Fenchel cuts and  compare it with the objective function cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In Section 4, we discuss Lagrangian relaxation as an  alternative approach for computing the DW bound and generating cutting planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Some techniques  for strengthening the proposed cuts are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Section 6 reports our computational  investigation on the Multiple Knapsack Assignment Problem, while Section 7 proposes and tests  the multi-thread hybrid algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Finally, some short conclusions are drawn in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  2  Preliminaries  We assume MIP (1) is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Throughout the paper, we call formulation (1) the original for-  mulation of the MIP and call constraints Ax ≥ b (potentially empty) linking constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In the  context of Dantzig-Wolfe decomposition, (1) is sometimes called the compact formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' MIPs  of this form with disjoint index sets I(j) are called loosely coupled MIPs [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Note that we do  not assume the original MIP (1) is loosely coupled and as such the supports of different blocks  can have overlaps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=', in MIPs with two-stage [17] or overlapping [18] block structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We call  the LP relaxation of (1), obtained by dropping the integrality constraints x ∈ X, the natural LP  relaxation, and denote its optimal objective value by zL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Throughout, we assume that all data is  rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='1  Dantzig-Wolfe Decomposition  We next reformulate (1) by replacing the constraints xI(j) ∈ P j with xI(j) ∈ conv(Qj), where  Qj = {y ∈ R|I(j)| : Gjy ≥ gj, y ∈ Xj},  and Xj has the integrality constraints inherited from X for the variables xI(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Replacing the  constraints xI(j) ∈ P j with xI(j) ∈ conv(Qj) in (1), one obtains the following Dantzig-Wolfe  reformulation of problem (1):  z∗ := min c⊤x  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' xI(j) ∈ conv(Qj),  j ∈ J,  Ax ≥ b,  x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (2)  We call the continuous relaxation of (2), obtained by relaxing the integrality constraints x ∈ X, the  DW relaxation of (1), and the bound zD obtained by solving the DW relaxation the DW bound  of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The DW bound zD is potentially stronger than the natural LP relaxation bound zL as  P j ⊇ conv(Qj) for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Consequently, we have  z∗ ≥ zD ≥ zL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  3  In practice, computing zD is not as straightforward as solving the continuous relaxation of (2)  since polyhedra  �  conv(Qj)  �q  j=1 are not given explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' One often has to apply the so-called DW  decomposition to compute zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  By the Minkowski-Weyl theorem and Meyer’s theorem [19] for  rational polyhedra, each conv(Qj) can be represented by its extreme points and extreme rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For  j ∈ J, let V j and Rj denote the set of extreme points and the set of extreme rays of conv(Qj),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The DW reformulation (2) can be further reformulated using (V j)j∈J and (Rj)j∈J,  leading to the following extended reformulation of (2):  z∗ = min c⊤x  (3a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' xI(j) =  �  v∈V j  λvv +  �  r∈Rj  µrr,  j ∈ J,  (3b)  �  v∈V j  λv = 1,  j ∈ J,  (3c)  λ ≥ 0, µ ≥ 0  (3d)  Ax ≥ b,  x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (3e)  The LP relaxation of (3) (obtained by dropping the integrality constraints x ∈ X, and equivalent  to DW relaxation) can be solved iteratively via column generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Specifically, at each iteration,  a restricted LP is solved by replacing V j and Rj with a small collection of extreme points ˆV j ⊆ V j  and extreme rays ˆRj ⊆ Rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' A new extreme point or a new extreme ray is generated by solving a  pricing subproblem  Dj(πj) := min  �  (πj)⊤v : v ∈ conv(Qj)  �  = min  �  (πj)⊤v : v ∈ Qj�  (4)  for each block j ∈ J, where πj above is the dual solution associated with constraints (3b) in the  restricted LP relaxation of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' By convention, we define Dj(πj) = −∞ if the pricing subproblem  (4) is unbounded, and when this happens one can obtain an extreme ray r ∈ Rj by finding an  unbounded ray of min{(πj)⊤v : v ∈ P j}, which is added to ˆRj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If, on the other hand, Dj(πj) is  finite, one obtains a solution of (4) as an extreme point v ∈ V j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' This point is added to ˆV j provided  that it has a negative reduced cost that is computed by subtracting the dual variable associated  with the j-th constraint of (3c) from Dj(πj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The restricted LP, with augmented vertices and rays,  is then solved again and this process is repeated until no such points or rays are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  The restricted LP at each iteration gives an upper bound for zD, and these upper bounds  converge to zD in a finite number of iterations as |V j| and |Rj| are finite for all j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The algorithm  iterates between the restricted LP master problem and pricing subproblems until the solutions of  the pricing subproblems all have nonnegative reduced costs, in which case it can be shown that the  optimal objective value of the restricted LP takes value zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Such an inner approximation approach  for solving the LP relaxation of (3) is called DW decomposition [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Algorithm 1 in Appendix A  presents a pseudocode describing the standard DW decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' When the algorithm terminates,  in addition to the lower bound zD, an optimal solution to the LP relaxation of (2) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  This solution does not necessarily satisfy the integrality constraints x ∈ X in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If this solution  is fractional, branching is necessary to obtain an optimal solution of the original problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The  subproblems in the branch-and-bound tree can again be solved using column generation, and this  procedure for finding the exact solution of (3) is called branch and price [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' A notable extension  of branch and price is the so-called branch-and-cut-and-price algorithm, where cutting planes that  are valid for the original formulation are added to LP relaxations at branch-and-price nodes to  further strengthen the formulation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=', [20, 21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  4  Figure 2: Solutions of DW Pricing Subproblems Build Both Inner (Blue, Dashed) and Outer (Red,  Solid) Approximations  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='2  Inner and Outer Approximations  In DW decomposition, the restricted LP at each iteration corresponds to an inner approximation of  the DW reformulation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The inner approximation is iteratively augmented by extreme points and  extreme rays generated from the pricing subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' At the same time, the pricing subproblems  can also be used to derive an outer approximation of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Specifically, by definition of the pricing  subproblems, for each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , q} and πj ∈ RI(j), inequality (πj)⊤y ≥ Dj(πj) is valid for  conv(Qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Therefore, given the particular dual multipliers (πj(τ))q  j=1 obtained at iteration τ of  DW decomposition, the following inequalities are valid for the DW relaxation of (2):  (πj(τ))⊤xI(j) ≥ Dj(πj(τ)),  j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (5)  Then, for each iteration t of DW decomposition, the following inequality holds:  zD ≥ min c⊤x  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  �  πj(τ)  �⊤xI(j) ≥ Dj  �  πj(τ)  �  ,  j ∈ J, τ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , t,  xI(j) ∈ P j,  j ∈ J,  Ax ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  The above inner approximation can be potentially stronger than the natural LP relaxation of (1)  as (5) may cut off solutions that are feasible to the natural LP relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The overall inner and  outer process is depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  3  Incorporating the DW Bound Into the Formulation  As anticipated, it has been shown in numerous applications that the DW bound can be much  stronger than the natural LP relaxation bound [22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' However, directly using formulation  (3) is not practical since formulation (3) has potentially exponentially many variables and requires  enumerating all vertices and extreme rays of  �  conv(Qj)  �q  j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Even when the LP relaxation of  formulation (3) is solved by column generation and the DW bound zD is effectively computed,  enforcing integrality of x ∈ X requires specialized branching and DW decompostion must be  applied at every node, leading to the branch-and-price algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' A straightforward approach for  enforcing the DW bound zD in the original space is to add to the LP relaxation of (1) a cut of the  form  c⊤x ≥ zD,  5  which we call the objective function cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' However, it is well known that adding such an objective  function cut often slows down the branch-and-cut solution process in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We first observe a basic property of the optimal face of the LP relaxation after adding the  objective function cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Let P be a polyhedron in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If neither c⊤x ≤ v nor c⊤x ≥ v is valid for P, then  dim(P ∩ {x : c⊤x = v}) = dim(P) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Define P + := P ∩ {x : c⊤x ≤ v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Note that c⊤x ≤ v is an irredundant inequality for P +  because c⊤x ≤ v is not valid for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' By [25, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='26], dim(P ∩ {x : c⊤x = v}) = dim(P +) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We next show that the affine hull of P + is equal to the affine hull of P, and thus dim(P +) = dim(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  By [25, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='17], we only need to show the following two statements:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Equality c⊤x = v is not valid for P +;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If a⊤x ≤ a0 is valid for P but a⊤x = a0 is not valid for P, then a⊤x = a0 is not valid for P +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Note that c⊤x ≥ v is not valid for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Therefore, there exists ˆx ∈ P such that c⊤ˆx < v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The  first statement then follows from the fact that ˆx ∈ P +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Similarly, there exists ¯x ∈ P such that  a⊤¯x < a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For λ ∈ (0, 1), define xλ := (1 − λ)ˆx + λ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Because a⊤¯x ≤ a0, we have xλ ∈ P + but  a⊤xλ < a0 for a small enough λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The second statement then follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  When zD > zL, adding the objective function cut into the original formulation would create an  optimal face almost as high-dimensional as the original LP relaxation polyhedron under the mild  assumption that the LP relaxation of (1) contains a point x′ with c⊤x′ > zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Consequently, adding  the objective function cut would often yield a formulation with a large optimal face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' This, in turn,  can cause not only performance variability [26], but also serious computational issues especially in  early stages of the branch and cut in terms of branching [27], as well as cutting plane generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='1  An alternative approach  In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='2, we observed that valid inequalities for the DW relaxation can be generated while  solving the pricing subproblems, and adding these valid inequalities into the natural LP relaxation  can at most yield DW bound zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We next show that relatively few of these cutting planes can  readily recover zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Specifically, assume at iteration t of DW decomposition, the restricted LP we  solve is of the form  zt  D = min c⊤x  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' xI(j) =  �  v∈ ˆV j  vλj  v +  �  r∈ ˆRj  rµj  r,  j ∈ J,  (πj)  Ax ≥ b,  (β)  �  v∈ ˆV j  λj  v = 1,  j ∈ J,  (θj)  λj ≥ 0, µj ≥ 0,  j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (6)  Let (π1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , πq,t, βt, θt  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , θt  q) denote the optimal values of dual variables (π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , πq, β, θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , θq)  for the restricted LP at iteration t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We then have the following result for the valid inequalities  derived at the last iteration of DW decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  6  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Assume DW decomposition terminates in T iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Then,  zD = min c⊤x  (7a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' (πj,T )⊤xI(j) ≥ Dj(πj,T ),  j ∈ J,  (7b)  Ax ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (7c)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The “≥” direction is implied by the definition of zD as inequality (7b) is valid for conv(Qj)  for j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We next show the “≤” direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Based on LP duality of (6) at iteration T and the  termination condition of DW decomposition, the following equalities hold:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' zD = b⊤βT + �q  j=1 θT  j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' ci = A⊤  i βT + �  j:i∈I(j) πj,T  i  , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Note that at the last iteration T, the DW pricing subproblems are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Let (vj,T )q  j=1 denote  the solutions of the DW pricing subproblems at iteration T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Note that the reduced costs associated  with points (vj,T )q  j=1 are nonnegative at iteration T of DW decomposition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=', (πj,T )⊤vj,T − θT  j =  Dj(πj,T ) − θT  j ≥ 0 for j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Therefore, for each solution x satisfying (7b) and (7c), we have the  following inequality:  c⊤x =  n  �  i=1  cixi =  n  �  i=1  �  xiA⊤  i βT +  �  j:i∈I(j)  xiπj,T  i  �  = (βT )  ����  ≥0  ⊤ Ax  ����  ≥b  +  q  �  j=1  (πj,T )⊤xI(j)  �  ��  �  ≥Dj(πj,T )  ≥ b⊤βT +  q  �  j=1  Dj(πj,T ) ≥ b⊤βT +  q  �  j=1  θT  j = zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (8)  We call inequalities (7b) last-iteration DW cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Theorem 2 shows that q last-iteration DW  cuts together with linking constraints recover the DW bound zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We remark that the last-iteration  DW cuts are not necessarily all nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' It is possible that πj,T = 0 for some j ∈ J, which implies  that the convexification of the j-th block has no impact on improving the dual bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  It is worth emphasizing that (7) is not a valid formulation for the MIP (1) even if we add  integrality constraints x ∈ X to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' One should use last-iteration DW cuts as cutting planes and  add them to the original formulation (1) to obtain a valid formulation whose LP relaxation bound  is precisely zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Also note that Theorem 2 does not imply that last-iteration DW cuts dominate  other cutting planes of the form (5) that can be generated at intermediate iterations τ < T, in the  sense that intermediate-iteration DW cuts may still cut off fractional points that do not violate  any of the last-iteration DW cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='2  Dual Degeneracy and LP Optimal Face  When comparing the strength of different collections of cutting planes or different formulations,  very often the LP relaxation bound is used as the sole criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' However, the effectiveness of two  formulations in branch and cut may differ significantly even if they have the same or very similar  LP relaxation bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' A particular measure that should also be taken into account is the dual  degeneracy level of the LP relaxation of the formulation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' A dual basic solution of an LP is  called dual degenerate if at least one of the dual basic variable is set to 0 in that solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Next, we  formally define the degeneracy level of a dual basic solution of an LP (given in inequality form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  7  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Consider an LP with n variables and m inequality constraints, and let α be a basic  feasible dual solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We define the degeneracy level of α to be n − ∥α∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  A highly dual degenerate LP relaxation is associated with many alternative LP basic primal  optimal solutions, which usually corresponds to a large optimal face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The following result shows  how the size of the optimal face, in particular, its dimension, is related to the degeneracy level of  a dual basic optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Assume α∗ ∈ Rm  + is a dual basic optimal solution of an LP with n variables and  m inequality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Then, the optimal face of the LP has dimension at most n − ∥α∗∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Fur-  thermore, if α∗ is the unique dual optimal solution, then the optimal face of the LP has dimension  exactly n − ∥α∗∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Assume the LP is of the form min{c⊤x : Gx ≥ h}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Let F denote the optimal face of the  LP, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=',  F = {x : Gx ≥ h, c⊤x ≤ h⊤α∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (9)  Let (gk)⊤ denote the k-th row of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  By complementary slackness of LP, (gk)⊤x = hk for all  x ∈ F for all k with α∗  k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Since α∗ is a dual basic optimal solution, {(gk, hk)}k:α∗  k>0 are linearly  independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Otherwise, there exists β ∈ Rm \\ {0} such that βk = 0 for all k with α∗  k = 0 and  �  k:α∗  k>0 βk(gk, hk) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Note that α∗ + ϵβ and α∗ − ϵβ are then both dual optimal solutions of the  LP for small enough positive ϵ, which contradicts the fact that α∗ is a dual basic optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Therefore, dim(F) ≤ n − rank({gk}k:α∗  k>0) = n − rank({(gk, hk)}k:α∗  k>0) = n − ∥α∗∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Here, the  first inequality follows from [25, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='17], the second equality follows from the consistency  of the linear system {(gk)⊤x = hk}k:α∗  k>0, and the third equality follows from linear independence  of {(gk, hk)}k:α∗  k>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  If α∗ is the unique dual optimal solution, by strict complementary slackness of LP [28], there  exists an optimal solution x∗ ∈ F of the LP, such that (gk)⊤x∗ > hk for all k with α∗  k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' It implies  that {(gk)⊤x = hk}k:α∗  k>0 and c⊤x = h⊤α∗ are exactly all the implicit equalities that hold in the  inequality description (9) of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' By LP duality, c⊤x = h⊤α∗ is implied by {(gk)⊤x = hk}k:α∗  k>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Therefore, by [25, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='17], dim(F) = n − rank({gk}k:α∗  k>0) = n − ∥α∗∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We remark that Proposition 3 does not extend to dual nonbasic optimal solutions, moreover,  the ℓ0-norm of dual nonbasic optimal solutions can be greater than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Note that the dual optimal  solution is unique when the primal solution is nondegenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For the primal degenerate case,  even if there exists a unique dual basic optimal solution, it is possible that the dimension of the  optimal face of the LP is strictly less than the degeneracy level of that dual basic optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  See Example 2 in Appendix C for an example where the unique dual basic optimal solution has a  strictly positive dual degeneracy level but the primal optimal solution is still unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Under some mild assumptions, Proposition 3 implies the dimension of the optimal face of the  LP (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Assume the LP (7) has a unique dual optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Then, the optimal face  of (7) has dimension n − q − ∥βT ∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Note that the proof of Theorem 2 implies that (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , 1, βT ) is a dual optimal solution of  (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The result then follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Proposition 4 also implies that if the dual optimal solution is unique, then none of the DW  last-iteration cuts can be redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If this is not the case, the dimension of the optimal face  8  after adding the last iteration cuts depends on the number of cuts that are active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Note that when  applying the last-iteration DW cuts in practice, we would add them to the original formulation  (1), resulting in an LP optimal face whose size can be even smaller due to constraints xI(j) ∈ P j,  j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  4  Bound Computation and Cut Generation via Lagrangian Re-  laxation  Lagrangian relaxation [29] is an alternative approach for computing the DW bound zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In La-  grangian relaxation, separate auxiliary variables are created for each block and these auxiliary  variables are related to the original variables using additional (copying) constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The copying  constraints together with the linking constraints Ax ≥ b are then dualized into the objective to  obtain a Lagrangian relaxation of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' More precisely, one writes  zD = min c⊤x  (10a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' yj ∈ conv(Qj),  j ∈ J,  (10b)  yj = xI(j),  j ∈ J,  (πj)  (10c)  Ax ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (β)  (10d)  After dualizing constraints (10c) and (10d) using variables π and β ≥ 0, one obtains the following  Lagrangian relaxation:  z(π, β) = min c⊤x +  q  �  j=1  (πj)⊤(yj − xI(j)) + β⊤(b − Ax),  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' yj ∈ Qj,  j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (11)  Note that when the index sets {I(j)}q  j=1 are nonoverlapping, (10) and (11) can be simplified by  properly removing the copying constraints and the associated dual variables π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  In general, it follows from Lagrangian duality [30] that the largest Lagrangian relaxation bound  matches zD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=',  zD = max  β≥0,π z(π, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (12)  Note that x is unconstrained in (11) and therefore z(π, β) = −∞ unless the coefficients of the x  variables in the objective function are zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=',  ci −  �  j:i∈I(j)  πj  i − β⊤Ai = 0  for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , n},  where Ai denotes the i-th column of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Consequently, (12) can also be written as  zD = max z(π, β)  (13a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  �  j:i∈I(j)  πj  i + β⊤Ai = ci,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , n,  (13b)  β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (13c)  Problem (13) is called the Lagrangian dual problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For (π, β) satisfying (13b) and (13c), it holds  that  z(π, β) =  q  �  j=1  Dj(πj) + b⊤β,  9  Figure 3: Comparison of the Cutting Plane Method (Left) and the Level Method (Right) on Solving  the Lagrangian Dual Problem  0  200  400  600  800  1000  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' of Iterations  1000  800  600  400  200  0  Relative Gap (%)  cutting plane method UB  cutting plane method LB  0  200  400  600  800  1000  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' of Iterations  1000  800  600  400  200  0  Relative Gap (%)  level method UB  level method LB  where Dj : R|I(j)| → R ∪ {−∞} is a piecewise linear concave function of the form  Dj(πj) = min{(πj)⊤v : v ∈ Qj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (14)  Therefore, (13) is a nonsmooth convex optimization problem with a separable objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' It  is worth emphasizing that the pricing problem (4) in DW decomposition has exactly the same form  as (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The function values and supergradients of the concave function Dj(·) can be evaluated by  solving (14) [29] (an optimal solution of (14) is a supergradient of Dj(·) at πj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' This alternative  way of viewing DW bound zD as the optimal value of the Lagrangian dual problem allows us  to use various convex optimization methods for computing DW bound zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  For example, DW  decomposition is equivalent to applying the classical cutting plane method [31] to solve (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Since  the description of Qj involves integer variables in general, functions Dj(·) are often piecewise  linear concave with exponentially many pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In that case, convex optimization methods with  some stabilization techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=', the level method [32]) often outperform the cutting plane  method, and the difference can be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Figure 3 is a representative example of the difference  in performance between the cutting plane method and the level method for the DW bound zD  (averaged over a set of multiple knapsack assignment problem instances that are used in Section  6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Details about our implementation of the level method are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='1  Dantzig-Wolfe Fenchel Cuts  The cutting planes we derived in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='2 belong to a more general class of cutting planes  called Fenchel cuts [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Fenchel cuts are cutting planes that can be derived from a subsystem of  constraints (including integrality constraints) of a given MIP formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Let the feasible region  corresponding to such a subsystem is given by  Q = {x ∈ Rn : Gx ≥ g, x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Given a direction µ ∈ Rn, a Fenchel cut (associated with Q in direction µ) is given by µ⊤x ≥ f(µ),  where f(µ) = minx{µ⊤x : x ∈ Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In other words, an inequality is a Fenchel cut (associated with  Q) if and only if it is valid and have the tightest possible right-hand side for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  In particular, the j-th inequality in (5) is a Fenchel cut associated with the subsystem defined  by Q = {x ∈ Rn : GjxI(j) ≥ gj, xI(j) ∈ Xj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We call such a Fenchel cut associated with a  particular block of the DW reformulation a Dantzig-Wolfe Fenchel cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  10  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We call an inequality a Dantzig-Wolfe Fenchel (DWF) cut for (1) if it is of the form  (πj)⊤xI(j) ≥ Dj(πj)  (15)  for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , q}, where Dj(·) is defined in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We have shown in Section 3 that DWF cuts associated with the last iteration of DW decompo-  sition can recover DW bound zD and applying these DWF cuts rather than the objective function  cut for enforcing DW bound yields root node relaxations with lower dual degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Another  advantage of using DW cuts is regarding the dimension of the cuts, which is often used as a mea-  sure of the strength of the cutting plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' A higher dimension of the associated face means that  the cutting plane is closer to being facet-defining and irredundant, and therefore stronger in some  sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Let S denote the feasible region of the original problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We next present a relationship  between the dimension of a DWF cut in conv(S) and its restriction in conv(Qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We say that MIP (1) has block relative feasibility if for each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , q} and  y ∈ Qj there exists x ∈ S such that xI(j) = y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=', projxI(j)(S) = Qj for j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  The block relative feasibility assumption holds for a broad class of MIP problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For example,  for two-stage stochastic integer programs, relatively complete recourse [33] implies block relative  feasibility if each block is defined by first-stage and second-stage constraints associated with a  particular scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Assume problem (1) has block relative feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If (πj)⊤y ≥ Dj(πj) defines a  d-dimensional face of conv(Qj) for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , q}, then (15) defines a face of conv(S) of  dimension at least d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Since (πj)⊤y ≥ Dj(πj) defines a d-dimensional face of conv(Qj), there exists d + 1 affinely  independent points {yk}d+1  k=1 ⊆ Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' By the block relative feasibility assumption, there exist points  {xk}d+1  k=1 ⊆ S such that xk  I(j) = yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Points {xk}d+1  k=1 are affinely independent from each other by  affine independence of {yk}d+1  k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The conclusion then follows from the fact that {xk}d+1  k=1 are all on  the face associated with (15) in conv(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Proposition 5 indicates that DWF cuts whose restrictions in the space of xI(j) correspond  to high-dimensional faces of conv(Qj) are likely to define high-dimensional faces in conv(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' It  motivates the idea of strengthening some DWF cuts to obtain higher-dimensional DWF cuts, which  will be discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In contrast to DWF cuts that potentially define high-dimensional  faces of conv(S), the objective function cut c⊤x ≥ zD usually corresponds to an empty face of  conv(S) unless zD = z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Even if zD = z∗, the face associated with the objective function cut may  still be low-dimensional unless the problem has many alternative optimal primal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='2  Cut Generation From the Lagrangian Dual  As discussed earlier, solving the Lagrangian dual problem can be computationally more efficient  than the standard DW decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' During the solution of the Lagrangian dual problem, DWF  cuts similar to (5) can also be generated every time we evaluate the function values of Dj(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  The following result demonstrates the strength of DWF cuts generated from the evaluation of the  Lagrangian dual function at any point (π, β) with z(π, β) > −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  11  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Let (π, β) be Lagrangian dual multipliers for (11) satisfying constraints (13b) and  (13c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Then,  z(π, β) ≤ min c⊤x  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' (πj)⊤xI(j) ≥ Dj(πj),  j ∈ J,  Ax ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (16)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Consider any x ∈ Rn feasible to the right-hand side LP of (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Since ci = �  j:i∈I(j) πj  i +  β⊤Ai for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , n by (13b), we have  c⊤x =  n  �  i=1  �  j:i∈I(j)  πj  i xi + β⊤Ax =  q  �  j=1  (πj)⊤xI(j)  �  ��  �  ≥Dj  �  πj�  + β  ����  ≥0  ⊤ Ax  ����  ≥b  ≥  p  �  j=1  Dj(πj) + b⊤β(τ) = z(π, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Note that, unlike Theorem 2, Proposition 6 does not depend how the dual multiplier is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  If one can solve the Lagrangian dual problem to optimality, then Proposition 6 implies that one  can recover DW bound using DWF cuts associated with an optimal solution of the Lagrangian  dual problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Let (¯π, ¯β) be an optimal solution of (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Then,  zD = min c⊤x  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' (¯πj)⊤xI(j) ≥ Dj(¯πj),  j ∈ J,  Ax ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (17)  Results similiar to Propostion 4 can be derived for the optimal face of the LP (17) that utilizes  DWF cuts associated with an optimal Lagrangian dual solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Even if the Lagrangian dual  problem is not solved to optimality, Proposition 6 still guarantees that DWF cuts can provide  strength at least as strong as the best Lagrangian dual bound by generating DWF cuts associated  with the dual solution that provides the strongest bound obtained so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Besides, there may be  values for adding DWF cuts obtained at different dual multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' See Example 1 in Appendix C  showing that DWF cuts can potentially provide stronger dual bounds than the best Lagrangian  dual bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  5  Generating a Stronger Relaxation  In this section we describe how to strengthen the DWF cuts to obtain a stronger relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The  strengthened cutting planes are still DWF cuts valid for blocks Qj, and therefore do not lead to  better bounds than zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' However, these strengthened cutting planes may potentially make more  constraints active in the LP relaxation, and therefore can help reduce the dual degeneracy level of  the formulation and improve the branch-and-cut performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Moreover, the strengthened cutting  planes are more likely to define high-dimensional faces of the original problem, as they define  higher-dimensional faces of the block polyhedra conv(Qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  12  Figure 4: Disjunctive Coefficient Strengthening for Three Cases  xi  0  1  xi  0  1  r  xi  0  1  x0  x1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='1  Disjunctive Coefficient Strengthening  We first describe a disjunctive coefficient strengthening technique for binary variables [34] that  strengthens the coefficients of a valid inequality one at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Given a valid inequality a⊤x ≥ f  for the set Q, let Q= := {x ∈ Q : a⊤x = f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For a binary variable xi, its coefficient ai in the cut  can be strengthened if one of the following two cases hold: (i) xi = 0 for all x ∈ Q=, or, (ii) xi = 1  for all x ∈ Q=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If there are points x0, x1 ∈ Q= such that x0  i = 0 and x1  i = 1, then this approach  does not improve (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=', decrease) the coefficient of the variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Figure 4 shows an example of all  three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Note that if we solve  ¯f = min{a⊤x : x ∈ Q, xi = 1}  (18)  and observe that ¯f > f, then we can strengthen the original inequality a⊤x ≥ f to be a⊤x ≥  f + ( ¯f − f)xi using the disjunction  Q = {x ∈ Q : xi = 0} ∪ {x ∈ Q : xi = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Similarly, if we solve  ¯f = min{a⊤x : x ∈ Q, xi = 0},  (19)  and observe that ¯f > f, then we can strengthen the original inequality a⊤x ≥ f to be a⊤x ≥  f + ( ¯f − f)(1 − xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If either problem (18) or (19) is infeasible, then one can simply fix variable xi  to 0 or 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' It is easy to verify that the original inequality is implied by the strengthened  inequality together with the bound constraint xi ≥ 0 or xi ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Therefore, if the original inequality  is active in the LP relaxation, then one round of coefficient strengthening (if applicable) would  potentially make a bound constraint active in the LP relaxation and reduce the dual degeneracy  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Note that this approach does not increase the size of the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  For strengthening a DWF cut obtained from a block j, we set Q = Qj and apply coefficient  strengthening sequentially to all coefficients of binary variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In practice, we keep a set L of  points that are known to be elements of Q=, generated from previous solutions of (14), (18) and  (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If there are x0, x1 ∈ L such that x0  i = 0 and x1  i = 1, then without solving (18) or (19) we  conclude that disjunctive strengthening cannot be applied to the i-th coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='2  Strengthening via Tilting  We next describe a tilting technique introduced in [35, 36] that starts with a valid inequality and  iteratively tilts it to obtain a facet-defining inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Let a⊤x ≥ f be a valid inequality for Q  and assume that it is not facet-defining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Also assume that conv(Q) is full dimensional and there  is a set Q′ ⊆ Q such that all points x ∈ Q′ satisfy a⊤x = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The algorithm first generates a point  13  ¯x ∈ Q \\ Q′ that satisfies a⊤¯x > f, and a vector (v, w) such that v⊤x = w for all x ∈ Q′ ∪ {¯x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The  algorithm then does the following:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If v⊤x ≥ w is valid for Q, then the algorithm outputs ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Otherwise the algorithm computes  the largest λ+ ∈ R+ such that (a + λ+v)⊤x ≥ f + λ+w is valid for Q and outputs this  inequality together with a point ¯x+ ∈ Q \\ Q′ satisfying (a + λ+v)⊤¯x+ = f + λ+w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If v⊤x ≤ w is valid for Q, then the algorithm outputs ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Otherwise the algorithm computes  the largest λ− ∈ R+ such that (a − λ−v)⊤x ≥ f − λ−w is valid for Q and outputs this  inequality together with a point ¯x− ∈ Q \\ Q′ satisfying (a − λ−v)⊤¯x− ≥ f − λ−w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Note that when conv(Q) is full dimensional, both v⊤x ≥ w and v⊤x ≤ w cannot be valid and  consequently we obtain two (possibly identical) valid inequalities whose conic combination implies  the original inequality a⊤x ≥ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Moreover, each one of the these inequalities has one more known  feasible point on its associated face than a⊤x ≥ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In [36], the authors show that recursively tilting  one of the two obtained valid inequalities leads to a facet-defining inequality for conv(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  In our context, we apply the tilting idea with the following modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For a DWF cut  associated with block j, we set Q to be Qj, and instead of picking one of the two tilted inequalities  for the subsequent tilting iteration, we apply tilting to both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' By doing so, we create a binary tree  where the root node corresponds to the original DWF cut and the remaining nodes correspond to  DWF cuts obtained by tilting the inequalities associated with their parent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For any such  tree, cuts associated with the leaf nodes imply the original DWF cut associated with the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We call the collection of inequalities associated with the leaf nodes of a depth-d tree depth-d tilted  DWF cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Using a depth-d tree means replacing one DWF cut with up to 2d DWF cuts, which  can be computationally expensive if d is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Therefore, it is often beneficial to choose a relatively  small value for d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Note that this process does not improve the DW bound but can potentially help  reduce the dimension of the LP optimal face as more constraints are likely to become active at the  optimal face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  To improve computational performance, we keep track of the feasible points for block j ∈ J  that we encounter during the overall algorithm and store them in a set ˆQj ⊆ Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Using this set of  points reduces the number of oracle calls needed in the following two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' First, we can check if  any point in ˆQj satisfies the condition for ¯x and use it for choosing (v, w), thus avoiding an extra  call to the optimization oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In addition, when computing λ+, we use ˆQj to obtain the following  upper bound on it:  λ+ :=  min  x∈Qj:v⊤x<w  a⊤x − f  w − v⊤x  ≤  min  x∈ ˆQj:v⊤x<w  a⊤x − f  w − v⊤x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  This upper bound can be very close to the final λ+ value and, in practice, we have observed a  significantly reduction in the number of oracle calls needed to compute λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' This in turn reduces  the time spent on tilting significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We use the same idea when computing λ− ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  6  Multiple Knapsack Assignment Problem: A Case Study  We test our approaches on the Multiple Knapsack Assignment Problem (MKAP) [37] that is an  extension of the well-studied multiple knapsack and assignment problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' A detailed description of  the MKAP problem, its DW reformulation as well as the test instances are presented in Appendix  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' To our knowledge, solving MKAP as an integer program by a branch-and-cut MIP solver is the  only exact solution approach that has been tried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We compare the performance of the following formulations:  14  Figure 5: Tilting One Valid Inequality Into Two Stronger Valid Inequalities  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' MIP: The original formulation (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' OBJ: Objective cut c⊤x ≥ zD added into the original formulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' DWF: Last-iteration DWF cuts added into the original formulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' STR: Last-iteration DWF cuts with disjunctive coefficient strengthening added into the orig-  inal formulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' DdT: Last-iteration DWF cuts with disjunctive coefficient strengthening and depth-d tilting  added into the original formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  All experiments are run on a computer with a Intel(R) Xeon(R) Gold 6258R processor running  at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='7GHz, with up to 4 threads used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' All optimization problems are solved using the optimization  solver Gurobi [38] version 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
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+page_content='  The parameters K, M, and N define the size of these instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In all the tables presented in  this section, statistics presented in a row correspond to averages over 30 instances for a particular  combination of (K, M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='1  Comparing Relaxations  We first present some results regarding lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We compare the DW bound zD with the LP  relaxation bound zL and the Gurobi root node bound (denoted by zR) obtained by adding solver  cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' With some abuse of notation, we use zD to denote the best Lagrangian dual bound obtained  by the level method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In theory, zD should be equal to the DW bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' However, numerical issues  happen occasionally when solving the quadratic program in the level method, especially when the  DW bound is close to the LP relaxation bound, in which case zD can be strictly less than zL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  In Table 1, we present the relative gaps rL := (zD −zL)/|zD| and rR := (zD −zR)/|zD| between  zD and natural LP relaxation bound zL as well as the Gurobi root bound zR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The  running time tR spent at the root node and running time tD spent on solving the Lagrangian dual  problem by the level method are also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Depending on the instance size, the relative difference between the DW bound and the natrual  LP relaxation bound varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' When the gap is small, Gurobi can often generate a root node relax-  ation that is almost as strong as or even slightly stronger than the DW relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' When the gap  is large, Gurobi can close some gap at the root node but the bound is often significantly weaker  than DW bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  In our preliminary experiments, we observe that DWF cuts are more likely to be effective  when DW bound is significantly stronger than the natrual LP relaxation bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Therefore, we  focus on instance classes (defined by the size parameters |K|, |M| and |N|) whose average natrual  15  Table 1: Comparison of Different Bounds and Their Computing Time  |K|  |M|  |N|  rL (%)  rR (%)  tR (s)  tD (s)  2  10  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
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+page_content='5  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='7  300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
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+page_content='0  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='8  LP relaxation bound is more than 1% weaker than average DW bound, especially the ones with  Gurobi root bound more than 1% worse than DW bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We denote these two sets of instances  by I1 and I2(⊆ I1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In Table 1, I1 instances correspond to rows with bold rL and  I2 instances correspond to rows with bold rR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Set I1 consists of 870 instances and set I2 consists  of 270 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Over the I1 instances, we observe that zD is very close to the DW bound with  relative gaps always less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='01% except on 3 instances (whose relative gaps are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='03%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='16%  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='34%, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We next investigate the dual degeneracy of different formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In Table 2, we report the  relative normalized degeneracy level of the optimal dual solution computed by Gurobi for LP  relaxations of different formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The relative degeneracy level is calculated as the degeneracy  level n−∥α∗∥0 divided by the dimension n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=', 1−∥α∗∥0/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Bold rows correspond to I2 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Due to numerical tolerances, the optimal values α∗  k of some dual variables can sometimes be very  close to but not equal to zero, especially the ones associated with bound constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We treat α∗  k  as 0 if |α∗  k| ≤ 10−8 for all entries α∗  k of α∗ when computing ∥α∗∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Although the strengthening  techniques in Section 5 do not guarantee a monotonic decrease in degeneracy level, it is clear from  Table 2 that formulations with stronger strengthening tend to have less degenerate optimal dual  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We observe that the degeneracy level of DWF is already comparable to MIP on I2  instances but is higher than MIP on I1 \\ I2 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' With coefficient strengthening, STR has  lower average dual degeneracy than MIP in almost all cases except on the instance class (5,40,50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  The dual degeneracy continues to decrease with further tilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Finally, it is worth noting that, as  expected, the relative degeneracy level of OBJ is extremely high, close to 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Time spent on  generating these different formulations is presented in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  16  Table 2: Relative Degeneracy Levels of LP Optimal Dual Solutions for Different Formulations  (|K|, |M|, |N|)  (1 − ∥α∗∥0/n) × 100%  MIP  OBJ  DWF  STR  D1T  D2T  D3T  D4T  D5T  D6T  ( 2,20, 50)  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
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+page_content=' of Nodes Explored  8  6  4  2  0  2  4  Relative Gap (%)  MIP UB  MIP LB  0  20000  40000  60000  80000  100000  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' of Nodes Explored  8  6  4  2  0  2  4  Relative Gap (%)  OBJ UB  OBJ LB  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='2  Objective Function Cut  We next emprically investigate how the objective function cut (formulation OBJ) may hurt the  branch-and-cut solution comparing to simply using the original formulation MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We consider MKAP instances with (|K|, |M|, |N|) = (25, 20, 300), for which DW bound is  much stronger than the natrual LP relaxation bound (with average gap 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='18%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Out of these  30 instances, Gurobi can solve 22 using MIP and 2 using OBJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We plot in Figure 6 the average  relative upper and lower bound gaps (with respect to the optimal value) obtained by Gurobi using  formulations MIP and OBJ on (25, 20, 300) instances as the number of branching nodes explored  increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We observe that although DW bound is almost equal to the optimal value for these  instances (with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='01% average relative gap), Gurobi can hardly produce good feasible solutions  to improve the primal bound when using the OBJ formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Instead, Gurobi can find better  17  feasible solutions much more efficiently when using MIP, where all optimal solutions are found  after exploring a bit more than 20,000 nodes, leaving a few instances unsolved because Gurobi  cannot prove optimality within the timelimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' This shows that a single objective function cut can  already significantly influence the performance of Gurobi’s primal heuristics, which is plausibly due  to worse branching decisions and cut generation for OBJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' A table reporting the average number of  different cutting planes generated when using different formulations is presented in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='3  Branch-and-Cut Performance  Table 3: Comparison of Branch-and-Cut Performance on I2 Instances  (|K|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' |M|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' |N|)  Number of Instances Solved  Average B&C Time (s)  MIP  OBJ  DWF  STR  D3T  D6T  MIP  OBJ  DWF  STR  D3T  D6T  (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='100)  28/30  18/30  30/30  30/30  30/30  30/30  ≥ 65  ≥298  2  0  0  0  (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='200)  11/30  10/30  30/30  30/30  30/30  30/30  ≥424  ≥457  6  1  2  2  (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='300)  7/30  11/30  30/30  30/30  30/30  30/30  ≥489  ≥454  41  2  12  7  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='100)  30/30  30/30  30/30  30/30  30/30  30/30  0  1  0  0  0  0  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='200)  30/30  30/30  30/30  30/30  30/30  30/30  2  25  0  0  0  0  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='300)  30/30  29/30  30/30  30/30  30/30  30/30  6  ≥ 40  0  0  1  3  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='200)  29/30  9/30  30/30  30/30  30/30  30/30  ≥ 44  ≥499  3  2  1  1  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='300)  22/30  2/30  29/30  30/30  30/30  30/30  ≥224  ≥590  ≥ 48  5  3  7  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='300)  1/30  0/30  0/30  2/30  1/30  3/30  ≥595  ≥600  ≥600  ≥590  ≥591  ≥575  Table 4: Comparison of Branch-and-Cut Performance on I2 Instances (Cont’d)  (|K|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' |M|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' |N|)  Average Optimality Gap (%)  Average Number of Nodes  MIP  OBJ  DWF  STR  D3T  D6T  MIP  OBJ  DWF  STR  D3T  D6T  (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='100)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  ≥  22761  ≥1698156  4481  220  2  1  (10,10,200)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  ≥2778880  ≥3898948  6828  407  33  3  (10,10,300)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  ≥2616971  ≥2781174  131139  1208  631  6  (25,10,100)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  93  1123  1  1  1  1  (25,10,200)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  3853  20272  1  1  1  1  (25,10,300)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  8488  ≥  25864  1  1  1  1  (25,20,200)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  ≥  27987  ≥ 438310  2255  426  3  2  (25,20,300)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='00  ≥ 113353  ≥ 410728  ≥ 13861  929  1  1  (25,30,300)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='22  ≥  81545  ≥ 379999  ≥105115  ≥35197  ≥20036  ≥6568  Finally, we compare the branch-and-cut performance of formulations MIP, OBJ, STR, D3T  and D6T on the selected MKAP instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We set a 10-minute branch-and-cut timelimit for  each formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For I2 instances, we summarize in Tables 3 and 4 some statistics regarding the  number of instances solved by different formulations, average branch-and-cut time (excluding time  for the Lagrangian dual problem and cut generation/strengthening), average ending optimality  gap and average number of branching nodes needed to solve the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We observe that  branch and cut on OBJ solves fewer instances than MIP, which is consistent with the folklore that  simply adding an objective function cut to improve dual bound is mostly ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The ending  optimality gap of OBJ might be better than that of MIP because OBJ has a much stronger lower  bound, whereas finding good feasible solutions can be hard for OBJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Without any strengthening,  formulation DWF already significantly outperforms both MIP and OBJ in branch and cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For  most instances, D6T requires the smallest number of branching nodes to solve the problem, while  STR has the best solution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' This can be explained by the fact that D6T has the strongest  LP relaxation while STR has a more compact formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Compared to D3T or D6T, the  processing of each branching node takes less time for STR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For the hardest (25, 30, 300) instances,  a stronger polyhedral relaxation becomes more important for closing the optimality gap, where  D6T has the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  In Figure 7 we plot gap closed as time or number of nodes  explored increases for different methods on the hardest (25, 30, 300) instances where the relative  18  gap is computed by comparing with the best feasible solution found by all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We observe  that although all methods except MIP start from the same root bound (DW bound zD), with  coefficient strengthening and tilting the solver can find good primal solutions more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We  also note that each branching node takes longer time to explore when coefficient strengthening and  tilting are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Additional tables comparing branch-and-cut performance on I1 \\ I2 instances  are presented in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Figure 7: Gap Closed as Time (Left) and Number of Nodes Explored (Right) Increases  0  100  200  300  400  500  600  Solution Time (s)  2  1  0  1  2  3  4  5  Relative Gap (%)  MIP UB  OBJ UB  DWF UB  STR UB  D6T UB  MIP LB  OBJ LB  DWF LB  STR LB  D6T LB  0  20000  40000  60000  80000  100000  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' of Nodes Explored  2  1  0  1  2  3  4  5  Relative Gap (%)  MIP UB  OBJ UB  DWF UB  STR UB  D6T UB  MIP LB  OBJ LB  DWF LB  STR LB  D6T LB  7  Hybrid Implementation in a Multi-Thread Context  Computationally, using the last iteration DWF cuts (and their strengthened versions) does not  always lead to a performance improvement as it (i) requires additional computational effort to  generate the cuts and (ii) increases the size of the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Similar to other MIP techniques,  one needs to decide whether or not to use this approach for a given problem instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Machine  learning (ML) techniques have been widely applied to make algorithmic decisions inside MIP  solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For example, FICO Xpress implements a random forest model to determine whether or  not local cuts should be generated in nodes other than the root of the branch-and-bound tree  [39], while CPLEX uses ML to decide whether to linearize the objective function of mixed-integer  quadratic programming problems [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  In the remainder of this section, we present some preliminary experiment results with a simple  ML model to predict whether or not branch and cut may benefit from DWF cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We consider  a computational environment with 4 threads where one starts with running branch and cut on  MIP (using 3 threads) and solves the Lagrangian dual problem (using 1 thread) in parallel until  the Lagrangian dual problem is solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' After this first phase, one decides to either allocate all 4  threads to the original formulation MIP or to the strengthened formulation STR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  To collect labels for the training phase, we use 20% of I1 instances as the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We first  run standard branch and cut (using 3 threads) and Lagrangian dual (using 1 thread) in parallel  until the Lagrangian dual problem is solved, and collect some simple features for each instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We then separately try allocating all 4 threads to (i) solving the original formulation MIP, and  (ii) solving the strengthened formulation STR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We label an instance as promising if one of the  following holds when both methods are run for 10 minutes:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The instance can be solved with STR but not with MIP, or,  19  Table 5: Comparing MIP, STR and HYB on Test Instances  MIP  STR  HYB  Number of Instances Solved  167/242  206/242  208/242  Average Optimality Gap (%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='18%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='05%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='05%  Average Solution Time (s)  ≥ 223  ≥ 117  ≥ 110  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Neither method can solve the instance, and STR has a smaller optimality gap (at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='01%  smaller), or,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Both methods can solve the instance and STR reduces the solution time by 10% or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Using this labeled data, we train a simple decision stump (depth-1 decision tree) to determine  whether or not we should restart the MIP using the formulation STR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The features we use are  composites of zL, zD, zR, zLB and zUB, where zLB and zUB denote the lower and upper bounds  obtained by the solver from solving the original formulation until the Lagrangian dual problem is  solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' In our experiments, zUB is always finite as the solver can easily find out that the all-zero  solution is feasible for MKAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The resulting decision stump obtained after training is as follows:  “If (zD − zLB)/zUB > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='05%, then switch to the strengthened formulation STR.”  Instances that can be solved using the original MIP formulation before we finish computing zD  are dropped from the training set and the test set as there is no algorithmic decision of interest  to make in that situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' This leaves us a training set with 62 instances and a test set with 242  instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Next we compare a hybrid implementation with default Gurobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' To further simplify the imple-  mentation, we replace the parallelization of branch and bound and Lagrangian dual by a simulated  parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Such a simulated hybrid implementation is described below:  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (a) Using 1 thread, solve the Lagrangian dual problem using the level method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Store the  solution time tD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (b) Using 3 threads, run a standard MIP solver using the original MIP formulation with  timelimit tD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If the MIP is solved to optimality in Step 1, then stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Else if (zD − zLB)/zUB > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='05%, then restart the MIP solver using all 4 thread with the  strengthened formulation STR and the upper bound (primal solution) obtained from the  MIP solver in Step 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Else, allocate all 4 threads to the MIP solver for continuing with the original formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  In Table 5 we summarize results obtained by directly solving the original formulation (‘MIP’),  directly generating and solving the strengthened formulation (‘STR’), and applying our simulated  hybrid implementation (‘HYB’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For each method, we report the number of instances solved within  the timelimit of 10 minutes, the ending optimality gap and average time spent on solving the  problem including cut generating time for STR and HYB in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Note that STR is likely to  outperform MIP on the test instances because the DW bound is already significantly stronger than  the LP relaxation bound on those instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We observe that both STR and HYB outperform  MIP on the test instances while HYB slightly outperforms MIP in terms of solution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' This is  20  because HYB avoids solving the more expensive STR formulation in some cases when the bound  provided by STR is not significantly stronger than MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Another interesting observation is that  HYB solves exactly all instances that can be solved by either MIP or STR, including two instances  that cannot be solved by STR but can be solved by MIP, demonstrating the value of having the  flexibility of switching between different formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We believe this hybrid version would be  relatively easy to implement by modern MIP solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  8  Conclusions  We investigated methods for generating and strengthening cutting planes that are derivable from  DW reformulation to accelerate branch and cut for MIPs with block structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Numerical exper-  iments have shown that adding these cutting planes is effective in cases when the DW bound is  strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We also proposed ways to incorporate our methodology into a MIP solver when multiple  threads are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Empirically, we observed that if many of the cutting planes are added into the formulation,  then the LP solution at each branching node can be significantly slowed down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' A potential fix is to  consider cut selection within our cut generation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' One may also consider some adaptive  tilting schemes rather than tilting each inequality to the same depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We also observed that the  Lagrangian dual problem often has nonunique optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  It is therefore interesting to  investigate whether using alternative dual solutions can lead to stronger cutting planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
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+page_content='  24  A  Dantzig-Wolfe Decomposition  Algorithm 1 Standard Dantzig-Wolfe Decomposition  1: Initialize:  ˆV j ← a subset of extreme points of conv(Qj), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , p  ˆRj ← a subset of extreme rays of conv(Qj), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , p  t ← 0  2: t ← t + 1, solve  zt  D = min c⊤x  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' xI(j) =  �  v∈ ˆ  V j  vλj  v +  �  r∈ ˆ  Rj  rµj  r,  j ∈ J,  (πj)  Ax ≥ b,  (β)  �  v∈ ˆV j  λj  v = 1,  j ∈ J,  (θj)  λj ≥ 0, µj ≥ 0,  j ∈ J  (20)  (assume { ˆV j}q  j=1 and { ˆRj}q  j=1 are initialized such that (20) is always feasible)  3: let (π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , πq, θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , θq) denote the values of optimal dual variables for (20)  4: for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , q do  5:  solve the following pricing problem:  Dj(πj) := min{(πj)⊤v : v ∈ conv(Qj)} = min{(πj)⊤v : v ∈ Qj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='(21)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='6:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='if the pricing problem (21) is bounded then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='7:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='let vj denote an optimal solution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='8:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='ζj ← (πj)⊤vj − θj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='9:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='if ζj < 0 then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='10:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='ˆV j ← ˆV j ∪ {vj}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='11:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='12:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='else  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='13:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='ζj ← −∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='14:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='let rj denote an extreme ray of conv(Qj) with (πj)⊤rj < 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='15:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='ˆRj ← ˆRj ∪ {rj}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='16:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='17:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='if ζj ≥ 0 for all j ∈ J then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='18:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='return zD = zt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='19:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='else  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='20:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='go to step 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='21:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='22: end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='The Level Method for the Lagrangian Dual Problem  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='The level method adds on top of the cutting plane method a regularization step that requires  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='solving a convex quadratic program to find a promising candidate multiplier (π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' β) = (¯π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' ¯β) to  explore while staying close to the previous multiplier that is already explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' A pseudocode of  the level method is given in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For generating the upper bound ¯z, we give the original  formulation to the solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We pick the second feasible solution ¯x found by the solver (often much  better than the first feasible solution) and set ¯z = c⊤¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Constraint (22d) is important in early  iterations of the algorithm to ensure that problem (13) is bounded, but becomes redundant when  UB< ¯z in later iterations of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Within the level method, LB and UB store the best  lower and upper bounds of zD found by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' If we set the termination condition to be  25  Algorithm 2 The Level Method for Solving (13)  1: Initialize:  ˆV j ← ∅, ˆRj ← ∅, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , p  ¯z ← an upper bound of zD  LB ← −∞, UB ← ∞, t ← 0  2: Main Loop: t ← t + 1, solve:  UB ← max  q  �  j=1  θj + b⊤β  (22a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' θj ≤ v⊤πj,  v ∈ ˆV j, j ∈ J,  (22b)  r⊤πj ≥ 0,  r ∈ ˆRj, j ∈ J,  (22c)  q  �  j=1  θj + b⊤β ≤ ¯z,  (22d)  �  j:i∈I(j)  πj  i + β⊤Ai = ci,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , n,  (22e)  β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (22f)  (22g)  3: if LB = −∞ then  4:  (¯π, ¯β) ← optimal value of (π, β) in (22)  5: else  6:  solve:  min  ���  π − ¯π, β − ¯β  ���2  2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  q  �  j=1  θj + b⊤β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='7 · UB + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='3 · LB  (22b) − (22f)  (23)  7:  (¯π, ¯β) ← optimal value of (π, β) in (23)  8: end if  9: for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , q do  10:  solve (14) for πj = ¯πj  11:  if (14) is bounded then  12:  let vj denote an optimal solution  13:  ˆV j ← ˆV j ∪ {vj}  14:  else  15:  let rj denote an extreme ray of conv(Qj) with (πj)⊤rj < 0  16:  ˆRj ← ˆRj ∪ {rj}  17:  end if  18: end for  19: LB ← max  �  LB, �q  j=1 Dj(¯πj) + b⊤ ¯β  �  20: if UB-LB is small enough then  21:  return LB  22: else  23:  go to step 2  24: end if  UB-LB=0, then the algorithm returns the exact DW bound zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' To avoid some numerical issues,  when implementing the level method we terminate the algorithm if difference between LB and UB  is within 10−6 relative tolerance, or if the solver fails to solve the quadratic master problem (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We observe that the second case rarely happens but can sometimes lead to a Lagrangian dual  bound weaker than the natrual LP relaxation bound since we do not solve the Lagranigan dual  problem to optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  26  C  Examples  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Consider the following MIP:  z∗ = min x1 + x2 + 2x3 + 2x4  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' x2 + x4 ≥ 3,  3x1 + x2 + 3x3 + x4 ≥ −12,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='5 ≤ xi ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='5, xi ∈ Z, i = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Define I1 = {1, 2}, I2 = {3, 4}, Qj = {y ∈ Z2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='5 ≤ y1 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='5 ≤ y2 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='5} for j ∈ {1, 2} and  the linking constraints Ax ≥ b to be  � 0  1  3  1  � � x1  x2  �  +  � 0  1  3  1  � � x3  x4  �  ≥  � 3  12  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  In this example the MIP has symmetric blocks but asymmetric objective coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The LP relax-  ation of the problem gives the lower bound 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Let π(1) = (1, 1/4, 2, 5/4)⊤, β(1) = (3/4, 0)⊤, π(2) =  (2/11, 8/11, 13/11, 19/11)⊤, β(2) = (0, 3/11)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Note that the dual multipliers  �  π(τ), β(τ)  �  τ∈{1,2}  satisfy the assumptions in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The best dual bound is given by  max  τ∈{1,2} z  �  π(τ), β(τ)  �  = max{27/4, 78/11} = 78/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Adding DWF cuts associated with π(1) and adding DWF cuts associated with π(2) into the LP  relaxation of the original formulation yield bounds 125/16 and 149/19, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' These bounds  are stronger than the corresponding Lagrangian relaxation bounds 27/4 and 78/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' On the other  hand, the LP  min c⊤x  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  �  πj(τ)  �⊤xI(j) ≥ Dj  �  πj(τ)  �  ,  j ∈ J, τ = 1, 2,  Ax ≥ b  has the optimal objective value equal to 8, which happens to be z∗ in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Consider the following primal and dual LP pair:  (Primal) : min x1 − x2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' x1 + x2 ≥ 1,  x2 ≥ 1,  −x2 ≥ −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (Dual) : max α1 + α2 − α3  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' α1  = 1,  α1 + α2 − α3 = −1,  α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  In this example, the primal LP has a unique optimal solution x = (0, 1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Therefore, the dimension  of the optimal face is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Also, the problem has a unique dual basic solution α = (1, 0, 2)⊤, whose  0-norm is 2, which is strictly less than n minus the dimension of the optimal face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' However, note  that all dual solutions of the form α = (1, λ, 2 + λ)⊤ with λ ≥ 0 are optimal for the dual LP, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=',  the problem has nonunique dual optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  27  D  MKAP and Test Instances  In MKAP, there are a finite number of items, knapsacks and item classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Each item belongs to  exactly one item class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' The decision maker has to pack a subset of items into knapsacks, subject  to constraints that only items belonging to the same item class with a total weight no larger than  the capacity of the knapsack can be packed into the same knapsack, with the aim of maximizing  the total profit of the packed items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Specifically, let N, M and K denote the index sets of items,  knapsacks and item classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For j ∈ N, let pj denote the profit of item j and wj  denote the weight of item j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For i ∈ M, let Ci denote the capacity of knapsack i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' For k ∈ K,  let Sk denote the set of items that belong to item class k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=', {Sk}k∈K is a partition of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We  use variables x ∈ {0, 1}M+N to represent the packing decisions, where xij = 1 if and only if item  j is packed into knapsack i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Variables y ∈ {0, 1}M+K represent the assignment decisions, where  yik = 1 if and only if knapsack i is used to pack items from item class k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' MKAP can then be  formulated as the following integer program (we flip the sign of the objective function to formulate  it as a minimization problem):  min −  �  i∈M  �  j∈N  pjxij  (24a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  �  j∈Sk  wjxij ≤ Ciyik,  i ∈ M, k ∈ K,  (24b)  �  i∈M  xij ≤ 1,  j ∈ N,  (24c)  �  k∈K  yik ≤ 1,  i ∈ M,  (24d)  x ∈ {0, 1}M×N, y ∈ {0, 1}M×K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  (24e)  It has been observed in [41] that applying Lagrangian relaxation to (24) with constraints (24c)  and (24d) dualized can lead to a dual bound potentially much stronger than the LP relaxation  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' By equivalence between DW decomposition and Lagrangian relaxation, this observation  also applies to DW decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Specifically, we define |M| × |K| blocks in our DW decom-  position, where for each i ∈ M and k ∈ K block Qi,k is defined by a knapsack constraint  �  j∈Sk wjxij ≤ Ciyik together with constraints forcing (xij)j∈Sk and yik to be binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  We generate the instances following the scheme of [37] but using different instance sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We  consider instances with |K| ∈ {2, 5, 10, 25}, |M| ∈ {10, 20, 30, 40} and |N| ∈ {50, 100, 200, 300}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  For each combination (|K|, |M|, |N|) of the parameters, we generate three types (uncorrelated,  weakly correlated, strongly correlated) of MKAP instances with 10 instances generated using 10  different random seeds for each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We refer readers to [37] for a more detailed description of  the instance generation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' It is worth mentioning that the item classes {Sk}k∈K all have  equal sizes |N|/|K| for our test instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  E  Time Spent on Obtaining Different Formulations  Since both OBJ and DWF can be obtained directly after solving the Lagrangian dual problem,  we only report in Table 6 the total time spent on applying strengthening and different levels of  tilting for STR and DdT with d ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' , 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' Bold rows correspond to I2 instances and other rows  correspond to I1 \\ I2 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  28  Table 6: Comparison of Time Spent on Obtaining Different Formulations  (|K|, |M|, |N|)  Generation Time Excluding Lagrangian Dual Time (s)  STR  D1T  D2T  D3T  D4T  D5T  D6T  ( 2,20, 50)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
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+page_content='0  At each row, the averages of 30 instances are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  F  Number of Cutting Planes Generated Using Different Formu-  lations  In Table 7, we report the number of cutting planes generated by Gurobi within 10 minutes for  formulations MIP, OBJ, DWF, STR, D3T and D6T averaged over the (25, 20, 300) MKAP  instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  G  Branch-and-Cut Results for I1 \\ I2 instances  In Tables 8 and 9 we report branch-and-cut results for I1\\I2 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' We observe that the original  MIP formulation is very competitive on these instances, whose performance is often better than  DWF and even STR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' This can be explained by the fact that although the natrual LP relaxation  is weak (rL is large), Gurobi can close much of the gap by presolve and cutting planes at the root  node (rR is small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  29  Table 7: Average Number of Cutting Planes Generated by Gurobi  MIP  OBJ  DWF  STR  D3T  D6T  Gomory  132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='1  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='4  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  Lift-and-project  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  Cover  616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='1  176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='5  121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='5  173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='3  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  Clique  362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='3  248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='5  148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='3  266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='4  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  MIR  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='2  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='6  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='7  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  StrongCG  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='3  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='8  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='7  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  Flow cover  228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='8  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='6  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  Zero half  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='9  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  RLT  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  Relax-and-lift  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='0  Table 8: Comparison of Branch-and-Cut Performance on I1 \\ I2 Instances  (|K|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' |M|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' |N|)  Number of Solved  Average B&C Time (s)  MIP  OBJ  DWF  STR  D3T  D6T  MIP  OBJ  DWF  STR  D3T  D6T  ( 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  23/30  30/30  30/30  30/30  30/30  2  ≥202  7  5  4  4  ( 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  29/30  30/30  30/30  30/30  30/30  0  ≥ 25  0  0  0  0  ( 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  0  0  0  0  0  0  ( 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  4  62  7  4  2  1  ( 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  1  42  4  2  1  0  ( 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  0  19  0  0  0  0  ( 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  0  0  0  0  0  0  (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  0  1  0  0  0  0  (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  0  16  0  0  0  0  (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  0  1  0  0  0  0  (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  0  0  0  0  0  0  (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='100)  30/30  1/30  29/30  30/30  30/30  30/30  40  ≥600  ≥103  72  15  12  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  0  0  0  0  0  0  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  0  0  0  0  0  0  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='100)  30/30  20/30  30/30  30/30  30/30  30/30  1  ≥204  0  0  0  0  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  0  0  0  0  0  0  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='100)  30/30  10/30  30/30  30/30  30/30  30/30  5  ≥432  1  1  0  0  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='200)  16/30  1/30  10/30  15/30  23/30  26/30  ≥377  ≥600  ≥475  ≥402  ≥255  ≥143  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' 50)  30/30  30/30  30/30  30/30  30/30  30/30  0  0  0  0  0  0  (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='100)  30/30  9/30  30/30  30/30  30/30  30/30  2  ≥464  1  1  0  0  At each row,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content=' the averages of 30 instances are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
+page_content='  Table 9: Comparison of Branch-and-Cut Performance on I1 \\ I2 Instances (Cont’d)  (|K|, |M|, |N|)  Average Optimality Gap (%)  Average Number of Nodes  MIP  OBJ  DWF  STR  D3T  D6T  MIP  OBJ  DWF  STR  D3T  D6T  ( 2,20, 50)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFPT4oBgHgl3EQfsjWt/content/2301.13149v1.pdf'}
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+arXiv:2301.13520v1  [cond-mat.supr-con]  31 Jan 2023
+Theoretical study of phonon-mediated superconductivity beyond Migdal-Eliashberg
+approximation and Coulomb pseudopotential
+Jie Huang,1 Zhao-Kun Yang,1 Xiao-Yin Pan,2, ∗ and Guo-Zhu Liu1, †
+1Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
+2Department of Physics, Ningbo University, Ningbo, Zhejiang 315211, China
+In previous theoretical studies of phonon-mediated superconductors, the electron-phonon coupling
+is treated by solving the Migdal-Eliashberg equations under the bare vertex approximation, whereas
+the effect of Coulomb repulsion is incorporated by introducing one single pseudopotential parameter.
+These two approximations become unreliable in low carrier-density superconductors in which the
+vertex corrections are not small and the Coulomb interaction is poorly screened. Here, we shall go
+beyond these two approximations and employ the Dyson-Schwinger equation approach to handle
+the interplay of electron-phonon interaction and Coulomb interaction in a self-consistent way. We
+first derive the exact Dyson-Schwinger integral equation of the full electron propagator. Such an
+equation contains several unknown single-particle propagators and fermion-boson vertex functions,
+and thus seems to be intractable. To solve this difficulty, we further derive a number of identities
+satisfied by all the relevant propagators and vertex functions and then use these identities to show
+that the exact Dyson-Schwinger equation of electron propagator is actually self-closed. This self-
+closed equation takes into account not only all the vertex corrections, but also the mutual influence
+between electron-phonon interaction and Coulomb interaction. Solving it by using proper numerical
+methods leads to the superconducting temperature Tc and other quantities. As an application of
+the approach, we compute the Tc of the interfacial superconductivity realized in the one-unit-cell
+FeSe/SrTiO3 system. We find that Tc can be strongly influenced by the vertex corrections and the
+competition between phonon-mediated attraction and Coulomb repulsion.
+I.
+INTRODUCTION
+Superconductivity develops in metals as the result of
+Cooper pairing instability when the attraction between
+electrons mediated by the exchange of phonons (or other
+types of bosons) overcomes the static Coulomb repulsion,
+which is the basic picture of Bardeen-Cooper-Schrieffer
+(BCS) theory [1]. In principle, the precise values of the
+pairing gap ∆ and the transition temperature Tc should
+be determined by performing a careful theoretical study
+of the complicated interplay of electron-phonon interac-
+tion (EPI) and Coulomb interaction. This is difficult to
+achieve. Traditionally, these two interactions are treated
+by using different methods [1]. The EPI is handled within
+the Migdal-Eliashberg (ME) theory, and ∆ and Tc are
+computed by solving a set of integral equations satisfied
+by the electrons’ renormalization function and the pair-
+ing function. In contrast, the Coulomb interaction is not
+handled at such a quantitative level: its impact on Tc
+is approximately measured by one single pseudopoten-
+tial parameter. Over the last decades, the ME theory
+and the pseudopotential have been jointly applied [2–5]
+to evaluate Tc and other quantities in various phonon-
+mediated superconductors.
+That EPI and Coulomb interaction are handled quite
+differently can be understood by making a field-theoretic
+analysis. Let us first consider EPI. The EPI describes
+the mutual influence of the dynamics of electrons and
+∗Corresponding author: panxiaoyin@nbu.edu.cn
+†Corresponding author: gzliu@ustc.edu.cn
+phonons on each other, and hence appears to be very
+complicated. Within quantum many-body theory [1, 2],
+one needs to compute an infinite number of Feynman di-
+agrams to accurately compute any observable quantity,
+which is apparently impractical. Fortunately, treatment
+of EPI can be greatly simplified as the Migdal theorem
+[6] indicates that the EPI vertex corrections are strongly
+suppressed by the small factor λ (ωD/EF ), where λ is
+a dimensionless coupling parameter, ωD is Debye fre-
+quency, and EF is Fermi energy.
+For normal metals,
+λ (ωD/EF ) ≪ 1, thus EPI vertex corrections can be
+safely ignored.
+Under the bare vertex approximation,
+Eliashberg [7] derived a set of coupled equations, called
+ME equations, to study EPI-induced superconducting
+transition.
+We then discuss the treatment of Coulomb repulsion.
+After defining an auxiliary scalar field A to represent
+the static Coulomb potential, one can map the Coulomb
+interaction into an equivalent fermion-boson interaction
+that has a similar field-theoretic structure to EPI. How-
+ever, one cannot naively use the ME theory to handle this
+fermion-boson interaction since there is no Migdal-like
+theorem to guarantee the smallness of its vertex correc-
+tions. Without a well-controlled scheme in hands, one is
+forced to investigate the Coulomb interaction by making
+approximations. Tolmachev [8] and Morel and Ander-
+son [9] introduced the pseudopoential µ∗ to include the
+impact of Coulomb repulsion. For a three-dimensional
+metal, the bare Coulomb interaction is described by
+V0(q) = 4πe2
+q2 .
+(1)
+This bare function is renormalized to become energy-
+
+2
+momentum dependent, namely
+VR(ω, q) =
+1
+V −1
+0
+(q) − Π(ω, q) =
+4πe2
+q2 − 4πe2Π(ω, q). (2)
+The full polarization function Π(ω, q) is hard to compute.
+A widely used approximation is to calculate Π(ω, q) at
+the lowest one-loop level, corresponding to the random
+phase approximation (RPA). The one-loop polarization
+Π1L(ω, q) is still very complex. Nevertheless, it is easy to
+reveal that Π1L(ω, q) approaches a constant in the limits
+of ω = 0 and q → 0, i.e., Π1L(ω = 0, q → 0) ∝ −N0,
+where N0 is the normal-state density of states (DOS) on
+Fermi surface.
+For metals with a large Fermi surface,
+both EF and N0 are fairly large.
+Thus the Coulomb
+interaction becomes short-ranged and can be roughly de-
+scribed by [9]
+Vsim(q) ∝
+1
+q2 + κD
+,
+(3)
+where the static screening factor κD ∝ 4πe2N0. Morel
+and Anderson [9] suggested to perform an average of
+Vsim(q) on the Fermi surface, which yields a parameter
+µ ∝ ⟨Vsim(q)⟩FS.
+The pseudopoential µ∗ is related to µ via the relation [9]
+µ∗ =
+µ
+1 + µ ln(EF /ωD).
+(4)
+Obviously, µ∗ ≪ µ in normal metals where ωD ≪ EF ,
+rendering the robustness of superconductivity against
+Coulomb repulsion.
+As the above analysis demonstrates, the ME theory of
+EPI and the Coulomb pseudopotential should be reliable
+if the condition ωD ≪ EF is fulfilled. This condition is
+violated in phonon-mediated superconductors that have
+a low carrier density, with dilute SrTiO3 being a famous
+example [10, 11]. In such superconductors, EF and N0
+are both small. There are no small factors to suppress
+the EPI vertex corrections, indicating the breakdown of
+Migdal theorem. Moreover, the Coulomb interaction is
+poorly screened due to the smallness of N0. The time-
+dependence and spatial variation of Coulomb potential
+cannot be well described by the oversimplified function
+Vsim(q) shown in Eq. (3). Accordingly, the pseugopoten-
+tial defined in Eq. (4) may no longer be valid as it comes
+directly from Eq. (3). It is more appropriate to adopt
+an energy-momentum dependent VR(ω, q) to replace the
+static Vsim(q). Another potentially important contribu-
+tion arises from the mutual influence between EPI and
+Coulomb interaction. This contribution was ignored in
+the original work of Morel and Anderson [9] and also in
+most, if not all, the subsequent studies on the Coulomb
+repulsion [12–15]. The validity of this approximation is
+not clear. In principle, we expect that EPI can affect
+Coulomb interaction and vice versa, because both EPI
+and Coulomb interaction result in a re-distribution of all
+the charges of the system. We should not simply discard
+their interplay if we cannot prove that such an interplay
+is negligible. In light of the above analysis, we consider
+it necessary to establish a more powerful approach to su-
+persede the ME theory of EPI and the pseudopotential
+treatment of Coulomb repulsion. To achieve this goal, we
+should take up the challenge of including all the higher
+order corrections.
+Recently, a non-perturbative Dyson-Schwinger (DS)
+equation approach [16] was developed to determine the
+EPI vertex corrections. At the core of this approach is
+the decoupling of the DS equation of the full electron
+propagator G(p) from all the rest DS equations with the
+help of several exact identities. It is found [16] that the
+DS equation of G(p) obtained by using this approach is
+self-closed and can be solved numerically. This approach
+was later extended [17] to deal with one single fermion-
+boson interaction, be it EPI or Coulomb interaction, in
+the context of Dirac fermion systems. More recently, the
+approach was further generalized [18] in order to treat the
+coupling of Dirac fermions to two different bosons. Ac-
+cording to the results of Refs. [17, 18], the DS equation
+of full Dirac fermion propagator is self-closed irrespective
+of whether the fermions are subjected to either EPI or
+Coulomb interaction, or both.
+In this paper, we shall combine the approaches of
+[16] and [18] to examine how the interplay of EPI and
+Coulomb interaction affects the transition temperature
+Tc of phonon-mediated superconductors.
+Our analysis
+will be based on an effective model that describes the
+couplings of the electron field ψ to a phonon field φ and
+an auxiliary boson A. The EPI is described by the ψ-φ
+coupling and the Coulomb interaction is described by the
+ψ-A coupling. Although there is not any direct coupling
+between φ and A, these two bosons can affect each other
+since they are both coupled to the same electrons. As a
+consequence, the DS equation of G(p) becomes formally
+very complicated. To solve this difficulty, we derive four
+exact identities after carrying out a series of analytical
+calculations and then use such identities to show that
+the exact DS equation of G(p) is still self-closed. The
+higher order corrections neglected in the ME theory and
+the pseudopotential method are properly taken into ac-
+count in this self-closed DS equation. Solving such an
+equation leads us to Tc and other quantities.
+We shall apply the approach to a concrete example -
+the interfacial superconductivity of one-unit-cell (1UC)
+FeSe/SrTiO3 system. After computing Tc by solving the
+self-closed DS equation of G(p), we show that the value
+of Tc depends strongly on the chosen approximations.
+In particular, Tc obtained under the bare vertex (ME)
+approximation is substantially modified when the vertex
+corrections are included.
+The rest of the paper is organized as follows. In Sec. II,
+we define the effective field theory for phonon-mediated
+superconductors. In Sec. III, we obtain the DS equation
+of G(p) and prove its self-closure with the help of four
+exact identities. In Sec. IV, we present the numerical re-
+
+3
+sults of Tc obtained by solving the self-consistent integral
+equations of two renormalization functions and the pair-
+ing function. In Sec. V, we summarize the results and
+discuss the limitations of our calculations.
+II.
+MODEL
+Our generic method is applicable to systems defined
+in any spatial dimension. However, for concreteness, we
+consider a model defined at two spatial dimensions, since
+later we shall apply the approach to 1UC FeSe/SrTiO3.
+The interplay of EPI and Coulomb interaction is de-
+scribed by the following effective Lagrangian density
+L = Lf + Lp + LA + Lfp + LfA,
+(5)
+Lf = ψ†(x) (i∂0σ0 − ξ∇σ3) ψ(x),
+(6)
+Lp = 1
+2φ†(x)D(x)φ(x),
+(7)
+LA = 1
+2A(x)F(x)A(x),
+(8)
+Lfp = −gφ(x)ψ†(x)σ3ψ(x),
+(9)
+LfA = −A(x)ψ†(x)σ3ψ(x).
+(10)
+The electrons are represented by the Nambu spinor [19]
+ψ(p) =
+�
+cp↑, c†
+−p↓
+�T
+along with four 2 × 2 matrices, in-
+cluding unit matrix σ0 and three Pauli matrices σ1,2,3.
+Although the system is non-relativistic, we choose to use
+a three-dimensional vector x ≡ (x0, x) = (x0, x1, x2) for
+the purpose of simplifying notations. The time x0 can
+be either real or imaginary (in Matsubara formalism),
+and the results hold in both cases.
+The fermion field
+ψ(x) is obtained by making Fourier transformation to
+ψ(p). For simplicity, here we assume that the kinetic en-
+ergy operator is ξ∇ = − 1
+2m
+�
+∂2
+1 + ∂2
+2
+�
+− µF , where µF is
+chemical potential.
+As demonstrated in Ref. [16], our
+generic approach remains valid if ξ∇ takes a different
+form. Phonons are represented by the scalar field φ(x),
+whose equation of free motion is expressed via the oper-
+ator D(x) as
+D(x)φ(x) = 0.
+(11)
+The EPI strength parameter g appearing in LfA is not
+necessarily a constant and could be a function of phonon
+momentum. A(x) is an auxiliary scalar field. Its equation
+of free motion is given by
+F(x)A(x) = 0.
+(12)
+The Coulomb interaction is effectively described by the
+coupling between ψ(x) and A(x) shown in LfA. Indeed,
+LA + LfA can be derived by performing a Hubbard-
+Stratonovich transformation to the following Hamilto-
+nian term for quartic Coulomb interaction
+e2
+4π
+�
+d2xd2x′ψ†(x)σ3ψσ(x)
+1
+|x − x′|ψ†(x′)σ3ψ(x′).(13)
+Notice that the model does not contain self-coupling
+terms of bosons.
+The Coulomb interaction originates
+from the Abelian U(1) gauge principle and the boson
+field A(x) can be regarded as the time component of U(1)
+gauge field (i.e., scalar potential). It is well established
+that an Abelian gauge boson does not interact with it-
+self. The situation is different for phonons. In principle
+phonons could interact with themselves.
+Ignoring the
+phonon self-couplings is justified only when the lattice
+vibration is well captured by the harmonic oscillating
+approximation.
+When the non-harmonic contributions
+are not negligible, the self-couplings of phonons need to
+be explicitly incorporated. Such non-harmonic contribu-
+tions might lead to a considerable influence on the value
+of Tc, as shown in a recent work [20]. In this paper, we
+shall not consider the non-harmonic contributions and
+therefore omit self-coupling terms of φ.
+Another notable feature of the model is that the two
+scalar fields φ(x) and A(x) do not directly interact with
+each other. Hence there are no such terms as φ(x)A(x)
+or φ2(x)A2(x) in the Lagrangian density. However, the
+mutual influence between φ(x) and A(x) cannot be sim-
+ply ignored since both of them are coupled to the same
+electrons. It will be shown later that the DS equation of
+electron propagator has a very complicated form due to
+the mutual influence between φ(x) and A(x). Moreover,
+φ(x) and A(x) are coupled to the same fermion density
+operator ψ†(x)σ3ψ(x). This implies that the vertex func-
+tion of ψ-φ coupling has a very similar structure to that
+of ψ-A coupling. The different behaviors of φ boson and
+A boson is primarily caused by the difference in the ex-
+pressions of the operators D(x) and F(x), or equivalently,
+the difference in the free propagators of φ boson and A
+boson.
+The Lagrangian density L respects the following two
+global U(1) symmetries [16, 19]
+ψ → eiασ3ψ,
+(14)
+ψ → eiασ0ψ.
+(15)
+Here, α is an infinitesimal constant. The first symmetry
+leads to charge conservation associated with a conserved
+current jc
+µ(x) ≡ (jc
+t (x), jc(x)), where
+jc
+t (x) = ψ†(x)σ3ψ(x),
+(16)
+jc(x) =
+i
+2m
+� �
+∇ψ†(x)
+�
+σ0ψ(x) − ψ†(x)σ0 (∇ψ(x))
+�
+.
+(17)
+The second symmetry leads to spin conservation asso-
+ciated with a conserved current js
+µ(x) ≡ (js
+t (x), js(x)),
+where
+js
+t (x) = ψ†(x)σ0ψ(x),
+(18)
+js(x) =
+i
+2m
+��
+∇ψ†(x)
+�
+σ3ψ(x) − ψ†(x)σ3 (∇ψ(x))
+�
+.
+(19)
+These two currents are conserved in the absence of exter-
+nal sources, i.e., ∂µjc,s
+µ (x) = 0. Each conserved current
+
+4
+generate one specific Ward-Takahashi identity (WTI).
+The detailed derivation of such WTIs can be found in
+Ref. [16].
+III.
+DYSON-SCHWINGER EQUATION OF
+ELECTRON PROPAGATOR
+After defining the effective model, we now are ready to
+perform a non-perturbative study of the superconducting
+transition. The following analysis will be largely based
+on the approaches previously developed in Ref. [16] and
+Ref. [18]. We shall not give all the derivational details
+and only outline the major steps.
+In order to examine the interaction-induced effects, we
+would like to investigate the properties of various n-point
+correlation functions. Such correlation functions can be
+generated from three generating functionals [21]. Adding
+four external sources J, K, η, and η† to the original La-
+grangian density leads to
+LT = L + Jφ + KA + ψ†η + η†ψ.
+(20)
+The partition function is defined via LT as follows
+Z[J, K, η†, η] ≡
+�
+DφDADψ†Dψei
+�
+dxLT .
+(21)
+Here,
+�
+dx ≡
+�
+d3x =
+�
+dtd2x. Z is the generating func-
+tional for all n-point correlation functions.
+To obtain
+connected correlation functions, we need another gener-
+ating functional:
+W ≡ W[J, K, η†, η] = −i ln Z[J, K, η†, η].
+(22)
+W can be used to generate three two-point correlation
+functions
+G(x − y) = −i⟨ψ(x)ψ†(y)⟩ =
+δ2W
+δη†(x)δη(y)
+���
+J=0, (23)
+D(x − y) = −i⟨φ(x)φ†(y)⟩ = −
+δ2W
+δJ(x)δJ(y)
+���
+J=0, (24)
+F(x − y) = −i⟨A(x)A(y)⟩ = −
+δ2W
+δK(x)δK(y)
+���
+J=0. (25)
+Here, G(x−y), D(x−y), and F(x−y) are the full propa-
+gators of electron ψ, phonon φ, and boson A, respectively.
+The system is supposed to have translation invariance,
+so the propagators depend solely on the difference x − y.
+Throughout this section, we use the abbreviated nota-
+tion J = 0 to indicate that all the external sources are
+removed. The mutual influence between the properties
+of two bosons are embodied in two additional two-point
+correlation functions
+DF (x − y) = −i⟨φ(x)A(y)⟩ = −
+δ2W
+δJ(x)δK(y)
+���
+J=0, (26)
+FD(x − y) = −i⟨A(x)φ(y)⟩ = −
+δ2W
+δK(x)δJ(y)
+���
+J=0. (27)
+As aforementioned, the model does not have such a term
+as φA, thus DF = FD = 0 at the classic tree-level. But
+the quantum (loop-level) corrections can induce non-zero
+contributions to DF and FD.
+The interaction vertex function for a fermion-boson
+coupling can also be generated from W. In the case of
+EPI, we consider the following connected three-point cor-
+relation function:
+⟨φ(x)ψ(y)ψ†(z)⟩c
+=
+δ3W
+δJ(x)δη†(y)δη(z)
+���
+J=0
+= −
+�
+dx′dy′dz′D(x − x′)G(y − y′)
+×
+δ3Ξ
+δφ(x′)δψ†(y′)δψ(z′)
+���
+J=0G(z′ − z)
+−
+�
+dx′dy′dz′DF (x − x′)G(y − y′)
+×
+δ3Ξ
+δA(x′)δψ†(y′)δψ(z′)
+���
+J=0G(z′ − z),
+(28)
+where the generating functional for proper (irreducible)
+vertices Ξ is defined via W as
+Ξ = W −
+�
+dx
+�
+J⟨φ⟩ + K⟨A⟩ + η†⟨ψ⟩ + ⟨ψ†⟩η
+�
+. (29)
+The interaction vertex function for EPI is defined as
+Γp(y − x, x − z) =
+δ3Ξ
+δφ(x)δψ†(y)δψ(z)
+���
+J=0,
+(30)
+and that for ψ-A coupling is defined as
+ΓA(y − x, x − z) =
+δ3Ξ
+δA(x)δψ†(y)δψ(z)
+���
+J=0.
+(31)
+It is necessary to emphasize that Γp and ΓA depend on
+two (not three) free variables, namely y−x and x−z. The
+propagators and interaction vertex functions appearing
+in Eq. (28) are Fourier transformed as follows:
+G(p) =
+�
+dxeip·xG(x),
+(32)
+D(q) =
+�
+dxeiq·xD(x),
+(33)
+DF (q) =
+�
+dxeiq·xDF (x),
+(34)
+Γp,A(q, p) =
+�
+dxdyei(p+q)·(y−x)eip·(x−z)
+×Γp,A(y − x, x − z),
+(35)
+Here, the three-momentum is p ≡ (p0, p) = (p0, p1, p2).
+Performing Fourier transformation to ⟨φ(x)ψ(y)ψ†(z)⟩c,
+we find �
+dxdyei(p+q)·(y−x)eip·(x−z)⟨φ(x)ψ(y)ψ†(z)⟩c
+= −D(q)G(p + q)Γp(q, p)G(p)
+−DF(q)G(p + q)ΓA(q, p)G(p).
+(36)
+
+5
+The φ-A coupling can be investigated using the same
+procedure. In this case, we need to study another three-
+point correlation function ⟨A(x)ψ(y)ψ†(z)⟩c. Following
+the calculational steps that lead Eq. (28) to Eq. (36), we
+obtain
+�
+dxdyei(p+q)·(y−x)eip·(x−z)⟨A(x)ψ(y)ψ†(z)⟩c
+= −F(q)G(p + q)ΓA(q, p)G(p)
+−FD(q)G(p + q)Γp(q, p)G(p),
+(37)
+where F(q) and FD(q) are transformed from F(x) and
+FD(x) respectively as
+F(q) =
+�
+dxeiq·xF(x),
+(38)
+FD(q) =
+�
+dxeiq·xFD(x).
+(39)
+Making use of derivational procedure presented in
+Ref. [16] and Ref. [18], we find that the full electron prop-
+agator G(p) satisfies the following DS equation
+G−1(p) = G−1
+0 (p) − i
+�
+d3q
+(2π)3 gσ3G(p + q)D(q)Γp(q, p)
+−i
+�
+d3q
+(2π)3 σ3G(p + q)F(q)ΓA(q, p)
+−i
+�
+d3q
+(2π)3 gσ3G(p + q)DF (q)ΓA(q, p)
+−i
+�
+d3q
+(2π)3 σ3G(p + q)FD(q)Γp(q, p).
+(40)
+The electron self-energy Σ(p) = G−1(p)−G−1
+0 (p) consists
+of four terms, as shown in the right-hand side (r.h.s.)
+of Eq. (40).
+The first two terms stem from pure EPI
+and pure Coulomb interaction, respectively. The last two
+terms arise from the mutual influence between these two
+interactions. The contributions of such mixing terms to
+the self-energy were entirely ignored in the original pseu-
+dopotential treatment of Morel and Anderson [9].
+To
+the best of our knowledge, such mixing terms have never
+been seriously incorporated in previous pseudopotential
+studies [12–15]. While ignoring them might be valid in
+some normal metal superconductors, this approximation
+is not necessarily justified in all cases. It would be better
+to keep them in calculations.
+Unfortunately, retaining all the contributions to the
+self-energy makes the DS equation of G(p) extremely
+complex. It appears that the equation (40) is not even
+self-closed since D(q), F(q), DF (q), FD(q), Γp(q, p), and
+ΓA(q, p) are unknown. Technically, one can derive the
+DS equations fulfilled by these six unknown functions
+by using the generic rules of quantum field theory [16–
+18, 21]. Nevertheless, such equations are coupled to the
+formally more complicated DS equations of innumerable
+multi-point correlation functions and hence of little use.
+Probably, one would have to solve an infinite number
+of coupled DS equations to completely determine G(p),
+which is apparently not a feasible scheme.
+In order to simplify Eq. (40) and make it tractable,
+it might be necessary to introduce some approximations
+by hands. For instance, one could: (1) neglect the last
+two (mixing) terms of the r.h.s.; (2) discard all the vertex
+corrections by assuming that Γp,A(q, p) → σ3; (3) replace
+the full phonon propagator D(q) with the bare one, i.e.,
+D(q) → D0(q); (4) replace the full A-boson propaga-
+tor F(q) with a significantly simplified expression, such
+as Fsim(q) =
+1
+q2+κD , or even with a phenomenological
+parameter (e.g., the pseudopotential µ∗). Under all of
+the above approximations, one finds that the original DS
+equation (40) becomes
+G−1(p) = G−1
+0 (p) − i
+�
+d3q
+(2π)3 gσ3G(p + q)D0(q)σ3
+−iµ∗
+�
+d3q
+(2π)3 σ3G(p + q)σ3,
+(41)
+which is self-closed and can be solved numerically. The
+free electron propagator has the form
+G0(p) =
+1
+iǫnσ0 − ξpσ3 + ∆pσ1
+,
+(42)
+and the full electron propagator is expanded as
+G(p) =
+1
+A1(ǫn, p)iǫnσ0 − A2(ǫn, p)ξpσ3 + ∆(ǫn, p)σ1
+,
+(43)
+where A1(ǫn, p) and A2(ǫn, p) are two renormalization
+functions and ∆(ǫn, p) is pairing function. Inserting G(p)
+and G0(p) into Eq. (41), one would obtain the standard
+ME equations of A1(ǫn, p), A2(ǫn, p), and ∆(ǫn, p) with
+the parameter µ∗ characterizing the impact of Coulomb
+repulsion. In the past sixty years, such simplified equa-
+tions have been extensively applied [1–5] to study a large
+number of phonon-mediated superconductors. However,
+the four approximations that lead to Eq. (41) are not
+justified under certain conditions. Some, or perhaps all,
+of them break down in superconductors having a small
+Fermi energy.
+Now we seek to find a more powerful method to deal
+with the original exact DS equation of G(p) given by
+Eq. (40) by going beyond the above approximations. We
+believe that one should not try to determine each of
+the six functions D(q), F(q), DF (q), FD(q), Γp(q, p),
+and ΓA(q, p) functions separately, which can never be
+achieved.
+Alternatively, one should make an effort to
+determine such products as D(q)Γp(q, p), F(q)ΓA(q, p),
+DF (q)ΓA(q, p), and FD(q)Γp(q, p). This is the key idea of
+the approach proposed in Ref. [18], where we have proved
+the self-closure of the DS equation of the Dirac fermion
+propagator G(p) in a model describing the coupling of
+Dirac fermions to two distinct bosons in graphene. Below
+we show that this same approach can be adopted to prove
+the self-closure of the DS equation given by Eq. (40). We
+
+6
+shall derive two exact identities satisfied by D(q), F(q),
+DF (q), FD(q), Γp(q, p), and ΓA(q, p).
+The derivation of the needed exact identities is based
+on the invariance of partition function Z under arbitrary
+infinitesimal changes of φ and A. The invariance of Z
+under an arbitrary infinitesimal change of φ gives rise to
+⟨D(x)φ(x) − gψ†(x)σ3ψ(x) + J(x)⟩ = 0.
+(44)
+Using ⟨φ(x)⟩ = δW/δJ(x), we perform functional deriva-
+tives with respect to η(z) and η†(y) in order and obtain
+the following relation
+D(x)⟨φ(x)ψ(y)ψ†(z)⟩c = g⟨ψ†(x)σ3ψ(x)ψ(y)ψ†(z)⟩c.
+(45)
+Making a Fourier transformation to the left-hand side
+(l.h.s.) of Eq. (45) yields
+D−1
+0 (q)
+�
+− D(q)G(p + q)Γp(q, p)G(p) − DF (q)G(p + q)ΓA(q, p)G(p)
+�
+,
+(46)
+where the free phonon propagator D0(q) comes from D(x). To handle the r.h.s. of Eq. (45), we use two bilinear
+operators jc
+t (x) = ψ†(x)σ3ψ(x) and js
+t (x) = ψ†(x)σ0ψ(x) to define two current vertex functions Γ0,3(x − z, z − y):
+⟨ψ†(x)σ0,3ψ(x)ψ(y)ψ†(z)⟩c = −
+�
+dξdξ′G(y − ξ)Γ0,3(ξ − x, x − ξ′)G(ξ′ − z).
+(47)
+Γ0,3(x − z, z − y) should be Fourier transformed [16, 22] as
+Γ0,3(ξ − x, x − ξ′) =
+�
+dqdpe−i(p+q)·(ξ−x)−ip·(x−ξ′)Γ0,3(q, p).
+(48)
+For more properties of such current vertex functions, see Refs. [16, 22]. Then the r.h.s. of Eq. (45) is turned into
+�
+dxdyei(p+q)·(y−x)eip·(x−z)g⟨ψ†(x)σ0,3ψ(x)ψ(y)ψ†(z)⟩c → −gG(p + q)Γ0,3(q, p)G(p).
+(49)
+After substituting Eq. (46) and Eq. (49) into Eq. (45), we obtain the following identity
+D(q)Γp(q, p) + DF (q)ΓA(q, p) = D0(q)gΓ3(q, p).
+(50)
+Similarly, the invariance of Z under an infinitesimal change of A field requires the following equation to hold
+⟨F(x)A(x) − gψ†(x)σ3ψ(x) + K(x)⟩ = 0.
+(51)
+Carrying out similar analytical calculations generates another identity
+FD(q)Γp(q, p) + F(q)ΓA(q, p) = F0(q)Γ3(q, p),
+(52)
+where the free propagator of A boson F0(q) is computed by performing Fourier transformation to F(x).
+Making use of the two identities given by Eq. (50) and Eq. (52), we re-write the original DS equation (40) as
+G−1(p) = G−1
+0 (p) − i
+�
+d3q
+(2π)3
+�
+g2D0(q) + F0(q)
+�
+σ3G(p + q)Γ3(q, p).
+(53)
+This equation is still not self-closed if the current vertex function Γ3(q, p) relies on unknown functions other than
+G(p). As shown in Ref. [16], the symmetry of Eq. (14) yields the following WTI
+q0Γ0(q, p) − (ξp+q − ξp) Γ3(q, p) = σ0G−1(p + q) − G−1(p)σ0.
+(54)
+However, it is not possible to entirely determine Γ3(q, p) purely based on this single WTI, since Γ0(q, p) is also
+unknown. Fortunately, the symmetry of Eq. (15) generates another WTI [16]
+q0Γ3(q, p) − (ξp+q − ξp) Γ0(q, p) = σ3G−1(p + q) − G−1(p)σ3.
+(55)
+These two WTIs are coupled to each other and can be used to express Γ3(q, p) and Γ0(q, p) purely in terms of G(p).
+Now Γ3(q, p) can be readily obtained by solving these two WTIs:
+Γ3(q, p) = ω
+�
+G−1(p + q)σ3 − σ3G−1(p)
+�
++ (ξp+q − ξp)
+�
+G−1(p + q)σ0 − σ0G−1(p)
+�
+ω2 − (ξp+q − ξp)2
+.
+(56)
+We can see that the DS equation (53) becomes entirely
+self-closed because it contains merely one unknown func-
+
+7
+tion G(p). This equation can be numerically solved to
+determine G(p), provided that the three free propagators
+G0(p), D0(q), and F0(q) and the EPI coupling parameter
+g are known.
+It is useful to make some remarks here:
+(1) In many existing publications, it is naively deemed
+that a symmetry-induced WTI imposes an exact rela-
+tion between fermion propagator G(p) and interaction
+vertex function. To understand why this is a misconcep-
+tion, let us take EPI as an example.
+The EPI vertex
+function Γp(q, p) is defined via a three-point correlation
+function ⟨φψψ†⟩, which in itself is not necessarily related
+to any conserved current. There is no reason to expect
+Γp(q, p) to naturally enter into any WTI. To reveal the
+impact of some symmetry, one should use the symmetry-
+induced conserved current, say jc
+µ, to define a special cor-
+relation function ⟨jc
+µψψ†⟩, which, according to Eq. (47)
+and Eq. (49), is expressed in terms of current vertex func-
+tions Γ0(q, p) and Γ3(q, p). After imposing the constraint
+of current conservation ∂µjc
+µ = 0 to ⟨jc
+µψψ†⟩, one would
+obtain a WTI satisfied by Γ0(q, p), Γ3(q, p), and G(p), as
+shown in Eq. (54).
+(2) Our ultimate goal is to determine G(p).
+Its DS
+equation (40) contains two interaction vertex functions
+Γp(q, p) and ΓA(q, p). On the other hand, it is Γ0(q, p)
+and Γ3(q, p), rather than Γp(q, p) and ΓA(q, p), that enter
+into the WTIs given by Eq. (54) and Eq. (55). Hence,
+at least superficially the DS equation of G(p) and the
+WTIs are not evidently correlated. To find out a natural
+way to combine the DS equation of G(p) with WTIs, one
+needs to obtain the relations between interaction vertex
+functions and current vertex functions. Such relations do
+exist and are shown in Eq. (50) and Eq. (52).
+(3) The appearance of two free boson propagators
+D0(q) and F0(q) in the final DS equation (53) is not
+an approximation. It should be emphasized that the re-
+placement of the full boson propagators D(q) and F(q)
+appearing in the original DS equation (40) with their free
+ones is implemented based on two exact identities given
+by Eq. (50) and Eq. (52). The interaction-induced effects
+embodied in such functions as D(q), F(q), DF (q), FD(q),
+Γp(q, p), and ΓA(q, p) are already incorporated in current
+vertex function Γ3(q, p).
+IV.
+Tc OF 1UC FESE/SRTIO3 SYSTEM
+In this section, we apply the self-closed DS equation
+of G(p) given by Eq. (53) along with Eq. (56) to evalu-
+ate the pair-breaking temperature Tc of the supercon-
+ductivity realized in 1UC FeSe/SrTiO3 system.
+This
+material is found [23–25] to possess a surprisingly high
+Tc. While it is believed by many [24–31] that interfacial
+optical phonons (IOPs) from the SrTiO3 substrate are
+responsible for the observed high Tc, other microscopic
+pairing mechanisms cannot be conclusively excluded. For
+instance, Gor’kov [32] argued that IOPs alone are not ca-
+pable of producing such a high Tc. If this conclusion (not
+necessarily the argument itself) is correct, one would be
+forced to invoke at least one additional pairing mecha-
+nism, such as magnetic fluctuation or nematic fluctua-
+tion, to cooperate with IOPs [25, 32, 33]. Considerable
+research efforts have been devoted to calculating IOPs-
+induced Tc by using the standard ME theory [26–31] and
+its slightly refined version [34]. Nevertheless, thus far no
+consensus has been reached and the accurate value of Tc
+produced by IOPs alone is still controversial. To get a
+definite answer, it is important to compute Tc with a suf-
+ficiently high precision. This is certainly not an easy task
+since Tc could be influenced by many factors.
+Among all the factors that can potentially affect Tc,
+the EPI vertex corrections play a major role. Since the
+ratio ωD/EF is at the order of unity, the Migdal theorem
+becomes invalid. As shown in Ref. [16], including EPI
+vertex corrections can drastically change the value of Tc
+obtained under the bare vertex approximation. However,
+the calculations of Ref. [16] still need to be improved as
+they were based on two approximations that might lead
+to inaccuracies. The first approximation is the omission
+of the influence of Coulomb repulsion [16]. As discussed
+in Sec. I, the traditional pseudopotential method may not
+work well in 1UC FeSe/SrTiO3 owing to the smallness of
+EF . The impact of Coulomb interaction on Tc should
+be examined more carefully. The second approximation
+is that the electron momentum p was supposed [16] to
+be fixed at Fermi momentum pF such that ξp = 0. Un-
+der this second approximation, the DS equation of G(p)
+has only one integral variable (i.e., frequency). Then the
+computational time is significantly shortened. For this
+reason, this kind of approximation has widely been used
+in the existing calculations of Tc. But the pairing gap ∆
+and the renormalization factors A1 and A2 obtained by
+solving their single-variable equations depend solely on
+frequency. The momentum dependence is entirely lost.
+Since the EPI strength depends strongly on the trans-
+ferred momentum q, it is important not to neglect the
+momentum dependence of the DS equation of G(p). We
+shall discard the two approximations adopted in Ref. [16]
+and directly deal with the self-closed DS equation given
+by Eq. (53).
+We are particularly interested in how Tc is affected
+by the interplay between EPI and Coulomb repulsion.
+To avoid the difficulty brought by analytical continu-
+ation, we choose to work in the Matsubara formalism
+and express the electron momentum as p = (iǫn, p),
+where ǫn = (2n + 1)πT , and the boson momentum as
+q = (iωn′, q), where ωn′ = 2n′πT .
+The free phonon
+propagator has the form
+D0(q) =
+2Ωq
+(iωn′)2 − Ω2q
+.
+(57)
+The interfacial optical phonons are almost dispersionless
+[24, 28], thus Ωq can be well approximated by a constant.
+We choose to use the value of Ωq = 81meV [24, 28]. The
+Fermi energy is roughly µF = 65meV [25].
+The EPI
+strength parameter g depends on the phonon momentum
+
+8
+q as follows [24, 29]
+g ≡ g(q) = g0 exp(−|q|/q0).
+(58)
+The value of dimensionless parameter λ can be estimated
+by carrying out first-principle calculations [29]. Here we
+regard λ as a tuning parameter and choose a representa-
+tive value λ = 0.2. Parameter q0 measures the interaction
+range in the momentum space [29]. The free propagator
+of the A-boson is
+F0(q) =
+α′
+vF |q|,
+(59)
+where the effective fine structure constant is α′ = e2/ε
+with ε being an effective dielectric constant. The prime
+symbol in α′ is added to indicate that α′ differs from
+the standard fine structure constant α = 1/137 defined
+in QED4. The value of ε depends on the surroundings
+(substrate) of the superconducting plane. For the sake
+of generality, we suppose that ε can be freely changed.
+As the next step, we wish to substitute the free phonon
+propagator D0(q) given by Eq. (57), the free A-boson
+propagator F0(q) given by Eq. (59), the free electron
+propagator G0(p) given by Eq. (42), and the full electron
+propagator G(p) given by Eq. (43) into the DS equation
+(53) and also into the function Γ3(q, p) given by Eq. (56).
+However, we cannot naively do so since here we encounter
+a fundamental problem.
+Recall that Γ3(q, p) given by
+Eq. (56) is derived from two symmetry-induced WTIs.
+Once the pairing gap ∆(p) develops a finite value, the
+system enters into the superconducting state. The U(1)
+symmetry of Eq. (15) is preserved in both the normal and
+superconducting states, thus the WTI of Eq. (55) is not
+changed. In contrast, the U(1) symmetry of Eq. (14) is
+spontaneously broken in the superconducting state. Does
+this symmetry breaking change the WTI of Eq. (54)?
+If yes, one could insert the expression of G(p) given by
+Eq. (43) into Γ3(q, p). Otherwise, one should explore the
+modification of a WTI owing to symmetry breaking. At
+present, there seems no conclusive answer. Nambu [19]
+adopted the WTI from charge conservation to prove the
+gauge invariance of electromagnetic response functions of
+a superconductor based on a ladder-approximation of the
+DS equation of vertex function. Following the scheme of
+Nambu, Schrieffer [1] assumed (without giving a proof)
+that this WTI is the same in the superconducting and
+normal phases and used this assumption to show the
+existence of a gapless Goldstone mode. Nakanishi [35]
+later demonstrated that, while the WTI for a U(1) gauge
+theory has the same form in symmetric and symmetry-
+broken phases, the situation of other field theories may be
+different. Yanagisawa [36] revisited this issue in a recent
+work and argued that spontaneous breaking of a continu-
+ous symmetry gives rise to an additional term to WTI due
+to the generation of Goldstone boson(s). But the expres-
+sion of such an additional term was not specified. It also
+remains unclear whether the approach of [36] works for
+superconductors. The superconducting transition is pro-
+foundly different from other symmetry-breaking driven
+transitions. We learn from the Anderson mechanism [37]
+that the Goldstone boson generated via U(1) symmetry
+breaking is eliminated by the long-range Coulomb inter-
+action, which lifts the originally gapless mode to a gapped
+plasmon mode. Thus, the WTI associated with charge
+conservation is not expected to acquire the additional
+term derived in [36] upon entering the superconducting
+phase. Nevertheless, the absence of a Goldstone-boson-
+induced term does not mean that the WTI is entirely
+unchanged after implementing the Anderson mechanism.
+In order to obtain a complete theoretical description
+of superconducting transition, one needs to establish a
+unified framework to reconcile the non-perturbative DS
+equation approach with the Anderson mechanism.
+At
+present, such a framework is unavailable. In the absence
+of a satisfactory solution to the above problem, we have
+to introduce an approximation so as to proceed with our
+calculations. Our purpose is to compute Tc. Near Tc,
+the gap ∆ vanishes and the symmetry of Eq. (14) is
+still (approximately) respected, thus the WTI of Eq. (54)
+holds. Then we substitute the full electron propagator
+G(p) given by Eq. (43) into Eq. (53) and assume that
+Γ3(q, p) takes the following form
+Γ3(q, p) = ω
+�
+G−1
+s (p + q)σ3 − σ3G−1
+s (p)
+�
++ (ξp+q − ξp)
+�
+G−1
+s (p + q)σ0 − σ0G−1
+s (p)
+�
+ω2 − (ξp+q − ξp)2
+,
+(60)
+where
+Gs(p) =
+1
+A1(ǫn, p)iǫnσ0 − A2(ǫn, p)ξpσ3
+(61)
+is independent of the pairing gap ∆. This manipulation leads to three coupled nonlinear integral equations:
+A1(ǫn, p) = 1 + T
+iǫn
+�
+d2q
+(2π)2
+�
+g2(q)D0(ωm, q) + F0(ωm, q)
+�
+×A1(ǫn + ωm, p + q)i(ǫn + ωm)Γ33 + A2(ǫn + ωm, p + q)ξp+qΓ30 − ∆(ǫn + ωm, p + q)Γ3d
+A2
+1(ǫn + ωm, p + q)(ǫn + ωm)2 + A2
+2(ǫn + ωm, p + q)ξ2
+p+q + ∆2(ǫn + ωm, p + q)
+,
+(62)
+
+9
+A2(ǫn, p) = 1 − T
+ξp
+�
+d2q
+(2π)2
+�
+g2(q)D0(ωm, q) + F0(ωm, q)
+�
+×A1(ǫn + ωm, p + q)i(ǫn + ωm)Γ30 + A2(ǫn + ωm, p + q)ξp+qΓ33 − ∆(ǫn + ωm, p + q)Γ31
+A2
+1(ǫn + ωm, p + q)(ǫn + ωm)2 + A2
+2(ǫn + ωm, p + q)ξ2
+p+q + ∆2(ǫn + ωm, p + q)
+,
+(63)
+∆(ǫn, p) = −T
+�
+d2q
+(2π)2
+�
+g2(q)D0(ωm, q) + F0(ωm, q)
+�
+×A1(ǫn + ωm, p + q)i(ǫn + ωm)Γ3d − A2(ǫn + ωm, p + q)ξp+qΓ31 − ∆(ǫn + ωm, p + q)Γ33
+A2
+1(ǫn + ωm, p + q)(ǫn + ωm)2 + A2
+2(ǫn + ωm, p + q)ξ2
+p+q + ∆2(ǫn + ωm, p + q)
+.
+(64)
+Here, we have defined several quantities:
+Γ30 =
+1
+ω2m + (ξp+q − ξp)2
+�
+iωm[A2(ǫn + ωm, p + q)ξp+q − A2(ǫn, p)ξp]
++(ξp+q − ξp)[−A1(ǫn + ωm, p + q)(iǫn + iωm) + A1(ǫn, p)iǫn]
+�
+,
+(65)
+Γ33 =
+1
+ω2m + (ξp+q − ξp)2
+�
+iωm[−A1(ǫn + ωm, p + q)(iǫn + iωm) + A1(ǫn, p)iǫn]
++(ξp+q − ξp)[A2(ǫn + ωm, p + q)ξp+q − A2(ǫn, p)ξp]
+�
+,
+(66)
+Γ31 = (ξp+q − ξp) [−∆(ǫn + ωm, p + q) + ∆(ǫn, p)]
+ω2m + (ξp+q − ξp)2
+,
+(67)
+Γ3d = iωm [∆(ǫn + ωm, p + q) + ∆(ǫn, p)]
+ω2m + (ξp+q − ξp)2
+.
+(68)
+It is easy to reproduce the ME equations if the full
+propagator G(p) appearing in Eq. (56) is replaced with
+the free propagator G0(p), which leads to the bare vertex
+approximation Γ3 → σ3. An alternative manipulation is
+to substitute A1 = A2 = 1 and ∆ = 0 into Eq. (65-68).
+This yields
+Γ30 = 0,
+Γ33 = 1,
+Γ31 = 0,
+Γ3d = 0.
+(69)
+Then Eqs. (62-64) become the standard ME equations.
+The above integral equations can be numerically solved
+using the iteration method (see Ref. [16] for a detailed
+illustration of this method).
+The computational time
+required to reach convergent results depends crucially
+on the number of integral variable: adding one variable
+would exponentially increase the computational time.
+The coupled equations given by (62-64) have only one
+variable ǫn if all electrons are assumed to reside exactly
+on the Fermi surface. Such an assumption simplifies the
+equations and dramatically decreases the computational
+time, but might not be well justified in the present case
+due to the strong momentum dependence of EPI coupling
+strength. There, here we consider all the possible values
+of p and directly solve Eqs. (62-64) without introducing
+further approximations. But these equations have three
+integral variables, namely ωm, q1, and q2. Solving them
+would consume too many computational resources.
+The burden of numerical computation can be greatly
+lightened if the number of integral variable is reduced.
+We suppose that the system is isotropic and then make
+an effort to integrate over the angle θ between p and q
+before starting the iterative process. After doing so, only
+two free variables, namely ωm and |q|, are involved in the
+process of performing iterations. This manipulation can
+greatly decrease the computational time. Generically, it
+is not easy to integrate over θ.
+In our case, however,
+although the current vertex function Γ3(q, p) is compli-
+cated, the free propagators D0(q) and F0(q) are simple
+functions of their variables. The phonon energy Ωq is
+a constant, thus D0(q) depends solely on the frequency,
+i.e., D0(q) = D0(ωm).
+In comparison, F0(q) depends
+solely on the momentum. To illustrate why θ can be in-
+tegrate out, it is more convenient to deal with the DS
+equation shown in Eq. (53) instead of the formally com-
+plicated equations (62-64). After re-defining p + q → k
+and
+�
+dq →
+�
+dk, we re-write Eq. (53) as
+G−1(ǫn, p) = G−1
+0 (ǫn, p) +
+�
+m
+� kdkdθ
+(2π) σ3G(ǫm, k)
+×
+�
+g2D0(ωm) +
+e2
+ε
+�
+p2 + k2 − 2|p||k| cos θ
+�
+×Γ3(ǫn, p, ǫm, k).
+Both G0(ǫn, p) and G(ǫn, p) are independent of θ, thus
+θ is not involved in the iterative process and can be nu-
+merically integrated at each step.
+
+10
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+10-3
+35
+40
+45
+50
+55
+60
+65
+Tc[K]
+ME
+DS
+FIG. 1: Critical temperatures Tc obtained by solving the com-
+plete DS equation of G(p) and the simplified ME equation of
+G(p) at a fixed coupling parameter λ = 0.2. The results of Tc
+for other values of λ can be similarly evaluated. ME results
+and DS results are shown using red squares and green circles,
+respectively.
+Notice that F0(q) is singular at q = 0, reflecting the
+long-range nature of bare Coulomb interaction. Within
+the framework of jellium model for the electron liquid, the
+contribution of q = 0 must be eliminated since it cancels
+out the static potential between negative and positive
+charges.
+This can be implemented by introducing an
+infrared cutoff δ. In our calculations the infrared cutoff
+is taken to be 10−6pF . We have already confirmed that
+the final results of Tc are nearly unchanged as δ varies
+within the range of
+�
+10−8pF , 10−3pF
+�
+.
+The numerical results of Tc for λ = 0.2 are presented in
+Fig. 1, where the blue and black lines correspond to the
+ME-level and DS-level approximations, respectively. The
+results for other values of λ can be similarly obtained.
+The pairing gap is supposed to have an isotropic s-wave
+symmetry.
+We first consider the simplest case in which the
+Coulomb interaction is absent, corresponding to α′ = 0
+or equivalently ε = ∞. In a previous work [16], it was
+found that including the EPI vertex corrections tends
+to promote Tc evaluated at the ME-level (bare vertex).
+This conclusion was reached based on a special assump-
+tion that the electrons always strictly reside on the Fermi
+surface [16]. Here we have re-solved Eqs. (62-64) without
+making this assumption. The numerical results plotted
+in Fig. 1 clearly show that the inclusion of EPI vertex
+corrections leads to a considerable reduction of Tc. We
+can infer from these results that the EPI vertex correc-
+tions and the momentum dependence of A1,2 and ∆ are
+both important and should be carefully incorporated.
+The influence of Coulomb repulsion on Tc can be read-
+ily examined by varying the tuning parameter ε or α′.
+For a realistic material, ε takes a specific value, so does
+Tc. According to Fig. 1, Tc is suppressed drastically as
+ε decreases.
+Tc is completely determined once all the
+model parameters are fixed.
+V.
+SUMMARY AND DISCUSSION
+In summary, we have performed a non-perturbative
+study of the interplay of EPI and Coulomb repulsion by
+using the DS equation approach. We have shown that
+the DS equation of the full electron propagator G(p) is
+self-closed provided that all the higher order corrections
+to EPI and Coulomb interaction are incorporated via a
+number of exact identities. This self-closed DS equation
+can be applied to study the superconducting transition
+beyond the ME approximation of EPI and the pseudopo-
+tential approximation of Coulomb repulsion.
+We have
+employed this approach to evaluate the pair-braking tem-
+perature Tc for the interfacial superconductivity in 1UC
+FeSe/SrTiO3 system and found that the value of Tc could
+be significantly miscalculated if the vertex corrections
+and the momentum-dependence of relevant quantities are
+not taken into account in a reliable way.
+The calculations of this work ignored several effects
+that might change the value of Tc and thus need to be
+improved in the future. First of all, the one-band model
+studied by us should be replaced with a more realistic
+multi-band model that embodies the actual electronic
+structure [31]. The phonon self-coupling terms [20] were
+entirely neglected in our calculations.
+Including such
+self-coupling terms invalidates the two identities given
+by Eq. (50) and Eq. (52).
+As a consequence, the DS
+equation of the electron propagator G(p) can no longer
+be made self-closed (see Ref. [16] and Ref. [17] for more
+details).
+In addition, as analyzed in Sec. IV, the role
+played by the Anderson mechanism has not yet been un-
+ambiguously determined. For these reasons, our results
+of Tc cannot be directly compared to the experimental re-
+sults in 1UC FeSe/SrTiO3 system. Therefore, although
+the present work provides a methodological advance in
+the non-perturbative study of superconducting transition
+driven by the interplay of EPI and Coulomb repulsion,
+we are still some way from a satisfactory theoretical de-
+scription of superconductors (including but not restricted
+to 1UC FeSe/SrTiO3).
+ACKNOWLEDGEMENTS
+We thank Jing-Rong Wang and Hao-Fu Zhu for helpful
+discussions.
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+Rev. B 103, 094501 (2021).
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+
diff --git a/Z9FRT4oBgHgl3EQfQDdI/content/tmp_files/load_file.txt b/Z9FRT4oBgHgl3EQfQDdI/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..377327e656ebbe5b248b1195eb895142924f5179
--- /dev/null
+++ b/Z9FRT4oBgHgl3EQfQDdI/content/tmp_files/load_file.txt
@@ -0,0 +1,808 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf,len=807
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='13520v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='supr-con]  31 Jan 2023  Theoretical study of phonon-mediated superconductivity beyond Migdal-Eliashberg  approximation and Coulomb pseudopotential  Jie Huang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='1 Zhao-Kun Yang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='1 Xiao-Yin Pan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' ∗ and Guo-Zhu Liu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' †  1Department of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' China  2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Ningbo University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Ningbo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Zhejiang 315211,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' China  In previous theoretical studies of phonon-mediated superconductors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' the electron-phonon coupling  is treated by solving the Migdal-Eliashberg equations under the bare vertex approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' whereas  the effect of Coulomb repulsion is incorporated by introducing one single pseudopotential parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  These two approximations become unreliable in low carrier-density superconductors in which the  vertex corrections are not small and the Coulomb interaction is poorly screened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Here, we shall go  beyond these two approximations and employ the Dyson-Schwinger equation approach to handle  the interplay of electron-phonon interaction and Coulomb interaction in a self-consistent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' We  first derive the exact Dyson-Schwinger integral equation of the full electron propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Such an  equation contains several unknown single-particle propagators and fermion-boson vertex functions,  and thus seems to be intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' To solve this difficulty, we further derive a number of identities  satisfied by all the relevant propagators and vertex functions and then use these identities to show  that the exact Dyson-Schwinger equation of electron propagator is actually self-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' This self-  closed equation takes into account not only all the vertex corrections, but also the mutual influence  between electron-phonon interaction and Coulomb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Solving it by using proper numerical  methods leads to the superconducting temperature Tc and other quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' As an application of  the approach, we compute the Tc of the interfacial superconductivity realized in the one-unit-cell  FeSe/SrTiO3 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' We find that Tc can be strongly influenced by the vertex corrections and the  competition between phonon-mediated attraction and Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  INTRODUCTION  Superconductivity develops in metals as the result of  Cooper pairing instability when the attraction between  electrons mediated by the exchange of phonons (or other  types of bosons) overcomes the static Coulomb repulsion,  which is the basic picture of Bardeen-Cooper-Schrieffer  (BCS) theory [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In principle, the precise values of the  pairing gap ∆ and the transition temperature Tc should  be determined by performing a careful theoretical study  of the complicated interplay of electron-phonon interac-  tion (EPI) and Coulomb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' This is difficult to  achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Traditionally, these two interactions are treated  by using different methods [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The EPI is handled within  the Migdal-Eliashberg (ME) theory, and ∆ and Tc are  computed by solving a set of integral equations satisfied  by the electrons’ renormalization function and the pair-  ing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In contrast, the Coulomb interaction is not  handled at such a quantitative level: its impact on Tc  is approximately measured by one single pseudopoten-  tial parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Over the last decades, the ME theory  and the pseudopotential have been jointly applied [2–5]  to evaluate Tc and other quantities in various phonon-  mediated superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  That EPI and Coulomb interaction are handled quite  differently can be understood by making a field-theoretic  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Let us first consider EPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The EPI describes  the mutual influence of the dynamics of electrons and  ∗Corresponding author: panxiaoyin@nbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='cn  †Corresponding author: gzliu@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='cn  phonons on each other, and hence appears to be very  complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Within quantum many-body theory [1, 2],  one needs to compute an infinite number of Feynman di-  agrams to accurately compute any observable quantity,  which is apparently impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Fortunately, treatment  of EPI can be greatly simplified as the Migdal theorem  [6] indicates that the EPI vertex corrections are strongly  suppressed by the small factor λ (ωD/EF ), where λ is  a dimensionless coupling parameter, ωD is Debye fre-  quency, and EF is Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  For normal metals,  λ (ωD/EF ) ≪ 1, thus EPI vertex corrections can be  safely ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Under the bare vertex approximation,  Eliashberg [7] derived a set of coupled equations, called  ME equations, to study EPI-induced superconducting  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  We then discuss the treatment of Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  After defining an auxiliary scalar field A to represent  the static Coulomb potential, one can map the Coulomb  interaction into an equivalent fermion-boson interaction  that has a similar field-theoretic structure to EPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' How-  ever, one cannot naively use the ME theory to handle this  fermion-boson interaction since there is no Migdal-like  theorem to guarantee the smallness of its vertex correc-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Without a well-controlled scheme in hands, one is  forced to investigate the Coulomb interaction by making  approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Tolmachev [8] and Morel and Ander-  son [9] introduced the pseudopoential µ∗ to include the  impact of Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' For a three-dimensional  metal, the bare Coulomb interaction is described by  V0(q) = 4πe2  q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (1)  This bare function is renormalized to become energy-  2  momentum dependent, namely  VR(ω, q) =  1  V −1  0  (q) − Π(ω, q) =  4πe2  q2 − 4πe2Π(ω, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (2)  The full polarization function Π(ω, q) is hard to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  A widely used approximation is to calculate Π(ω, q) at  the lowest one-loop level, corresponding to the random  phase approximation (RPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The one-loop polarization  Π1L(ω, q) is still very complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Nevertheless, it is easy to  reveal that Π1L(ω, q) approaches a constant in the limits  of ω = 0 and q → 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=', Π1L(ω = 0, q → 0) ∝ −N0,  where N0 is the normal-state density of states (DOS) on  Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  For metals with a large Fermi surface,  both EF and N0 are fairly large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Thus the Coulomb  interaction becomes short-ranged and can be roughly de-  scribed by [9]  Vsim(q) ∝  1  q2 + κD  ,  (3)  where the static screening factor κD ∝ 4πe2N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Morel  and Anderson [9] suggested to perform an average of  Vsim(q) on the Fermi surface, which yields a parameter  µ ∝ ⟨Vsim(q)⟩FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The pseudopoential µ∗ is related to µ via the relation [9]  µ∗ =  µ  1 + µ ln(EF /ωD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (4)  Obviously, µ∗ ≪ µ in normal metals where ωD ≪ EF ,  rendering the robustness of superconductivity against  Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  As the above analysis demonstrates, the ME theory of  EPI and the Coulomb pseudopotential should be reliable  if the condition ωD ≪ EF is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' This condition is  violated in phonon-mediated superconductors that have  a low carrier density, with dilute SrTiO3 being a famous  example [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In such superconductors, EF and N0  are both small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' There are no small factors to suppress  the EPI vertex corrections, indicating the breakdown of  Migdal theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Moreover, the Coulomb interaction is  poorly screened due to the smallness of N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The time-  dependence and spatial variation of Coulomb potential  cannot be well described by the oversimplified function  Vsim(q) shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Accordingly, the pseugopoten-  tial defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (4) may no longer be valid as it comes  directly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' It is more appropriate to adopt  an energy-momentum dependent VR(ω, q) to replace the  static Vsim(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Another potentially important contribu-  tion arises from the mutual influence between EPI and  Coulomb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' This contribution was ignored in  the original work of Morel and Anderson [9] and also in  most, if not all, the subsequent studies on the Coulomb  repulsion [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The validity of this approximation is  not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In principle, we expect that EPI can affect  Coulomb interaction and vice versa, because both EPI  and Coulomb interaction result in a re-distribution of all  the charges of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' We should not simply discard  their interplay if we cannot prove that such an interplay  is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In light of the above analysis, we consider  it necessary to establish a more powerful approach to su-  persede the ME theory of EPI and the pseudopotential  treatment of Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' To achieve this goal, we  should take up the challenge of including all the higher  order corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Recently, a non-perturbative Dyson-Schwinger (DS)  equation approach [16] was developed to determine the  EPI vertex corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' At the core of this approach is  the decoupling of the DS equation of the full electron  propagator G(p) from all the rest DS equations with the  help of several exact identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' It is found [16] that the  DS equation of G(p) obtained by using this approach is  self-closed and can be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' This approach  was later extended [17] to deal with one single fermion-  boson interaction, be it EPI or Coulomb interaction, in  the context of Dirac fermion systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' More recently, the  approach was further generalized [18] in order to treat the  coupling of Dirac fermions to two different bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Ac-  cording to the results of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [17, 18], the DS equation  of full Dirac fermion propagator is self-closed irrespective  of whether the fermions are subjected to either EPI or  Coulomb interaction, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  In this paper, we shall combine the approaches of  [16] and [18] to examine how the interplay of EPI and  Coulomb interaction affects the transition temperature  Tc of phonon-mediated superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Our analysis  will be based on an effective model that describes the  couplings of the electron field ψ to a phonon field φ and  an auxiliary boson A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The EPI is described by the ψ-φ  coupling and the Coulomb interaction is described by the  ψ-A coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Although there is not any direct coupling  between φ and A, these two bosons can affect each other  since they are both coupled to the same electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' As a  consequence, the DS equation of G(p) becomes formally  very complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' To solve this difficulty, we derive four  exact identities after carrying out a series of analytical  calculations and then use such identities to show that  the exact DS equation of G(p) is still self-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The  higher order corrections neglected in the ME theory and  the pseudopotential method are properly taken into ac-  count in this self-closed DS equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Solving such an  equation leads us to Tc and other quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  We shall apply the approach to a concrete example -  the interfacial superconductivity of one-unit-cell (1UC)  FeSe/SrTiO3 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' After computing Tc by solving the  self-closed DS equation of G(p), we show that the value  of Tc depends strongly on the chosen approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  In particular, Tc obtained under the bare vertex (ME)  approximation is substantially modified when the vertex  corrections are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' II,  we define the effective field theory for phonon-mediated  superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' III, we obtain the DS equation  of G(p) and prove its self-closure with the help of four  exact identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' IV, we present the numerical re-  3  sults of Tc obtained by solving the self-consistent integral  equations of two renormalization functions and the pair-  ing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' V, we summarize the results and  discuss the limitations of our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  MODEL  Our generic method is applicable to systems defined  in any spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' However, for concreteness, we  consider a model defined at two spatial dimensions, since  later we shall apply the approach to 1UC FeSe/SrTiO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The interplay of EPI and Coulomb interaction is de-  scribed by the following effective Lagrangian density  L = Lf + Lp + LA + Lfp + LfA,  (5)  Lf = ψ†(x) (i∂0σ0 − ξ∇σ3) ψ(x),  (6)  Lp = 1  2φ†(x)D(x)φ(x),  (7)  LA = 1  2A(x)F(x)A(x),  (8)  Lfp = −gφ(x)ψ†(x)σ3ψ(x),  (9)  LfA = −A(x)ψ†(x)σ3ψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (10)  The electrons are represented by the Nambu spinor [19]  ψ(p) =  �  cp↑, c†  −p↓  �T  along with four 2 × 2 matrices, in-  cluding unit matrix σ0 and three Pauli matrices σ1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Although the system is non-relativistic, we choose to use  a three-dimensional vector x ≡ (x0, x) = (x0, x1, x2) for  the purpose of simplifying notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The time x0 can  be either real or imaginary (in Matsubara formalism),  and the results hold in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The fermion field  ψ(x) is obtained by making Fourier transformation to  ψ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' For simplicity, here we assume that the kinetic en-  ergy operator is ξ∇ = − 1  2m  �  ∂2  1 + ∂2  2  �  − µF , where µF is  chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  As demonstrated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [16], our  generic approach remains valid if ξ∇ takes a different  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Phonons are represented by the scalar field φ(x),  whose equation of free motion is expressed via the oper-  ator D(x) as  D(x)φ(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (11)  The EPI strength parameter g appearing in LfA is not  necessarily a constant and could be a function of phonon  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' A(x) is an auxiliary scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Its equation  of free motion is given by  F(x)A(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (12)  The Coulomb interaction is effectively described by the  coupling between ψ(x) and A(x) shown in LfA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Indeed,  LA + LfA can be derived by performing a Hubbard-  Stratonovich transformation to the following Hamilto-  nian term for quartic Coulomb interaction  e2  4π  �  d2xd2x′ψ†(x)σ3ψσ(x)  1  |x − x′|ψ†(x′)σ3ψ(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (13)  Notice that the model does not contain self-coupling  terms of bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The Coulomb interaction originates  from the Abelian U(1) gauge principle and the boson  field A(x) can be regarded as the time component of U(1)  gauge field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=', scalar potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' It is well established  that an Abelian gauge boson does not interact with it-  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The situation is different for phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In principle  phonons could interact with themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Ignoring the  phonon self-couplings is justified only when the lattice  vibration is well captured by the harmonic oscillating  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  When the non-harmonic contributions  are not negligible, the self-couplings of phonons need to  be explicitly incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Such non-harmonic contribu-  tions might lead to a considerable influence on the value  of Tc, as shown in a recent work [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In this paper, we  shall not consider the non-harmonic contributions and  therefore omit self-coupling terms of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Another notable feature of the model is that the two  scalar fields φ(x) and A(x) do not directly interact with  each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Hence there are no such terms as φ(x)A(x)  or φ2(x)A2(x) in the Lagrangian density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' However, the  mutual influence between φ(x) and A(x) cannot be sim-  ply ignored since both of them are coupled to the same  electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' It will be shown later that the DS equation of  electron propagator has a very complicated form due to  the mutual influence between φ(x) and A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Moreover,  φ(x) and A(x) are coupled to the same fermion density  operator ψ†(x)σ3ψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' This implies that the vertex func-  tion of ψ-φ coupling has a very similar structure to that  of ψ-A coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The different behaviors of φ boson and  A boson is primarily caused by the difference in the ex-  pressions of the operators D(x) and F(x), or equivalently,  the difference in the free propagators of φ boson and A  boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The Lagrangian density L respects the following two  global U(1) symmetries [16, 19]  ψ → eiασ3ψ,  (14)  ψ → eiασ0ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (15)  Here, α is an infinitesimal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The first symmetry  leads to charge conservation associated with a conserved  current jc  µ(x) ≡ (jc  t (x), jc(x)), where  jc  t (x) = ψ†(x)σ3ψ(x),  (16)  jc(x) =  i  2m  � �  ∇ψ†(x)  �  σ0ψ(x) − ψ†(x)σ0 (∇ψ(x))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (17)  The second symmetry leads to spin conservation asso-  ciated with a conserved current js  µ(x) ≡ (js  t (x), js(x)),  where  js  t (x) = ψ†(x)σ0ψ(x),  (18)  js(x) =  i  2m  ��  ∇ψ†(x)  �  σ3ψ(x) − ψ†(x)σ3 (∇ψ(x))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (19)  These two currents are conserved in the absence of exter-  nal sources, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=', ∂µjc,s  µ (x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Each conserved current  4  generate one specific Ward-Takahashi identity (WTI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The detailed derivation of such WTIs can be found in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  DYSON-SCHWINGER EQUATION OF  ELECTRON PROPAGATOR  After defining the effective model, we now are ready to  perform a non-perturbative study of the superconducting  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The following analysis will be largely based  on the approaches previously developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [16] and  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' We shall not give all the derivational details  and only outline the major steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  In order to examine the interaction-induced effects, we  would like to investigate the properties of various n-point  correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Such correlation functions can be  generated from three generating functionals [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Adding  four external sources J, K, η, and η† to the original La-  grangian density leads to  LT = L + Jφ + KA + ψ†η + η†ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (20)  The partition function is defined via LT as follows  Z[J, K, η†, η] ≡  �  DφDADψ†Dψei  �  dxLT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (21)  Here,  �  dx ≡  �  d3x =  �  dtd2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Z is the generating func-  tional for all n-point correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  To obtain  connected correlation functions, we need another gener-  ating functional:  W ≡ W[J, K, η†, η] = −i ln Z[J, K, η†, η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (22)  W can be used to generate three two-point correlation  functions  G(x − y) = −i⟨ψ(x)ψ†(y)⟩ =  δ2W  δη†(x)δη(y)  ���  J=0, (23)  D(x − y) = −i⟨φ(x)φ†(y)⟩ = −  δ2W  δJ(x)δJ(y)  ���  J=0, (24)  F(x − y) = −i⟨A(x)A(y)⟩ = −  δ2W  δK(x)δK(y)  ���  J=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (25)  Here, G(x−y), D(x−y), and F(x−y) are the full propa-  gators of electron ψ, phonon φ, and boson A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The system is supposed to have translation invariance,  so the propagators depend solely on the difference x − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Throughout this section, we use the abbreviated nota-  tion J = 0 to indicate that all the external sources are  removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The mutual influence between the properties  of two bosons are embodied in two additional two-point  correlation functions  DF (x − y) = −i⟨φ(x)A(y)⟩ = −  δ2W  δJ(x)δK(y)  ���  J=0, (26)  FD(x − y) = −i⟨A(x)φ(y)⟩ = −  δ2W  δK(x)δJ(y)  ���  J=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (27)  As aforementioned, the model does not have such a term  as φA, thus DF = FD = 0 at the classic tree-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' But  the quantum (loop-level) corrections can induce non-zero  contributions to DF and FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The interaction vertex function for a fermion-boson  coupling can also be generated from W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In the case of  EPI, we consider the following connected three-point cor-  relation function:  ⟨φ(x)ψ(y)ψ†(z)⟩c  =  δ3W  δJ(x)δη†(y)δη(z)  ���  J=0  = −  �  dx′dy′dz′D(x − x′)G(y − y′)  ×  δ3Ξ  δφ(x′)δψ†(y′)δψ(z′)  ���  J=0G(z′ − z)  −  �  dx′dy′dz′DF (x − x′)G(y − y′)  ×  δ3Ξ  δA(x′)δψ†(y′)δψ(z′)  ���  J=0G(z′ − z),  (28)  where the generating functional for proper (irreducible)  vertices Ξ is defined via W as  Ξ = W −  �  dx  �  J⟨φ⟩ + K⟨A⟩ + η†⟨ψ⟩ + ⟨ψ†⟩η  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (29)  The interaction vertex function for EPI is defined as  Γp(y − x, x − z) =  δ3Ξ  δφ(x)δψ†(y)δψ(z)  ���  J=0,  (30)  and that for ψ-A coupling is defined as  ΓA(y − x, x − z) =  δ3Ξ  δA(x)δψ†(y)δψ(z)  ���  J=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (31)  It is necessary to emphasize that Γp and ΓA depend on  two (not three) free variables, namely y−x and x−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The  propagators and interaction vertex functions appearing  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (28) are Fourier transformed as follows:  G(p) =  �  dxeip·xG(x),  (32)  D(q) =  �  dxeiq·xD(x),  (33)  DF (q) =  �  dxeiq·xDF (x),  (34)  Γp,A(q, p) =  �  dxdyei(p+q)·(y−x)eip·(x−z)  ×Γp,A(y − x, x − z),  (35)  Here, the three-momentum is p ≡ (p0, p) = (p0, p1, p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Performing Fourier transformation to ⟨φ(x)ψ(y)ψ†(z)⟩c,  we find �  dxdyei(p+q)·(y−x)eip·(x−z)⟨φ(x)ψ(y)ψ†(z)⟩c  = −D(q)G(p + q)Γp(q, p)G(p)  −DF(q)G(p + q)ΓA(q, p)G(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (36)  5  The φ-A coupling can be investigated using the same  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In this case, we need to study another three-  point correlation function ⟨A(x)ψ(y)ψ†(z)⟩c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Following  the calculational steps that lead Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (28) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (36), we  obtain  �  dxdyei(p+q)·(y−x)eip·(x−z)⟨A(x)ψ(y)ψ†(z)⟩c  = −F(q)G(p + q)ΓA(q, p)G(p)  −FD(q)G(p + q)Γp(q, p)G(p),  (37)  where F(q) and FD(q) are transformed from F(x) and  FD(x) respectively as  F(q) =  �  dxeiq·xF(x),  (38)  FD(q) =  �  dxeiq·xFD(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (39)  Making use of derivational procedure presented in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [16] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [18], we find that the full electron prop-  agator G(p) satisfies the following DS equation  G−1(p) = G−1  0 (p) − i  �  d3q  (2π)3 gσ3G(p + q)D(q)Γp(q, p)  −i  �  d3q  (2π)3 σ3G(p + q)F(q)ΓA(q, p)  −i  �  d3q  (2π)3 gσ3G(p + q)DF (q)ΓA(q, p)  −i  �  d3q  (2π)3 σ3G(p + q)FD(q)Γp(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (40)  The electron self-energy Σ(p) = G−1(p)−G−1  0 (p) consists  of four terms, as shown in the right-hand side (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=')  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The first two terms stem from pure EPI  and pure Coulomb interaction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The last two  terms arise from the mutual influence between these two  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The contributions of such mixing terms to  the self-energy were entirely ignored in the original pseu-  dopotential treatment of Morel and Anderson [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  To  the best of our knowledge, such mixing terms have never  been seriously incorporated in previous pseudopotential  studies [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' While ignoring them might be valid in  some normal metal superconductors, this approximation  is not necessarily justified in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' It would be better  to keep them in calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Unfortunately, retaining all the contributions to the  self-energy makes the DS equation of G(p) extremely  complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' It appears that the equation (40) is not even  self-closed since D(q), F(q), DF (q), FD(q), Γp(q, p), and  ΓA(q, p) are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Technically, one can derive the  DS equations fulfilled by these six unknown functions  by using the generic rules of quantum field theory [16–  18, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Nevertheless, such equations are coupled to the  formally more complicated DS equations of innumerable  multi-point correlation functions and hence of little use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Probably, one would have to solve an infinite number  of coupled DS equations to completely determine G(p),  which is apparently not a feasible scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  In order to simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (40) and make it tractable,  it might be necessary to introduce some approximations  by hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' For instance, one could: (1) neglect the last  two (mixing) terms of the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (2) discard all the vertex  corrections by assuming that Γp,A(q, p) → σ3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (3) replace  the full phonon propagator D(q) with the bare one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=',  D(q) → D0(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (4) replace the full A-boson propaga-  tor F(q) with a significantly simplified expression, such  as Fsim(q) =  1  q2+κD , or even with a phenomenological  parameter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=', the pseudopotential µ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Under all of  the above approximations, one finds that the original DS  equation (40) becomes  G−1(p) = G−1  0 (p) − i  �  d3q  (2π)3 gσ3G(p + q)D0(q)σ3  −iµ∗  �  d3q  (2π)3 σ3G(p + q)σ3,  (41)  which is self-closed and can be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The  free electron propagator has the form  G0(p) =  1  iǫnσ0 − ξpσ3 + ∆pσ1  ,  (42)  and the full electron propagator is expanded as  G(p) =  1  A1(ǫn, p)iǫnσ0 − A2(ǫn, p)ξpσ3 + ∆(ǫn, p)σ1  ,  (43)  where A1(ǫn, p) and A2(ǫn, p) are two renormalization  functions and ∆(ǫn, p) is pairing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Inserting G(p)  and G0(p) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (41), one would obtain the standard  ME equations of A1(ǫn, p), A2(ǫn, p), and ∆(ǫn, p) with  the parameter µ∗ characterizing the impact of Coulomb  repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In the past sixty years, such simplified equa-  tions have been extensively applied [1–5] to study a large  number of phonon-mediated superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' However,  the four approximations that lead to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (41) are not  justified under certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Some, or perhaps all,  of them break down in superconductors having a small  Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Now we seek to find a more powerful method to deal  with the original exact DS equation of G(p) given by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (40) by going beyond the above approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' We  believe that one should not try to determine each of  the six functions D(q), F(q), DF (q), FD(q), Γp(q, p),  and ΓA(q, p) functions separately, which can never be  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Alternatively, one should make an effort to  determine such products as D(q)Γp(q, p), F(q)ΓA(q, p),  DF (q)ΓA(q, p), and FD(q)Γp(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' This is the key idea of  the approach proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [18], where we have proved  the self-closure of the DS equation of the Dirac fermion  propagator G(p) in a model describing the coupling of  Dirac fermions to two distinct bosons in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Below  we show that this same approach can be adopted to prove  the self-closure of the DS equation given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' We  6  shall derive two exact identities satisfied by D(q), F(q),  DF (q), FD(q), Γp(q, p), and ΓA(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The derivation of the needed exact identities is based  on the invariance of partition function Z under arbitrary  infinitesimal changes of φ and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The invariance of Z  under an arbitrary infinitesimal change of φ gives rise to  ⟨D(x)φ(x) − gψ†(x)σ3ψ(x) + J(x)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (44)  Using ⟨φ(x)⟩ = δW/δJ(x), we perform functional deriva-  tives with respect to η(z) and η†(y) in order and obtain  the following relation  D(x)⟨φ(x)ψ(y)ψ†(z)⟩c = g⟨ψ†(x)σ3ψ(x)ψ(y)ψ†(z)⟩c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (45)  Making a Fourier transformation to the left-hand side  (l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=') of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (45) yields  D−1  0 (q)  �  − D(q)G(p + q)Γp(q, p)G(p) − DF (q)G(p + q)ΓA(q, p)G(p)  �  ,  (46)  where the free phonon propagator D0(q) comes from D(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' To handle the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (45), we use two bilinear  operators jc  t (x) = ψ†(x)σ3ψ(x) and js  t (x) = ψ†(x)σ0ψ(x) to define two current vertex functions Γ0,3(x − z, z − y):  ⟨ψ†(x)σ0,3ψ(x)ψ(y)ψ†(z)⟩c = −  �  dξdξ′G(y − ξ)Γ0,3(ξ − x, x − ξ′)G(ξ′ − z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (47)  Γ0,3(x − z, z − y) should be Fourier transformed [16, 22] as  Γ0,3(ξ − x, x − ξ′) =  �  dqdpe−i(p+q)·(ξ−x)−ip·(x−ξ′)Γ0,3(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (48)  For more properties of such current vertex functions, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [16, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Then the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (45) is turned into  �  dxdyei(p+q)·(y−x)eip·(x−z)g⟨ψ†(x)σ0,3ψ(x)ψ(y)ψ†(z)⟩c → −gG(p + q)Γ0,3(q, p)G(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (49)  After substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (46) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (49) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (45), we obtain the following identity  D(q)Γp(q, p) + DF (q)ΓA(q, p) = D0(q)gΓ3(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (50)  Similarly, the invariance of Z under an infinitesimal change of A field requires the following equation to hold  ⟨F(x)A(x) − gψ†(x)σ3ψ(x) + K(x)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (51)  Carrying out similar analytical calculations generates another identity  FD(q)Γp(q, p) + F(q)ΓA(q, p) = F0(q)Γ3(q, p),  (52)  where the free propagator of A boson F0(q) is computed by performing Fourier transformation to F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Making use of the two identities given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (50) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (52), we re-write the original DS equation (40) as  G−1(p) = G−1  0 (p) − i  �  d3q  (2π)3  �  g2D0(q) + F0(q)  �  σ3G(p + q)Γ3(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (53)  This equation is still not self-closed if the current vertex function Γ3(q, p) relies on unknown functions other than  G(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [16], the symmetry of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (14) yields the following WTI  q0Γ0(q, p) − (ξp+q − ξp) Γ3(q, p) = σ0G−1(p + q) − G−1(p)σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (54)  However, it is not possible to entirely determine Γ3(q, p) purely based on this single WTI, since Γ0(q, p) is also  unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Fortunately, the symmetry of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (15) generates another WTI [16]  q0Γ3(q, p) − (ξp+q − ξp) Γ0(q, p) = σ3G−1(p + q) − G−1(p)σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (55)  These two WTIs are coupled to each other and can be used to express Γ3(q, p) and Γ0(q, p) purely in terms of G(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Now Γ3(q, p) can be readily obtained by solving these two WTIs:  Γ3(q, p) = ω  �  G−1(p + q)σ3 − σ3G−1(p)  �  + (ξp+q − ξp)  �  G−1(p + q)σ0 − σ0G−1(p)  �  ω2 − (ξp+q − ξp)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (56)  We can see that the DS equation (53) becomes entirely  self-closed because it contains merely one unknown func-  7  tion G(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' This equation can be numerically solved to  determine G(p), provided that the three free propagators  G0(p), D0(q), and F0(q) and the EPI coupling parameter  g are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  It is useful to make some remarks here:  (1) In many existing publications, it is naively deemed  that a symmetry-induced WTI imposes an exact rela-  tion between fermion propagator G(p) and interaction  vertex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' To understand why this is a misconcep-  tion, let us take EPI as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The EPI vertex  function Γp(q, p) is defined via a three-point correlation  function ⟨φψψ†⟩, which in itself is not necessarily related  to any conserved current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' There is no reason to expect  Γp(q, p) to naturally enter into any WTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' To reveal the  impact of some symmetry, one should use the symmetry-  induced conserved current, say jc  µ, to define a special cor-  relation function ⟨jc  µψψ†⟩, which, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (47)  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (49), is expressed in terms of current vertex func-  tions Γ0(q, p) and Γ3(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' After imposing the constraint  of current conservation ∂µjc  µ = 0 to ⟨jc  µψψ†⟩, one would  obtain a WTI satisfied by Γ0(q, p), Γ3(q, p), and G(p), as  shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (2) Our ultimate goal is to determine G(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Its DS  equation (40) contains two interaction vertex functions  Γp(q, p) and ΓA(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' On the other hand, it is Γ0(q, p)  and Γ3(q, p), rather than Γp(q, p) and ΓA(q, p), that enter  into the WTIs given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (54) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Hence,  at least superficially the DS equation of G(p) and the  WTIs are not evidently correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' To find out a natural  way to combine the DS equation of G(p) with WTIs, one  needs to obtain the relations between interaction vertex  functions and current vertex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Such relations do  exist and are shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (50) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (3) The appearance of two free boson propagators  D0(q) and F0(q) in the final DS equation (53) is not  an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' It should be emphasized that the re-  placement of the full boson propagators D(q) and F(q)  appearing in the original DS equation (40) with their free  ones is implemented based on two exact identities given  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (50) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The interaction-induced effects  embodied in such functions as D(q), F(q), DF (q), FD(q),  Γp(q, p), and ΓA(q, p) are already incorporated in current  vertex function Γ3(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Tc OF 1UC FESE/SRTIO3 SYSTEM  In this section, we apply the self-closed DS equation  of G(p) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (53) along with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (56) to evalu-  ate the pair-breaking temperature Tc of the supercon-  ductivity realized in 1UC FeSe/SrTiO3 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  This  material is found [23–25] to possess a surprisingly high  Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' While it is believed by many [24–31] that interfacial  optical phonons (IOPs) from the SrTiO3 substrate are  responsible for the observed high Tc, other microscopic  pairing mechanisms cannot be conclusively excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' For  instance, Gor’kov [32] argued that IOPs alone are not ca-  pable of producing such a high Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' If this conclusion (not  necessarily the argument itself) is correct, one would be  forced to invoke at least one additional pairing mecha-  nism, such as magnetic fluctuation or nematic fluctua-  tion, to cooperate with IOPs [25, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Considerable  research efforts have been devoted to calculating IOPs-  induced Tc by using the standard ME theory [26–31] and  its slightly refined version [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Nevertheless, thus far no  consensus has been reached and the accurate value of Tc  produced by IOPs alone is still controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' To get a  definite answer, it is important to compute Tc with a suf-  ficiently high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' This is certainly not an easy task  since Tc could be influenced by many factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Among all the factors that can potentially affect Tc,  the EPI vertex corrections play a major role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Since the  ratio ωD/EF is at the order of unity, the Migdal theorem  becomes invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [16], including EPI  vertex corrections can drastically change the value of Tc  obtained under the bare vertex approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' However,  the calculations of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [16] still need to be improved as  they were based on two approximations that might lead  to inaccuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The first approximation is the omission  of the influence of Coulomb repulsion [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' As discussed  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' I, the traditional pseudopotential method may not  work well in 1UC FeSe/SrTiO3 owing to the smallness of  EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The impact of Coulomb interaction on Tc should  be examined more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The second approximation  is that the electron momentum p was supposed [16] to  be fixed at Fermi momentum pF such that ξp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Un-  der this second approximation, the DS equation of G(p)  has only one integral variable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=', frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Then the  computational time is significantly shortened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' For this  reason, this kind of approximation has widely been used  in the existing calculations of Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' But the pairing gap ∆  and the renormalization factors A1 and A2 obtained by  solving their single-variable equations depend solely on  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The momentum dependence is entirely lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Since the EPI strength depends strongly on the trans-  ferred momentum q, it is important not to neglect the  momentum dependence of the DS equation of G(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' We  shall discard the two approximations adopted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [16]  and directly deal with the self-closed DS equation given  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  We are particularly interested in how Tc is affected  by the interplay between EPI and Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  To avoid the difficulty brought by analytical continu-  ation, we choose to work in the Matsubara formalism  and express the electron momentum as p = (iǫn, p),  where ǫn = (2n + 1)πT , and the boson momentum as  q = (iωn′, q), where ωn′ = 2n′πT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The free phonon  propagator has the form  D0(q) =  2Ωq  (iωn′)2 − Ω2q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (57)  The interfacial optical phonons are almost dispersionless  [24, 28], thus Ωq can be well approximated by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  We choose to use the value of Ωq = 81meV [24, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The  Fermi energy is roughly µF = 65meV [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The EPI  strength parameter g depends on the phonon momentum  8  q as follows [24, 29]  g ≡ g(q) = g0 exp(−|q|/q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (58)  The value of dimensionless parameter λ can be estimated  by carrying out first-principle calculations [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Here we  regard λ as a tuning parameter and choose a representa-  tive value λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Parameter q0 measures the interaction  range in the momentum space [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The free propagator  of the A-boson is  F0(q) =  α′  vF |q|,  (59)  where the effective fine structure constant is α′ = e2/ε  with ε being an effective dielectric constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The prime  symbol in α′ is added to indicate that α′ differs from  the standard fine structure constant α = 1/137 defined  in QED4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The value of ε depends on the surroundings  (substrate) of the superconducting plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' For the sake  of generality, we suppose that ε can be freely changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  As the next step, we wish to substitute the free phonon  propagator D0(q) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (57), the free A-boson  propagator F0(q) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (59), the free electron  propagator G0(p) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (42), and the full electron  propagator G(p) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (43) into the DS equation  (53) and also into the function Γ3(q, p) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  However, we cannot naively do so since here we encounter  a fundamental problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Recall that Γ3(q, p) given by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (56) is derived from two symmetry-induced WTIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Once the pairing gap ∆(p) develops a finite value, the  system enters into the superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The U(1)  symmetry of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (15) is preserved in both the normal and  superconducting states, thus the WTI of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (55) is not  changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In contrast, the U(1) symmetry of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (14) is  spontaneously broken in the superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Does  this symmetry breaking change the WTI of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (54)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  If yes, one could insert the expression of G(p) given by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (43) into Γ3(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Otherwise, one should explore the  modification of a WTI owing to symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' At  present, there seems no conclusive answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Nambu [19]  adopted the WTI from charge conservation to prove the  gauge invariance of electromagnetic response functions of  a superconductor based on a ladder-approximation of the  DS equation of vertex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Following the scheme of  Nambu, Schrieffer [1] assumed (without giving a proof)  that this WTI is the same in the superconducting and  normal phases and used this assumption to show the  existence of a gapless Goldstone mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Nakanishi [35]  later demonstrated that, while the WTI for a U(1) gauge  theory has the same form in symmetric and symmetry-  broken phases, the situation of other field theories may be  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Yanagisawa [36] revisited this issue in a recent  work and argued that spontaneous breaking of a continu-  ous symmetry gives rise to an additional term to WTI due  to the generation of Goldstone boson(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' But the expres-  sion of such an additional term was not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' It also  remains unclear whether the approach of [36] works for  superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The superconducting transition is pro-  foundly different from other symmetry-breaking driven  transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' We learn from the Anderson mechanism [37]  that the Goldstone boson generated via U(1) symmetry  breaking is eliminated by the long-range Coulomb inter-  action, which lifts the originally gapless mode to a gapped  plasmon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Thus, the WTI associated with charge  conservation is not expected to acquire the additional  term derived in [36] upon entering the superconducting  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Nevertheless, the absence of a Goldstone-boson-  induced term does not mean that the WTI is entirely  unchanged after implementing the Anderson mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  In order to obtain a complete theoretical description  of superconducting transition, one needs to establish a  unified framework to reconcile the non-perturbative DS  equation approach with the Anderson mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  At  present, such a framework is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In the absence  of a satisfactory solution to the above problem, we have  to introduce an approximation so as to proceed with our  calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Our purpose is to compute Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Near Tc,  the gap ∆ vanishes and the symmetry of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (14) is  still (approximately) respected, thus the WTI of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (54)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Then we substitute the full electron propagator  G(p) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (43) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (53) and assume that  Γ3(q, p) takes the following form  Γ3(q, p) = ω  �  G−1  s (p + q)σ3 − σ3G−1  s (p)  �  + (ξp+q − ξp)  �  G−1  s (p + q)σ0 − σ0G−1  s (p)  �  ω2 − (ξp+q − ξp)2  ,  (60)  where  Gs(p) =  1  A1(ǫn, p)iǫnσ0 − A2(ǫn, p)ξpσ3  (61)  is independent of the pairing gap ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' This manipulation leads to three coupled nonlinear integral equations:  A1(ǫn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p) = 1 + T  iǫn  �  d2q  (2π)2  �  g2(q)D0(ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' q) + F0(ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' q)  �  ×A1(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)i(ǫn + ωm)Γ33 + A2(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)ξp+qΓ30 − ∆(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)Γ3d  A2  1(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)(ǫn + ωm)2 + A2  2(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)ξ2  p+q + ∆2(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (62)  9  A2(ǫn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p) = 1 − T  ξp  �  d2q  (2π)2  �  g2(q)D0(ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' q) + F0(ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' q)  �  ×A1(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)i(ǫn + ωm)Γ30 + A2(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)ξp+qΓ33 − ∆(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)Γ31  A2  1(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)(ǫn + ωm)2 + A2  2(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)ξ2  p+q + ∆2(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (63)  ∆(ǫn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p) = −T  �  d2q  (2π)2  �  g2(q)D0(ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' q) + F0(ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' q)  �  ×A1(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)i(ǫn + ωm)Γ3d − A2(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)ξp+qΓ31 − ∆(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)Γ33  A2  1(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)(ǫn + ωm)2 + A2  2(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)ξ2  p+q + ∆2(ǫn + ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' p + q)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (64)  Here, we have defined several quantities:  Γ30 =  1  ω2m + (ξp+q − ξp)2  �  iωm[A2(ǫn + ωm, p + q)ξp+q − A2(ǫn, p)ξp]  +(ξp+q − ξp)[−A1(ǫn + ωm, p + q)(iǫn + iωm) + A1(ǫn, p)iǫn]  �  ,  (65)  Γ33 =  1  ω2m + (ξp+q − ξp)2  �  iωm[−A1(ǫn + ωm, p + q)(iǫn + iωm) + A1(ǫn, p)iǫn]  +(ξp+q − ξp)[A2(ǫn + ωm, p + q)ξp+q − A2(ǫn, p)ξp]  �  ,  (66)  Γ31 = (ξp+q − ξp) [−∆(ǫn + ωm, p + q) + ∆(ǫn, p)]  ω2m + (ξp+q − ξp)2  ,  (67)  Γ3d = iωm [∆(ǫn + ωm, p + q) + ∆(ǫn, p)]  ω2m + (ξp+q − ξp)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (68)  It is easy to reproduce the ME equations if the full  propagator G(p) appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (56) is replaced with  the free propagator G0(p), which leads to the bare vertex  approximation Γ3 → σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' An alternative manipulation is  to substitute A1 = A2 = 1 and ∆ = 0 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (65-68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  This yields  Γ30 = 0,  Γ33 = 1,  Γ31 = 0,  Γ3d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  (69)  Then Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (62-64) become the standard ME equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The above integral equations can be numerically solved  using the iteration method (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [16] for a detailed  illustration of this method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The computational time  required to reach convergent results depends crucially  on the number of integral variable: adding one variable  would exponentially increase the computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The coupled equations given by (62-64) have only one  variable ǫn if all electrons are assumed to reside exactly  on the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Such an assumption simplifies the  equations and dramatically decreases the computational  time, but might not be well justified in the present case  due to the strong momentum dependence of EPI coupling  strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' There, here we consider all the possible values  of p and directly solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (62-64) without introducing  further approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' But these equations have three  integral variables, namely ωm, q1, and q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Solving them  would consume too many computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The burden of numerical computation can be greatly  lightened if the number of integral variable is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  We suppose that the system is isotropic and then make  an effort to integrate over the angle θ between p and q  before starting the iterative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' After doing so, only  two free variables, namely ωm and |q|, are involved in the  process of performing iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' This manipulation can  greatly decrease the computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Generically, it  is not easy to integrate over θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  In our case, however,  although the current vertex function Γ3(q, p) is compli-  cated, the free propagators D0(q) and F0(q) are simple  functions of their variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The phonon energy Ωq is  a constant, thus D0(q) depends solely on the frequency,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=', D0(q) = D0(ωm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  In comparison, F0(q) depends  solely on the momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' To illustrate why θ can be in-  tegrate out, it is more convenient to deal with the DS  equation shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (53) instead of the formally com-  plicated equations (62-64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' After re-defining p + q → k  and  �  dq →  �  dk, we re-write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (53) as  G−1(ǫn, p) = G−1  0 (ǫn, p) +  �  m  � kdkdθ  (2π) σ3G(ǫm, k)  ×  �  g2D0(ωm) +  e2  ε  �  p2 + k2 − 2|p||k| cos θ  �  ×Γ3(ǫn, p, ǫm, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Both G0(ǫn, p) and G(ǫn, p) are independent of θ, thus  θ is not involved in the iterative process and can be nu-  merically integrated at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='6  10-3  35  40  45  50  55  60  65  Tc[K]  ME  DS  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' 1: Critical temperatures Tc obtained by solving the com-  plete DS equation of G(p) and the simplified ME equation of  G(p) at a fixed coupling parameter λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The results of Tc  for other values of λ can be similarly evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' ME results  and DS results are shown using red squares and green circles,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Notice that F0(q) is singular at q = 0, reflecting the  long-range nature of bare Coulomb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Within  the framework of jellium model for the electron liquid, the  contribution of q = 0 must be eliminated since it cancels  out the static potential between negative and positive  charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  This can be implemented by introducing an  infrared cutoff δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In our calculations the infrared cutoff  is taken to be 10−6pF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' We have already confirmed that  the final results of Tc are nearly unchanged as δ varies  within the range of  �  10−8pF , 10−3pF  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The numerical results of Tc for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='2 are presented in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' 1, where the blue and black lines correspond to the  ME-level and DS-level approximations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The  results for other values of λ can be similarly obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The pairing gap is supposed to have an isotropic s-wave  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  We first consider the simplest case in which the  Coulomb interaction is absent, corresponding to α′ = 0  or equivalently ε = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' In a previous work [16], it was  found that including the EPI vertex corrections tends  to promote Tc evaluated at the ME-level (bare vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  This conclusion was reached based on a special assump-  tion that the electrons always strictly reside on the Fermi  surface [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Here we have re-solved Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (62-64) without  making this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The numerical results plotted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' 1 clearly show that the inclusion of EPI vertex  corrections leads to a considerable reduction of Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' We  can infer from these results that the EPI vertex correc-  tions and the momentum dependence of A1,2 and ∆ are  both important and should be carefully incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The influence of Coulomb repulsion on Tc can be read-  ily examined by varying the tuning parameter ε or α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  For a realistic material, ε takes a specific value, so does  Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' 1, Tc is suppressed drastically as  ε decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Tc is completely determined once all the  model parameters are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  SUMMARY AND DISCUSSION  In summary, we have performed a non-perturbative  study of the interplay of EPI and Coulomb repulsion by  using the DS equation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' We have shown that  the DS equation of the full electron propagator G(p) is  self-closed provided that all the higher order corrections  to EPI and Coulomb interaction are incorporated via a  number of exact identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' This self-closed DS equation  can be applied to study the superconducting transition  beyond the ME approximation of EPI and the pseudopo-  tential approximation of Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  We have  employed this approach to evaluate the pair-braking tem-  perature Tc for the interfacial superconductivity in 1UC  FeSe/SrTiO3 system and found that the value of Tc could  be significantly miscalculated if the vertex corrections  and the momentum-dependence of relevant quantities are  not taken into account in a reliable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  The calculations of this work ignored several effects  that might change the value of Tc and thus need to be  improved in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' First of all, the one-band model  studied by us should be replaced with a more realistic  multi-band model that embodies the actual electronic  structure [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' The phonon self-coupling terms [20] were  entirely neglected in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Including such  self-coupling terms invalidates the two identities given  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (50) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  As a consequence, the DS  equation of the electron propagator G(p) can no longer  be made self-closed (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [16] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' [17] for more  details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  In addition, as analyzed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' IV, the role  played by the Anderson mechanism has not yet been un-  ambiguously determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' For these reasons, our results  of Tc cannot be directly compared to the experimental re-  sults in 1UC FeSe/SrTiO3 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Therefore, although  the present work provides a methodological advance in  the non-perturbative study of superconducting transition  driven by the interplay of EPI and Coulomb repulsion,  we are still some way from a satisfactory theoretical de-  scription of superconductors (including but not restricted  to 1UC FeSe/SrTiO3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  We thank Jing-Rong Wang and Hao-Fu Zhu for helpful  discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Schrieffer, Theory of Superconductivity (Taylor and  Francis, 1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Scalapino, in Superconductivity, edited by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  11  Parks (Marcel Dekker, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' New York, 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Allen and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Mitrovi´c, Theory of Superconducting  Tc, Solid State Physics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=',  1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' Carbotte, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' Marsiglio, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' 417, 168102 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' Tolmachev, Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' Nauk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' SSSR 140:3, 563  (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Katsnelson,  and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' B 103, 094501 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  [17] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' :Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content='  Matter 35, 075601 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' Nambu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' 117, 648 (1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' Dee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Coulter, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content=' Johnston,  Communications Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' Zuber, Quantum Field Theory,  (McGraw-Hill Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
+page_content=' 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FRT4oBgHgl3EQfQDdI/content/2301.13520v1.pdf'}
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diff --git a/ZdFJT4oBgHgl3EQf7i0F/content/tmp_files/2301.11678v1.pdf.txt b/ZdFJT4oBgHgl3EQf7i0F/content/tmp_files/2301.11678v1.pdf.txt
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+arXiv:2301.11678v1  [math.OC]  27 Jan 2023
+GENERALIZING QUASI-NEWTON UPDATES TO HIGHER-ORDER
+DERIVATIVE TENSORS∗
+KARL WELZEL† AND RAPHAEL A. HAUSER†
+Abstract. At the heart of all quasi-Newton methods is an update rule that enables us to gradu-
+ally improve the Hessian approximation using the already available gradient evaluations. Theoretical
+results show that the global performance of optimization algorithms can be improved with higher-
+order derivatives. This motivates an investigation of generalizations of quasi-Newton update rules to
+obtain for example third derivatives (which are tensors) from Hessian evaluations. Our generalization
+is based on the observation that quasi-Newton updates are least-change updates satisfying the secant
+equation, with different methods using different norms to measure the size of the change. We present
+a full characterization for least-change updates in weighted Frobenius norms (satisfying an analogue
+of the secant equation) for derivatives of arbitrary order. Moreover, we establish convergence of the
+approximations to the true derivative under standard assumptions and explore the quality of the
+generated approximations in numerical experiments.
+Key words. quasi-Newton methods, tensors, approximate derivatives, higher-order optimiza-
+tion
+MSC codes. 90C53, 65D25
+1. Introduction. Consider a nonconvex unconstrained optimization problem
+for a function f : Rn → R
+(1.1)
+min
+x∈Rn f(x).
+Recent work on worst-case global convergence rates discovered that the more de-
+rivatives are provided at each iteration the fewer iterations are required to find an
+approximate minimizer. Birgin et al. [2] showed that if the function is p times con-
+tinuously differentiable with a Lipschitz continuous pth derivative and an oracle to
+compute the first p derivatives at any point is provided, then an algorithm exists that
+finds a point with ∥∇f(x)∥ ≤ ε in at most O(ε−(p+1)/p) oracle calls. This result
+generalized the known cases for p = 1 [20] and p = 2 [23] and was later extended by
+Cartis, Gould and Toint [5], who also proved that this bound is sharp for algorithms
+that minimize regularized Taylor models in each step such as the one used in [2].
+These new insights (although largely theoretical as yet) motivate us to consider
+ways of estimating higher-order derivatives from lower-order ones. A solid starting
+point for formulas that estimate pth derivatives from (p − 1)st are quasi-Newton
+update formulas, which estimate second derivatives from first derivatives (p = 2).
+The corresponding optimization methods use these Hessian approximations to build
+approximate Taylor models, which enable rapid convergence to the minimum.
+Note that this paper only considers the updating formulas and their properties
+without proposing a method that uses them (for optimization or otherwise). This
+method-agnostic approach enables anyone wishing to incorporate approximate higher-
+order derivatives into their algorithm to adapt the general results to their needs. Pre-
+vious papers concerned with approximate derivatives and higher-order models are ei-
+ther using higher-order Taylor expansions to derive better conventional quasi-Newton
+∗This work is supported by the Hong Kong Innovation and Technology Commission (InnoHK
+Project CIMDA)
+†Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United King-
+dom (welzel@maths.ox.ac.uk, hauser@maths.ox.ac.uk).
+1
+
+2
+KARL WELZEL AND RAPHAEL A. HAUSER
+methods or use approximate derivatives in a higher-order optimization scheme. As
+part of the first group, Wei, Li and Qi [29] considered a modification of the secant
+equation, so that the new approximate second-order Taylor expansion interpolates
+the function at the previous iterate, and show that this update predicts the curvature
+information at the new iterate better, at least when evaluated in the direction of the
+previous step. Their work was extended by Biglari, Hassan and Leong [1], who used a
+fourth-order Taylor expansion to derive an even better modification, and by Enshaei
+et al. [10], who applied the same idea to diagonal Hessian approximations.
+One of the very early papers on approximate higher-order models for optimization
+is [26]. In it, Schnabel and Chow constructed a fourth-order model that interpolates
+the last few function and gradient values at every iteration and imposed a simple
+low-rank structure on the third- and fourth-order tensor. This makes it possible to
+efficiently minimize the model. A different approach was taken by Shi and Pan [28].
+They developed a predictor–corrector approach for the descent trajectory defined by
+Newton’s method. First a predictor is determined using linesearch along the New-
+ton direction, then a quadratic curve ϕ: R → Rn is computed that interpolates the
+iterate, the predictor and their gradients (thereby approximating the descent trajec-
+tory), and lastly the function is minimized along this curve to obtain the next iterate.
+They showed that an implementation of this idea using the approximate Newton direc-
+tions generated by quasi-Newton methods is able to outperform regular quasi-Newton
+methods in numerical experiments. Mart´ınez and Raydan [18] combined a third-order
+model with a trust-region approach. The third-order term is restricted to be a cer-
+tain sum of n rank-one tensors which allows them to solve the model minimization
+subproblem by minimizing n independent cubic polynomials in one dimension. Their
+approach to computing this diagonal third-order term is similar to ours in that they
+choose one which is closest to the difference of the previous two Hessian evaluations.
+Most recently, Nesterov [22, 21] considered specialized algorithms for unconstrained
+convex optimization based on a third-order Taylor expansion with a fourth-order reg-
+ularization term. He mentions that an explicit calculation of the third derivative can
+be avoided since his algorithms only need the gradients of the model to solve the
+subproblem, and the contribution of the tensor term to the gradients can either be
+calculated through automatic differentiation or approximated with a finite-difference
+formula using only first derivatives of f.
+Our paper will be structured as follows: After introducing the tensor notation
+we use in section 2, we will derive the tensor analogues of quasi-Newton updates in
+section 3 and give a full characterization of these updates in section 4. The char-
+acterization includes an explicit formula and a useful recursive relationship between
+successive approximations. Moreover, we will show that these updates exhibit a cer-
+tain low-rank structure. Section 5 contains results on the convergence of the approx-
+imations to the exact derivative in the limit under certain conditions on the steps.
+Lastly, in section 6 we present limited numerical experiments to verify the behaviour
+predicted by the theory and discuss numerical limitations of this method.
+2. Notation. A p-tensor T of dimensions n1 × · · · × np is a multilinear map
+Rn1 × · · · × Rnp → R, so that its evaluation
+(2.1)
+T[s1, . . . , sp],
+si ∈ Rni
+is linear in each component. We denote the space of such p-tensors by Rn1⊗···⊗np. If
+n1 = · · · = np = n the space is denoted R⊗pn and T[s, . . . , s] is abbreviated as T[s]p.
+The notation also allows to apply the tensor to q < p vectors, which then results
+
+GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS
+3
+in a (p − q)-tensor. Moreover, we define the application of matrices W1, . . . , Wp of
+appropriate dimensions to a tensor by
+(2.2)
+�
+T[W1, . . . , Wp]
+�
+[s1, . . . , sp] = T[W1s1, . . . , Wpsp].
+The outer product of a p1-tensor T1 with a p2-tensor T2 is defined as
+(2.3)
+(T1 ⊗ T2)[s1, . . . , sp1+p2] = T1[s1, . . . , sp1] · T2[sp1+1, . . . , sp1+p2].
+In particular, tensors of the form T = v1 ⊗ · · · ⊗ vp for vectors v1, . . . , vp ∈ Rn, i.e.
+those where T[s1, . . . , sp] = �n
+i=1 vT
+i si, are called elementary or rank-one tensors.
+If all vectors are the same (v1 = · · · = vp = v), we abbreviate the notation above
+to ⊗pv. (Note that this notation is slightly inconsistent since 1-tensors should be
+row vectors, but are represented by standard column vectors to simplify the notation.
+This is why v[W ] = W T v.)
+For any T ∈ R⊗pn and any permutation σ ∈ Sp, let σ(T) ∈ R⊗pn be defined by
+(2.4)
+σ(T)[s1, . . . , sp] = T[sσ(1), . . . , sσ(p)].
+If σ(T) = T for all σ ∈ Sp, then T is called symmetric. The space of all symmetric
+p-tensors is denoted R⊗pn
+sym . The projection of R⊗pn onto R⊗pn
+sym is given by
+(2.5)
+Psym(T) = 1
+p!
+�
+σ∈Sp
+σ(T),
+see [14, Proposition 3.76].
+Just like matrices, tensors are fully characterized by their actions on basis vec-
+tors. This can be used to represent a p-tensor T ∈ R⊗pn as a p-dimensional array
+(ti1,...,ip)1≤ij≤n,1≤j≤p where
+(2.6)
+ti1,...,ip = T[ei1, . . . , eip]
+and
+T =
+n
+�
+i1,...,ip=1
+ti1,...,ipei1 ⊗ · · · ⊗ eip.
+There is a Frobenius inner product and a corresponding norm on these p-dimensional
+arrays and by extension on R⊗pn, which we will denote by ⟨T1, T2⟩F and ∥T∥F . Note
+that ⟨T, s1 ⊗ · · · ⊗ sp⟩F = T[s1, . . . , sp]. Another norm is the one induced by the
+2-norm on Rn (Hackbusch [14] calls it the injective norm) which is defined by
+(2.7)
+∥T∥2 =
+max
+∥si∥2=1, 1≤i≤p|T[s1, . . . , sp]|.
+Both norms are invariant under orthogonal transformations, so that if Q1, . . . , Qp ∈
+Rn×n are orthogonal matrices, then T[Q1, . . . , Qp] has the same Frobenius- and 2-
+norm as T. As for matrices, the 2-norm is bounded by the Frobenius norm:
+(2.8)
+∥T∥2 =
+max
+∥si∥2=1, 1≤i≤p|⟨T, s1 ⊗ · · · ⊗ sp⟩F | ≤ ∥T∥F .
+In the last inequality we used Cauchy-Schwarz and the fact that ∥s1 ⊗ · · ·⊗ sp∥F = 1
+if all si have unit 2-norm.
+Using this setup we can now introduce the notation for higher order derivatives.
+Let f : Rn → R be a smooth function. The pth total derivative of f at x ∈ Rn is
+denoted Dpf(x) and is recursively defined as the total derivative of Dp−1f where
+
+4
+KARL WELZEL AND RAPHAEL A. HAUSER
+D0f = f. This gives a chain of linear maps which we can regard as one multilinear
+map
+(2.9)
+Dpf(x): Rn → (Rn → (. . . (Rn → R))) = Rn × · · · × Rn
+�
+��
+�
+p times
+→ R
+making Dpf(x) a p-tensor of dimensions n×· · ·×n. By this definition the evaluation
+of this p-tensor Dpf(x)[s1, . . . , sp] is equal to the directional derivative of f at x
+along directions s1, . . . , sp ∈ Rn. For example, denoting the gradient by ∇f and the
+Hessian by ∇2f we have
+(2.10)
+D1f(x)[s1] = ∇f(x)T s1
+and
+D2f(x)[s1, s2] = sT
+1 ∇2f(x)s2.
+Moreover, Dpf(x) is symmetric, because partial derivatives commute (Schwarz’s the-
+orem). The pth-order Taylor expansion of f at x evaluated at an offset s ∈ Rn can
+be expressed in this notation as
+(2.11)
+Tf,p(x, s) =
+p
+�
+k=0
+1
+k!Dkf(x)[s]k ≈ f(x + s).
+3. Derivation. We now turn to our derivation of the generalized quasi-Newton
+update for higher derivatives by first introducing a few important conventional quasi-
+Newton methods. The most well-known update rule for quasi-Newton methods is the
+BFGS method, which is often described as a rank-two update to the current Hessian
+approximation. To motivate the generalization in this paper we take a different view
+and describe BFGS and similar methods as choosing minimal updates that satisfy the
+secant equation [9, 24]. Let f ∈ C2(Rn) be twice continuously differentiable function
+with gradient ∇f and Hessian ∇2f.
+Assume that we are given some sequence of
+points xk ∈ Rn for k ∈ N (possibly from minimizing f), the gradients ∇f(xk) at
+each iterate and some symmetric initial Hessian approximation B0 ∈ Rn×n
+sym . Quasi-
+Newton methods then update Bk at each step such that the new approximation
+correctly predicts the change in gradients of the previous iteration, that is
+(3.1)
+Bk+1(xk+1 − xk) = ∇f(xk+1) − ∇f(xk).
+This is called the secant equation. We can write (3.1) more succinctly if we define
+sk = xk+1 − xk and let �
+Bk =
+� 1
+0 ∇2f(xk + tsk) dt be the Hessian of f averaged over
+all points on the line from xk to xk+1. This gives the equivalent equation
+(3.2)
+Bk+1sk = �
+Bksk.
+Most quasi-Newton methods then prescribe that among all possible choices of
+Bk+1 that are symmetric and satisfy the secant equation we take the one that is
+closest to Bk in some norm. The simplest such rule is called the Powell-symmetric-
+Broyden (PSB) update and is defined as
+(PSB)
+Bk+1 = arg minB∈Rn×n
+sym ∥B − Bk∥F s.t. Bsk = �
+Bksk.
+It is derived from Broyden’s method [3] by adding the symmetry constraint.
+The Davidon-Fletcher-Powell (DFP) method [7, 12], even though it has been
+proposed before PSB, can be understood as a way to make the PSB method scale-
+invariant by choosing a weighted Frobenius norm. Let Wk = �
+B−1/2
+k
+(or, in fact, any
+
+GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS
+5
+nonsingular matrix with W −T
+k
+W −1
+k
+sk = �
+Bksk) be the weight matrix, then the DFP
+update is given by
+(DFP)
+Bk+1 = arg minB∈Rn×n
+sym ∥W T
+k (B − Bk)Wk∥F s.t. Bsk = �
+Bksk.
+If we rescale the input to f with the nonsingular matrix A, so that ¯f(x) = f(Ax),
+and also rescale the iterates using ¯xk = A−1xk, then the corresponding Hessian
+approximations of ¯f determined by the DFP method satisfy ¯
+Bk = AT BkA as long
+as it holds for the initial choice ¯
+B0.
+Finally, the famous BFGS method named after Broyden, Fletcher, Goldfarb and
+Shanno [4, 11, 13, 27], is the dual of DFP in the sense that the new approximation
+minimizes the difference between inverse matrices in a weighted Frobenius norm:
+(BFGS)
+Bk+1 = arg minB∈Rn×n
+sym ∥W −T
+k
+�
+B−1 − B−1
+k
+�
+W −1
+k
+∥F s.t. Bsk = �
+Bksk
+The weight matrices Wk are the same as above and in the same way they make the
+method scale invariant. For more on these updating rules, consult the textbook by
+Dennis and Schnabel [9, Chapter 9]
+Using this characterization as least-change updates allows a straightforward gen-
+eralization to tensors, except for the BFGS update. Since tensors lack the concept of
+an inverse tensor, (BFGS) cannot be used, and we will focus on (PSB) and (DFP). We
+now need to assume that f : Rn → R is p times continuously differentiable and that as
+before a sequence of points xk with a corresponding sequence of steps sk = xk+1 −xk
+is given. We will denote the approximations to Dpf(xk) by Ck ∈ R⊗pn
+sym and the true
+pth derivative averaged over all points on the line from xk to xk+1 by
+(3.3)
+�Ck =
+� 1
+0
+Dpf(xk + tsk) dt ∈ R⊗pn
+sym .
+This means that, in particular, �Ck[sk] = Dp−1f(xk+1) − Dp−1f(xk). The pth-order
+analogue of the secant equation (3.2) is given by
+(3.4)
+Ck+1[sk] = Dp−1f(xk+1) − Dp−1f(xk) = �Ck[sk]
+and the generalized quasi-Newton update formula for Ck reads
+(GQN)
+Ck+1 = arg minC∈R⊗pn
+sym ∥(C − Ck)[Wk]p∥F s.t. C[sk] = �Ck[sk].
+It provides a sequence of approximations of the pth derivative of f based solely on
+evaluations of the (p − 1)st derivative at the iterates xk, given some initial approxi-
+mation C0.
+Note that unlike for the DFP update we will not assume any specific choice
+of weight matrices, but rather consider them to be a given sequence of nonsingu-
+lar matrices. In particular, that covers the generalized PSB (Wk = I) and DFP
+(W −T
+k
+W −1
+k
+sk = ∇f(xk+1) − ∇f(xk)) updates, which simplify to (PSB) and (DFP)
+for p = 2.
+4. Characterization of generalized quasi-Newton updates. The following
+theorem provides a full characterization of the update in (GQN).
+Theorem 4.1. Let C• ∈ R⊗pn
+sym (the current approximation), �C ∈ R⊗pn
+sym (the inte-
+grated true derivative), a nonsingular matrix W ∈ Rn×n (the weight matrix) and a
+nonzero s ∈ Rn (the step) be given. The following equations all have the same unique
+solution C+ ∈ R⊗pn
+sym (the new approximation):
+
+6
+KARL WELZEL AND RAPHAEL A. HAUSER
+(a) C+ = arg minC∈R⊗pn
+sym ∥(C − C•)[W ]p∥F s.t. C[s] = �C[s]
+(b) C+ = C• + �p
+j=1(−1)j�p
+j
+�
+(vT s)−jPsym
+��
+⊗jv
+�
+⊗ (C• − �C)[s]j�
+(c) C+ = C• + Psym(A ⊗ v) for a (unique) A ∈ R⊗p−1n
+sym
+s.t. C+[s] = �C[s]
+(d) (C+ − �C)[W ]p = (C• − �C)[W ]p�
+I − W −1ssT W −T
+sT W −T W −1s
+�p
+where v = W −T W −1s.
+Proof. We will first prove the result for W = I and then see how that implies
+the full result for any nonsingular weight matrix W .
+The first characterization can be rewritten as
+(4.1)
+C+ = C• + arg minU∈R⊗pn
+sym ∥U∥F s.t. U[s] = (�C − C•)[s].
+Let Q ∈ Rn×n be an orthogonal matrix that maps e1 to some scalar multiple of
+s.
+This means U[Q]p has the same Frobenius norm as U and U[Q]p[ei1, . . . , eip]
+is fully determined by the equality constraint in (4.1) if 1 ∈ {i1, . . . , ip} because of
+symmetry. On the other hand, if 1 /∈ {i1, . . . , ip} there are no constraints on the
+values of U[Q]p[ei1, . . . , eip], so the unique choice that minimizes ∥U[Q]p∥F = ∥U∥F
+is clearly to set all of these values to zero. Therefore, there is a unique solution to the
+minimization problem in (a), and it is fully characterized by the fact that the update
+tensor U = C+ − C• is symmetric and satisfies the following two properties:
+U[s] = (�C − C•)[s]
+(4.2a)
+U[u1, . . . , up] = 0 if all ui are orthogonal to s
+(4.2b)
+We will use this as the basis to show the equivalence with all other characterization.
+In (b) we claim that the update U has the form
+(4.3)
+p
+�
+j=1
+(−1)j�p
+j
+�
+∥s∥−2j
+2
+Psym
+��
+⊗js
+�
+⊗ (C• − �C)[s]j�
+This tensor is clearly symmetric since it is a sum of symmetric tensors. To show that
+it satisfies property (4.2a) consider
+p
+�
+j=1
+(−1)j�p
+j
+�
+∥s∥−2j
+2
+Psym
+��
+⊗js
+�
+⊗ (C• − �C)[s]j�
+[s]
+(4.4a)
+=
+p
+�
+j=1
+(−1)j∥s∥−2j
+2
+��p−1
+j−1
+�
+Psym
+��
+⊗j−1s
+�
+⊗ (C• − �C)[s]j�
+∥s∥2
+2
++
+�p−1
+j
+�
+Psym
+��
+⊗js
+�
+⊗ (C• − �C)[s]j+1��
+(4.4b)
+= −Psym
+�
+(C• − �C)[s]
+�
+= (�C − C•)[s].
+(4.4c)
+The first equality uses the fact that each summand is the symmetric projection of an
+outer product of two symmetric tensors, a j-tensor and a (p − j)-tensor. Therefore,
+there are
+�p
+j
+�
+distinct ways to orient this outer product and for
+�p−1
+j−1
+�
+of them the vector
+s is applied to the j-tensor and for
+�p−1
+j
+�
+to the (p − j)-tensor. A close examination
+of the expression on the second line shows that it is a telescoping sum where all terms
+except the ones with coefficients
+�p−1
+0
+�
+and
+�p−1
+p
+�
+cancel out. Because
+�p−1
+p
+�
+= 0 the
+
+GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS
+7
+only remaining term is the negative symmetric projection of (C• − �C)[s]. Property
+(4.2b) follows from a similar consideration to the one above. If u1, . . . , up are all
+orthogonal to s, then
+(4.5)
+Psym
+��
+⊗js
+�
+⊗ (C• − �C)[s]j�
+[u1, . . . , up] = 0
+for j ≥ 1 because no matter how the outer product is oriented there is always a factor
+of sT ui = 0 in the result.
+For (c) we need to show that we can always write the update in the form Psym(A⊗
+s) for some symmetric (p−1)-tensor A and that A is unique. Note that the expression
+for the update in (b) is already of the form Psym(A ⊗ s):
+p
+�
+j=1
+(−1)j�p
+j
+�
+∥s∥−2j
+2
+Psym
+��
+⊗js
+�
+⊗ (C• − �C)[s]j�
+(4.6a)
+= Psym
+
+Psym
+
+
+p
+�
+j=1
+(−1)j�p
+j
+�
+∥s∥−2j
+2
+�
+⊗j−1s
+�
+⊗ (C• − �C)[s]j
+
+ ⊗ s
+
+
+(4.6b)
+It remains to show that among all updates of the form Psym(A ⊗ s) there is only one
+choice of A ∈ R⊗pn
+sym such that (4.2a) holds. Consider the linear map
+φ: R⊗p−1n
+sym
+→ R⊗p−1n
+sym
+A �→ Psym(A ⊗ s)[s]
+which maps a finite-dimensional vector space to itself. Combining what we already
+showed for (b) with the observation that the update in (b) is of the desired form, this
+map is surjective (we can prescribe any (�C−C•)[s]) and so it must be bijective, which
+shows uniqueness of A.
+Lastly, (d) claims that the update can be written as
+(4.7)
+(�C − C•) − (�C − C•)
+�
+I − ssT
+∥s∥2
+2
+�p
+.
+Clearly, property (4.2a) is satisfied because applying this tensor to s makes the second
+term vanish, leaving only the desired result.
+Moreover, for any vector u that is
+orthogonal to s the matrix I − ssT
+∥s∥2
+2 maps u to itself. This means applying the tensor
+above to u1, . . . , up, all of which are orthogonal to s, will give zero because both
+terms cancel out. This is property (4.2b).
+Now that the equivalence has been established for W = I, we consider the general
+case where W ∈ Rn×n is any nonsingular matrix. The minimization in (a)
+(4.8)
+C+ = arg minC∈R⊗pn
+sym ∥(C − C•)[W ]p∥F s.t. C[s] = �C[s]
+can be rewritten using D• = C•[W ]p, �D = �C[W ]p, D+ = C+[W ]p and r = W −1s as
+(4.9)
+D+ = arg minD∈R⊗pn
+sym ∥D − D•∥F s.t. D[r] = �D[r].
+Applying the existing characterizations and the fact that C+ = D+[W −1]p we get the
+claim after some algebraic manipulations.
+
+8
+KARL WELZEL AND RAPHAEL A. HAUSER
+Note that although the theorem assumes �C ∈ R⊗pn and W ∈ Rn×n to be given
+the resulting C+ only depends on �C[s] ∈ R⊗p−1n
+sym
+and W −T W −1s ∈ Rn. This makes
+the method applicable even when no pth derivatives exists.
+Moreover, the equivalence between (a) and (c) shows that there is a one-to-one
+correspondence between updates that exhibit a certain low-rank structure Psym(A⊗v),
+where v is any vector which is not orthogonal to s, and those updates that are minimal
+in some weighted Frobenius norm. If we apply this result to the matrix case (p = 2)
+both A and v are 1-tensors, so that Psym(A ⊗ v) is the symmetric projection of a
+rank-one matrix. Any minimal update (such as PSB or DFP) is therefore a symmetric
+matrix of rank at most two and any update matrix of type vwT + wvT is minimal in
+a certain weighted Frobenius norm.1
+5. Convergence of approximate derivates. Now that we know how the up-
+dates look like, we want to show that the approximate pth derivatives converge to the
+true derivative under certain assumptions. This means that, even though we previ-
+ously pointed out that the update is applicable when no pth derivatives are available,
+we will assume in this section that f is p-times continuously differentiable. The main
+tool of this section is the characterization Theorem 4.1 (d) which, when applied to
+(GQN), becomes
+(5.1) (Ck+1 − �Ck)[Wk]p = (Ck − �Ck)[Wk]p[Pk]p where Pk = I − W −1
+k
+sksT
+k W −T
+k
+sT
+k W −T
+k
+W −1
+k
+sk
+Note that Pk is the orthogonal projection onto the orthogonal complement of W −1
+k
+sk.
+5.1. Convergence for pth-order polynomials. When Wk and �Ck are con-
+stant, convergence is quite straightforward to show, and we do not need to assume
+convergence of the iterates xk:
+Theorem 5.1. Let C0 ∈ R⊗pn
+sym be given and update the approximations Ck ac-
+cording to (GQN). Assume Dpf(x) = C∗ everywhere (which makes f a pth order
+polynomial) and Wk = W∗ for every k ∈ N. After n steps sk such that W −1
+∗
+sk are
+orthogonal, we have Cn = C∗.
+Proof. Under the assumptions of the theorem, repeated application of (5.1) gives
+(5.2)
+(Cn − C∗)[W∗]p = (C0 − C∗)[W∗]p
+�n−1
+�
+k=0
+Pk
+�p
+.
+Because W −1
+0
+s0, . . . , W −1
+n−1sn−1 are orthogonal and Pk are orthogonal projections
+on their orthogonal complements, �n−1
+k=0 Pk = 0, so that (Cn − C∗)[W∗]p = 0. This
+implies Cn = C∗ because W∗ is nonsingular.
+5.2. Bounded deterioration property. For a convergence results for general
+functions f we first need to establish two lemmas that will help us when xk → x∗ and
+Wk → W∗. The first one shows that �Ck converges to the pth derivative at x∗, which
+we denote by C∗ = Dpf(x∗), and the second one gives a bound on the error term that
+we incur if we replace �Ck by C∗ in (5.1). Combining the two gives what Dennis and
+Schnabel [9] call the bounded deterioration principle in that Ck+1 can only be slightly
+worse than Ck at approximating C∗ as long as xk and xk+1 are close enough to x∗.
+1The symmetric rank-one (SR1) update fits the latter description with v = w, but the corre-
+sponding weight matrix depends on the current Hessian approximation, so it does not generalize to
+higher orders.
+
+GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS
+9
+Lemma 5.2. For xk → x∗ we have �Ck → C∗ as k → ∞. Moreover, if Dpf is
+Lipschitz continuous with constant L, then
+(5.3)
+∥�Ck − C∗∥2 ≤ L
+2 (∥xk − x∗∥2 + ∥xk+1 − x∗∥2)
+for all k ∈ N.
+Proof. By definition in (3.3) we have
+∥�Ck − C∗∥2 = ∥
+� 1
+0
+Dpf(xk + tsk) dt − Dpf(x∗)∥2
+(5.4a)
+≤
+� 1
+0
+∥Dpf(xk + tsk) − Dpf(x∗)∥2 dt.
+(5.4b)
+Since Dpf is continuous the right-hand side will become arbitrarily small as xk and
+xk+1 converge to x∗.
+This shows �Ck → C∗.
+If we additionally assume Lipschitz
+continuity of Dpf, we can bound the integrand in (5.4b):
+∥�Ck − C∗∥2 ≤
+� 1
+0
+L∥xk + tsk − x∗∥2 dt
+(5.5a)
+≤ L
+� 1
+0
+(1 − t)∥xk − x∗∥2 + t∥xk − x∗∥2 dt
+(5.5b)
+= L
+2 (∥xk − x∗∥2 + ∥xk+1 − x∗∥2)
+(5.5c)
+This gives the second claim.
+Lemma 5.3. If we define the error tensor Ek by
+(5.6)
+(Ck+1 − C∗)[Wk]p = (Ck − C∗)[Wk]p[Pk]p + Ek[Wk]p
+then ∥Ek∥2 ≤ (1 + κ2(Wk)p)∥�Ck − C∗∥2.
+Proof. Subtracting (5.1) from (5.6) gives
+(5.7)
+(�Ck − C∗)[Wk]p = (�Ck − C∗)[Wk]p[Pk]p + Ek[Wk]p.
+Multiply both sides by W −1
+k
+from all sides and rearrange to find
+∥Ek∥2 = ∥(�Ck − C∗) − (�Ck − C∗)[Wk]p[Pk]p[W −1
+k
+]p∥2
+(5.8a)
+≤
+�
+1 + ∥WkPkW −1
+k
+∥p
+2
+�
+∥�Ck − C∗∥2
+(5.8b)
+≤ (1 + κ2(Wk)p)∥�Ck − C∗∥2
+(5.8c)
+as required. In the last step we used ∥Pk∥2 ≤ 1.
+5.3. Convergence for weakly orthogonal steps. Since the updates only de-
+termine Ck[sk] in each step, we can never hope to recover the true pth derivative if
+(asymptotically) all steps lie in a low-dimensional subspace of Rn, so we must assume
+a weak orthogonality condition on the steps, namely
+(5.9)
+�����
+k0+m−1
+�
+k=k0
+�
+I − W −1
+∗
+sksT
+k W −T
+∗
+sT
+k W −T
+∗
+W −1
+∗
+sk
+������
+2
+≤ c < 1
+
+10
+KARL WELZEL AND RAPHAEL A. HAUSER
+for some fixed m ∈ N, c ∈ R≥0 and all k0 ∈ N large enough. Here, W∗ = limk→∞ Wk
+which we assume to be nonsingular as well. As Mor´e and Trangenstein [19] showed,
+this assumption is equivalent to uniform linear independence of the scaled steps
+W −1
+∗
+sk which is in turn equivalent to the standard assumption of uniform linear
+independence of the steps sk themselves.
+This leads to the following convergence
+theorem:
+Theorem 5.4. Let C0 ∈ R⊗pn
+sym be given and update the approximations Ck accord-
+ing to (GQN). Assume xk converge to x∗ ∈ Rn, Wk converge to some nonsingular
+matrix W∗ ∈ Rn×n and the steps are uniformly linearly independent. Then Ck con-
+verges to C∗ := Dpf(x∗).
+We need a technical lemma for this result, which shows that the projection ma-
+trices Pk get closer and closer to the projection matrices that appear in (5.9), which
+we will abbreviate with P ∗
+k = I − W −1
+∗
+sksT
+k W −T
+∗
+sT
+k W −T
+∗
+W −1
+∗
+sk .
+Lemma 5.5. Let the nonsingular matrices Wk ∈ Rn×n converge to the nonsingu-
+lar matrix W∗ ∈ Rn×n then
+(5.10)
+∥Pk − P ∗
+k ∥2 → 0
+and
+∥W −1
+∗
+WkPkW −1
+k
+W∗ − P ∗
+k ∥2 → 0
+holds for any sequence of steps (sk)k∈N as k → ∞.
+Proof. Without loss of generality we can assume that the steps are scaled such
+that ∥W −1
+∗
+sk∥2 = 1 for all k ∈ N. That means
+(5.11)
+∥W −1
+k
+sk − W −1
+∗
+sk∥2 ≤ ∥W −1
+k W∗ − I∥2
+�
+��
+�
+→0
+∥W −1
+∗
+sk∥2
+�
+��
+�
+=1
+→ 0
+as k → ∞.
+Since Pk projects onto the subspace that is orthogonal to W −1
+k
+sk and P ∗
+k projects onto
+the subspace that is orthogonal to W −1
+∗
+sk, the difference between the two projection
+matrices converges to zero. This is the first claim.
+For the second one we find that
+(5.12)
+W −1
+∗
+WkPkW −1
+k
+W∗ − P ∗
+k = (W −1
+∗
+Wk − I)
+�
+��
+�
+→0
+Pk W −1
+k
+W∗
+�
+��
+�
+→I
++ (Pk − P ∗
+k )
+�
+��
+�
+→0
+W −1
+k
+W∗
+�
+��
+�
+→I
++P ∗
+k (W −1
+k
+W∗ − I)
+�
+��
+�
+→0
+converges to 0 since the norms of Pk and P ∗
+k are at most one.
+Proof of Theorem 5.4. It suffices to consider the case when W∗ = I. Otherwise,
+let ¯f(x) = f(W∗x), ¯xk = W −1
+∗
+xk, ¯x∗ = W −1
+∗
+x∗, ¯
+Wk = W −1
+∗
+Wk and ¯C0 = C0[W∗]p
+and update ¯Ck according to the adapted (GQN). By construction, we then have
+Ck = ¯Ck[W −1
+∗
+]p so that ¯Ck → Dp ¯f(¯x∗) implies Ck → Dpf(x∗).
+Let ε > 0 be arbitrary. To prove convergence, we will show ∥Ck − C∗∥2 ≤ ε for
+every k large enough. Consider the recurrence relation for Ck − C∗ established in
+Lemma 5.3. We can rearrange it to read
+(5.13)
+(Ck+1 − C∗) = (Ck − C∗)[WkPkW −1
+k
+]p + Ek.
+Abbreviate WkPkW −1
+k
+by P ′
+k and note that P ′
+k is no longer an orthogonal projection
+but gets closer and closer to one as k → ∞. In particular, its norm approaches one
+
+GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS
+11
+since ∥P ′
+k∥2 ≤ κ2(Wk) → 1.
+To use the assumption (5.9) later on we apply the
+previous equality m times and get
+(5.14)
+(Ck0+m − C∗) = (Ck0 − C∗)
+�k0+m−1
+�
+k=k0
+P ′
+k
+�p
++
+k0+m−1
+�
+k=k0
+Ek
+�k0+m−1
+�
+l=k+1
+P ′
+l
+�p
+.
+Let K1 ∈ N be such that for all k ≥ K1 we have κ2(Wk) ≤ 2. This means using
+Lemma 5.3 we can bound the second term on the right-hand side of the previous
+equation by
+(5.15)
+�����
+k0+m−1
+�
+k=k0
+Ek
+�k0+m−1
+�
+l=k+1
+P ′
+l
+������
+2
+≤
+k0+m−1
+�
+k=k0
+2p(1 + 2p)∥�Ck − C∗∥
+for all k0 ≥ K1. Since Lemma 5.2 showed that �Ck → C∗ the bound above is smaller
+than ε/2 · (1 − ( 1+c
+2 )p) for k0 large enough, say k0 ≥ K2.
+Next, consider first term on the right-hand side of (5.14). Let P ∗
+k be the orthog-
+onal projection matrices from the assumption (5.9) as before. Note that P ′
+k ̸= P ∗
+k , so
+we cannot apply the assumption directly. Instead, we look at the difference between
+the product of P ′
+k and P ∗
+k and find
+�����
+k0+m−1
+�
+k=k0
+P ′
+k −
+k0+m−1
+�
+k=k0
+P ∗
+k
+�����
+2
+=
+�����
+k0+m−1
+�
+k=k0
+P ∗
+k0 · · · P ∗
+k−1(P ′
+k − P ∗
+k )P ′
+k+1 · · · P ′
+k0+m−1
+�����
+2
+(5.16a)
+≤
+k0+m−1
+�
+k=k0
+2m−1∥P ′
+k − P ∗
+k ∥2
+(5.16b)
+for k0 ≥ K1 using ∥P ′
+k∥ ≤ 2 and ∥P ∗
+k ∥ ≤ 1 ≤ 2. For k0 large enough, say k0 ≥ K3,
+the right-hand side of (5.16) is smaller than (1 − c)/2 by Lemma 5.5. Similarly, for k0
+large enough, say k0 ≥ K4, we have ∥�k0+m−1
+k=k0
+P ∗
+k ∥ ≤ c by assumption. Therefore,
+for k0 ≥ max{K3, K4}
+(5.17)
+�����
+k0+m−1
+�
+k=k0
+P ′
+k
+����� ≤
+�����
+k0+m−1
+�
+k=k0
+P ′
+k −
+k0+m−1
+�
+k=k0
+P ∗
+k
+�����
+2
++
+�����
+k0+m−1
+�
+k=k0
+P ∗
+k
+����� ≤ 1 + c
+2
+.
+In the previous two paragraphs we have established that asymptotically for every
+m steps the norm of Ck − C∗ is first multiplied by a factor that is at most slightly
+larger than c and then increased by an arbitrarily small error term. In particular for
+k0 ≥ K = max{K2, K3, K4} we have
+(5.18)
+∥Ck0+m − C∗∥2 ≤ ∥Ck0 − C∗∥2
+�1 + c
+2
+�p
++ ε/2 ·
+�
+1 −
+�1 + c
+2
+�p�
+.
+Repeatedly applying this inequality shows that ∥Ck0+im − C∗∥2 is bounded by a
+sequence (ai)i∈N which converges to ε/2. Use this observation for all k0 ∈ {K, K +
+1, . . . , K + m − 1} to see that ∥Ck − C∗∥2 ≤ ε for all k large enough, as claimed.
+Corollary 5.6. Let C0 ∈ R⊗pn
+sym be given and update the approximations Ck ac-
+cording to (GQN). Assume the steps all lie in a d-dimensional subspace spanned by
+
+12
+KARL WELZEL AND RAPHAEL A. HAUSER
+the columns of V ∈ Rn×d, so that the iterates are of the form xk = V ¯xk + x⊥
+where V T x⊥ = 0.
+Moreover, assume that ¯xk converge to ¯x∗ ∈ Rd (or equiva-
+lently xk converge to x∗ = V ¯x∗ + x⊥), Wk converge to some nonsingular matrix
+W∗ ∈ Rn×n and ¯sk = ¯xk+1 − ¯xk are uniformly linearly independent. Then Ck con-
+verges to C∗ := Dpf(x∗) in the subspace: Ck[V ]p → C∗[V ]p as k → ∞.
+Proof. We will construct a sequence of ¯Ck to which we can apply Theorem 5.4
+and then show ¯Ck = Ck[V ]p. Let ¯f : Rd → R be defined by ¯f(¯x) = f(V ¯x + x⊥)
+which implies
+(5.19)
+Dp ¯f(¯x) = Dpf(V ¯x + x⊥)[V ]p.
+Additionally, let the weight matrices ¯
+Wk be defined by ¯
+Wk = (V T W −T
+k
+W −1
+k
+V )−1/2.
+The operation B �→ B1/2 maps a symmetric positive semidefinite matrix B to the
+unique symmetric semidefinite matrix H such that HT H = B, and is continuous
+(see [17, Section 7.2]). Here, we know that V T W −T
+k
+W −1
+k
+V is positive definite (V
+is full-rank), so ¯
+Wk is also positive definite. Since the construction of ¯
+Wk depends
+continuously on Wk, the newly constructed weight matrices converge to a positive
+definite matrix ¯
+W∗ given by the same formula applied to W∗. That means if we now
+define ¯C0 = C0[V ]p and apply (GQN) using ¯f, ¯xk and ¯
+Wk, the resulting sequence
+¯Ck will converge to Dp ¯f(¯x∗) by Theorem 5.4.
+Clearly, the limit point Dp ¯f(¯x∗) = Dpf(x∗)[V ]p is already correct, it just remains
+to show that Ck[V ]p = ¯Ck for all k. By construction of ¯
+Wk we get
+(5.20)
+¯
+W −T
+k
+¯
+W −1
+k
+¯sk = V T W −T
+k
+W −1
+k
+V ¯sk = V T W −T
+k
+W −1
+k
+sk
+where sk = V ¯sk are steps in Rn. Theorem 4.1 (c) shows that Ck+1 = Ck + Psym(A ⊗
+W −T W −1sk) for some (p − 1)-tensor A ∈ R⊗p−1n
+sym
+, where A is chosen such that
+Ck+1[sk] = Dp−1f(xk+1) − Dp−1f(xk). This implies
+Ck+1[V ]p = Ck[V ]p + Psym(A ⊗ W −T W −1sk)[V ]p
+(5.21a)
+= Ck[V ]p + Psym(A[V ]p−1 ⊗ V T W −T W −1sk)
+(5.21b)
+= Ck[V ]p + Psym(A[V ]p−1 ⊗ ¯
+W −T
+k
+¯
+W −1
+k
+¯sk)
+(5.21c)
+and Ck+1[V ]p[¯sk] = Dp−1 ¯f(¯xk+1) − Dp−1 ¯f(¯xk). Again by Theorem 4.1 (c) ¯Ck and
+Ck[V ]p are updated in the same way, so inductively they define the same sequence.
+This concludes the proof.
+5.4. Generalized Dennis–Mor´e condition. Dennis and Mor´e [8] showed that
+if optimization methods choose their iterates using xk+1 = xk − B−1
+k ∇f(xk) and
+converge to x∗ then this convergence is Q-superlinear if and only if
+(5.22)
+lim
+k→∞
+∥(Bk − ∇2f(x∗))sk∥F
+∥sk∥2
+= 0.
+This equation is called the Dennis–Mor´e condition and is weaker than convergence of
+Bk to ∇2f(x∗). Instead, it suffices when Bk converges in the step directions.
+We are not concerned with convergence rates for any particular optimization
+method in this paper, but it seems natural to ask whether the approximations Ck
+satisfy a generalized Dennis–Mor´e condition
+(5.23)
+lim
+k→∞
+∥(Ck − Dpf(x∗))[sk]∥F
+∥sk∥2
+= 0
+when they are updated according to (GQN).
+
+GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS
+13
+Theorem 5.7. Let C0 ∈ R⊗pn
+sym be given and update the approximations Ck ac-
+cording to (GQN) where the function has a Lipschitz continuous pth derivative Dpf.
+Assume xk converge to x∗ ∈ Rn and Wk converge to some nonsingular matrix
+W∗ ∈ Rn×n fast enough such that
+(5.24)
+�
+k≥0
+∥xk − x∗∥2 < ∞
+and
+�
+k≥0
+∥Wk − W∗∥2 < ∞.
+Then the generalized Dennis–Mor´e condition (5.23) holds.
+Proof. As in the proof of Theorem 5.4 we assume W∗ = I without loss of gen-
+erality.
+Since the condition number is locally Lipschitz continuous around I, the
+assumption �
+k≥0∥Wk − I∥2 < ∞ also implies Cκ := �
+k≥0(κ2(Wk) − 1) < ∞.
+As a first step we need to use the bounded deterioration principle to show that
+∥Ck − C∗∥2 stays bounded. For this we again consider the m-step recursive formula
+established in (5.14):
+(5.25)
+(Cm − C∗) = (C0 − C∗)
+�m−1
+�
+k=0
+P ′
+k
+�p
++
+m−1
+�
+k=0
+Ek
+� m−1
+�
+l=k+1
+P ′
+l
+�p
+where P ′
+k = WkPkW −1
+k
+and Ek is defined in Lemma 5.3. Clearly,
+(5.26)
+ln
+������
+m−1
+�
+k=0
+P ′
+k
+�����
+2
+�
+≤
+m−1
+�
+k=0
+ln(∥P ′
+k∥2) ≤
+m−1
+�
+k=0
+ln(κ2(Wk)) ≤
+m−1
+�
+k=0
+(κ2(Wk) − 1) ≤ Cκ
+is uniformly bounded for all m, which means the same is true for ∥�m−1
+k=0 P ′
+k∥2 itself.
+Therefore,
+(5.27)
+∥Cm − C∗∥2 ≤ exp(Cκ)p
+�
+∥C0 − C∗∥2 +
+m−1
+�
+k=0
+∥Ek∥2
+�
+.
+Because Dpf is Lipschitz continuous and κ2(Wk) stays bounded, Lemmas 5.2 and 5.3
+show that there is a constant CE < ∞ such that
+(5.28)
+∥Ek∥2 ≤ CE/2(∥xk − x∗∥ + ∥xk+1 − x∗∥).
+This implies �m−1
+k=0 ∥Ek∥2 ≤ CE
+�m
+k=0∥xk − x∗∥2 is uniformly bounded for all m and
+therefore ∥Cm − C∗∥2 is as well.
+The next step is crucial for this proof and similar to the proof of Theorem 8.2.2
+in [9]. We note that in the Frobenius norm for any tensor T ∈ R⊗pn and any nonzero
+vector v ∈ Rn we have
+(5.29)
+∥T∥2
+F =
+����T
+�vvT
+vT v
+�����
+2
+F
++
+����T
+�
+I − vvT
+vT v
+�����
+2
+F
+= ∥T[v]∥2
+F
+∥v∥2
+2
++
+����T
+�
+I − vvT
+vT v
+�����
+2
+F
+because the matrices in brackets are orthogonal projections. We can apply this to the
+
+14
+KARL WELZEL AND RAPHAEL A. HAUSER
+case where I − vvT /(vT v) is Pk to get
+∥(Ck − C∗)[Wk]p[Pk]p∥
+2
+F
+(5.30a)
+≤ ∥(Ck − C∗)[Wk]p[Pk]∥2
+F
+(5.30b)
+= ∥(Ck − C∗)[Wk]p∥2
+F −
+��(Ck − C∗)[Wk]p�
+W −1
+k
+sk
+���2
+F
+∥W −1
+k
+sk∥2
+2
+(5.30c)
+= ∥(Ck − C∗)[Wk]p∥2
+F −
+���(Ck − C∗)[sk][Wk]p−1���
+2
+F
+∥W −1
+k
+sk∥2
+2
+(5.30d)
+≤ ∥(Ck − C∗)[Wk]p∥2
+F − ∥(Ck − C∗)[sk]∥2
+F
+∥sk∥2
+2
+∥W −1
+k ∥−2p
+2
+.
+(5.30e)
+Adding Ek[Wk]p into the Frobenius norm on the left-hand side gives
+∥(Ck+1 − C∗)[Wk]p∥2
+F = ∥(Ck − C∗)[Wk]p[Pk]p + Ek[Wk]∥2
+F
+(5.31a)
+= ∥(Ck − C∗)[Wk]p[Pk]p∥2
+F + ∥Ek[Wk]p∥2
+F.
+(5.31b)
+The inner product term ⟨(Ck − C∗)[Wk]p[Pk]p, Ek[Wk]p⟩F is missing in (5.31b)
+because it is zero. To show that, note that we can rewrite (5.7) from the proof of
+Lemma 5.3 as
+(5.32)
+(�Ck − Ek − C∗)[Wk]p = (�Ck − C∗)[Wk]p[Pk]p.
+Using the equivalence between Theorem 4.1 (d) and (c) the error tensor can be written
+explicitly as −Ek = Psym(A ⊗ W −T
+k
+W −1
+k
+sk) for some (p − 1)-tensor A and
+(5.33)
+Ek[Wk]p = −Psym(A[Wk]p−1 ⊗ W −1
+k
+sk).
+In other words, Ek[Wk]p can be expressed as a sum of outer products between W −1
+k
+sk
+and some (p−1)-tensor. Any inner product of such a tensor with (Ck − C∗)[Wk]p[Pk]p
+must be zero as Pk maps W −1
+k
+sk to zero.
+Combining (5.30) and (5.31) and rearranging gives
+∥(Ck − C∗)[sk]∥2
+F
+∥sk∥2
+2
+≤ ∥W −1
+k
+∥2p
+2
+�
+∥(Ck − C∗)[Wk]p∥2
+F
+− ∥(Ck+1 − C∗)[Wk]p∥2
+F + ∥Ek[Wk]p∥2
+F
+�
+(5.34a)
+≤ ∥(Ck − C∗)∥2
+Fκ2(Wk)2p − ∥(Ck+1 − C∗)∥2
+F + ∥Ek∥2
+Fκ2(Wk)2p
+(5.34b)
+Lastly, we wish to sum up both sides of the previous inequality over all k ≥ 0 and
+show that the right-hand side stays bounded. This immediately gives the claim. A
+few technicalities are needed. We already showed that Ck − C∗ stays bounded, so let
+C∆ < ∞ be a constant such that ∥Ck−C∗∥F ≤ C∆ for all k ∈ N. As established above,
+�
+k≥0(κ2(Wk) − 1) < ∞. This also implies that �
+k≥0
+�
+κ2(Wk)2p − 1
+�
+= Cκ,2p < ∞
+for some constant Cκ,2p since κ2(Wk)2p − 1 ≤ 4p(κ2(Wk) − 1) for κ2(Wk) small
+enough.
+
+GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS
+15
+Consider the Ck − C∗ terms on the right-hand side of (5.34) first:
+m−1
+�
+k=0
+�
+∥(Ck − C∗)∥2
+F κ2(Wk)2p − ∥(Ck+1 − C∗)∥2
+F
+�
+(5.35a)
+= ∥C0 − C∗∥2
+F κ2(W0)2p
+�
+��
+�
+constant
++
+m−1
+�
+k=1
+�
+κ2(Wk)2p − 1
+�
+�
+��
+�
+=Cκ,2p
+∥Ck − C∗∥2
+F
+�
+��
+�
+≤C2
+∆
+− ∥Cm − C∗∥2
+F
+�
+��
+�
+≥0
+(5.35b)
+is uniformly bounded for all m.
+The same is true for the Ek term.
+Because the
+Frobenius norm and the 2-norm for p-tensors are both norms on finite-dimensional
+vector spaces they are equivalent.
+Moreover, κ2(Wk) stays bounded, so for some
+constant C
+m−1
+�
+k=0
+∥Ek∥2
+F κ2(Wk)2p ≤ C
+m−1
+�
+k=0
+∥Ek∥2
+2 ≤ C
+�m−1
+�
+k=0
+∥Ek∥2
+�2
+(5.36a)
+holds, and the term is uniformly bounded. Therefore,
+(5.37)
+�
+k≥0
+∥(Ck − C∗)[sk]∥2
+F
+∥sk∥2
+2
+< ∞
+and
+∥(Ck − C∗)[sk]∥F
+∥sk∥2
+→ 0 as k → ∞
+as claimed.
+6. Numerical experiments. While the previous sections investigated the the-
+oretical properties of the generalized quasi-Newton updates, we now turn to numerical
+experiments to understand how quickly convergence of the approximations sets in and
+how the algorithm behaves for different kinds of iterates. This is not supposed to be
+a comprehensive treatment of the numerical performance of the algorithm, but rather
+give an idea of its general behaviour by considering a small toy problem.
+The implementation was done in Python using NumPy [15] and uses the explicit
+formula in Theorem 4.1 (b) at its core:
+(6.1)
+Ck+1 = Ck +
+p
+�
+j=1
+(−1)j�p
+j
+��
+vT
+k sk
+�−jPsym
+��
+⊗jvk
+�
+⊗ (Ck[sk] − Dk)[sk]j−1�
+where vk = W −T
+k
+W −1
+k
+sk and Dk = Dp−1f(xk+1) − Dp−1f(xk).
+This makes it
+particularly easy to implement the analogues of (PSB) (where vk = sk) and (DFP)
+(where vk = (∇f(xk+1) − ∇f(xk))/∥sk∥2). To increase legibility and since these two
+choices produce roughly similar approximations, only the PSB variant will be shown
+below.
+We chose to use the two-dimensional Rosenbrock function f(x, y) = (1 − x)2 +
+100(y − x2)2 as our objective function because it is a simple, yet widely used test
+function with nonconstant third derivative
+(6.2)
+D3f(x, y) =
+��
+−2400x
+−400
+−400
+0
+�
+�
+−400
+0
+0
+0
+��
+.
+The iterates xk are generated by different minimization algorithms starting at (0, 0)
+and converging to the global minimum of f at x∗ = (1, 1). Lastly, the initial approx-
+imation C0 is just the zero 3-tensor in the following.
+
+16
+KARL WELZEL AND RAPHAEL A. HAUSER
+0
+10
+20
+30
+40
+Iteration k
+10−14
+10−12
+10−10
+10−8
+10−6
+10−4
+10−2
+100
+∥Ck − D3f(xk)∥F /∥D3f(xk)∥F
+∥xk − x∗∥2/∥x∗∥2
+Fig. 6.1. Convergence of iterates and approximations for nonlinear CG
+6.1. Numerical limitations. In the first experiment, a nonlinear CG method2
+was used to minimize the Rosenbrock function.
+Although nonlinear CG methods
+with restarts can achieve superlinear local convergence, this implementation does not
+include restarts, and we observe roughly linear convergence of xk to x∗ in Figure 6.1.
+The relative error of each Ck is not measured with respect to C∗ = D3f(1, 1) here
+but instead with respect to the true third derivative at each iterate D3f(xk), since
+this is the more relevant metric in practice. From the convergence theorems in the
+previous section we expect that this quantity will also converge to zero, since both
+Ck and D3f(xk) converge to C∗.
+Unfortunately, although at first this seems to be true and the two error curves
+roughly coincide, from iteration 27 onwards the error in Ck increases quite consid-
+erably.
+The issue, as it turns out, stems from rounding errors in finite precision
+arithmetic, which we did not consider in the theory. Specifically the computation of
+Dk becomes more ill-conditioned the smaller the step sk is.
+In each iteration, the current approximation Ck moves closer to the integrated
+derivative �Ck, so we cannot expect Ck to approximate D3f(xk) better than �Ck. Of
+course, the only part of �Ck which is used is its component in the direction of sk, i.e.
+Dk/∥sk∥2. In exact arithmetic the proof of Lemma 5.2 also shows that
+(6.3)
+Dk
+∥sk∥2
+= �Ck[s→
+k ] = D3f(xk)[s→
+k ] + ∆Dk
+with
+∥∆Dk∥2 ≤ L
+2 ∥sk∥2
+where s→
+k is the normed step sk/∥sk∥2, i.e. the unit norm vector pointing in the same
+direction as sk, and L is the (local) Lipschitz constant of Dpf.
+Now, let �Dk be the computed Dk = Dp−1f(xk+1)−Dp−1f(xk) under the influence
+of rounding errors.
+As Higham [16, p.
+9] explains, if we subtract two numbers
+2scipy.optimize.minimize(method="CG") in SciPy version 1.9.3, implementing the Polak–
+Ribi`ere variant of nonlinear CG [24, p. 122]
+
+GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS
+17
+0
+10
+20
+30
+40
+Iteration k
+10−14
+10−12
+10−10
+10−8
+10−6
+10−4
+10−2
+100
+∥Ck − D3f(xk)∥F /∥D3f(xk)∥F
+∥xk − x∗∥2/∥x∗∥2
+max{∥sk−1∥2, εmach/∥sk−1∥2}
+√εmach
+Fig. 6.2. Convergence of iterates and approximations for nonlinear CG (extended)
+ˆa = a(1 + ∆a) and ˆb = b(1 + ∆b) from each other, and assume the relative errors ∆a
+and ∆b are bounded by δ, the absolute error in the result is bounded by
+(6.4)
+|−a∆a + b∆b| ≤ δ(|a| + |b|).
+The a and b in our case are the entries of Dp−1f(xk+1) and Dp−1f(xk). They are
+computed with the exact formulas, but stored in finite precision, so the best possible
+error bound δ is the machine precision εmach ≈ 10−16. This gives
+(6.5)
+�Dk
+∥sk∥2
+=
+Dk
+∥sk∥2
++ ∆�Dk = D3f(xk)[s→
+k ] + ∆Dk + ∆�Dk
+where ∥∆�Dk∥F ≤
+√
+2εmach
+�
+∥Dp−1f(xk+1)∥F + ∥Dp−1f(xk)∥F
+�
+/∥sk∥2.
+Equation (6.5) shows that there are two sources of error, one from using the
+secant equation and one from calculating Dk. The former is proportional to ∥sk∥2
+whereas the latter is proportional to 1/∥sk∥2.
+This leads to the V-shaped graph
+in Figure 6.1. If we assume that Dp−1f, Dpf and L have roughly the same scale,
+we can estimate that the lowest possible relative error in �Dk/∥sk∥2 (compared to
+D3f(xk)[sk/∥sk∥2]) is √εmach and is achieved when ∥sk∥2 ≈ √εmach. This analysis
+is analogous to one for numerical differentiation schemes where the same lower bound
+is derived, see for example [25, Section 5.7]. One notable exception to the rule occurs
+when Dp−1f(x∗) = 0. In that case ∥∆�Dk∥F stays close to machine precision and
+the approximations get better and better as sk converges to 0. This is one of the
+reasons that regular quasi-Newton methods (p = 2) work very well for optimization
+algorithms as they converge to stationary points.
+In Figure 6.2 one can see that indeed the best relative error achieved is roughly
+√εmach and that the maximum of ∥sk−1∥2 and εmach/∥sk−1∥2 is a pretty good proxy
+for a lower bound on the error.
+
+18
+KARL WELZEL AND RAPHAEL A. HAUSER
+0
+2
+4
+6
+8
+10
+12
+14
+16
+Iteration k
+10−14
+10−12
+10−10
+10−8
+10−6
+10−4
+10−2
+100
+∥Ck − D3f(xk)∥F /∥D3f(xk)∥F
+∥(Ck − D3f(xk))[s→
+k ]∥F /∥D3f(xk)∥F
+∥xk − x∗∥2/∥x∗∥2
+max{∥sk−1∥2, εmach/∥sk−1∥2}
+√εmach
+Fig. 6.3. Convergence of iterates and approximations for exact trust region
+Note that these numerical issues cannot be overcome with a different imple-
+mentation but are inherent in this approach of extracting third-order information
+from successive evaluations of second-order derivatives, since computing Dk is an
+ill-conditioned problem. A practical way to avoid losing accuracy in the last few it-
+erations would be to employ a heuristic that skips updating Ck when the expected
+size of errors ∆�Dk exceeds the size of the update or simply when ∥sk∥2 becomes too
+small.
+6.2. Convergence in a subspace. In the second experiment, we used a trust
+region approach with exact Hessian evaluations to generate the iterates.3
+As pre-
+dicted by the theory for these methods, we observe local quadratic convergence to the
+minimizer (and much fewer iterations in general). The dotted line shows that there
+are very few iterations in which accurate information about the third derivatives can
+be obtained, but even then the relative error in Ck is several orders of magnitude
+larger than the lower bound.
+A key difference in the two experiments is how the directions of the steps are
+distributed. Figure 6.4 plots the angle of each sk with the x-axis, normalized between
+0° and 180° so that opposite directions coincide. Whereas the steps generated by
+the nonlinear CG algorithm cover multiple well-separated directions during the main
+part of the algorithm, the steps generated by the trust region method tend to fall into
+a one-dimensional subspace, especially towards the end when convergence happens.
+This directly explains why the relative Frobenius error stays high and even increases
+towards the end in the second experiment: All the information we can extract from
+the (averaged) true derivative �Ck is its evaluation in the direction sk and since most
+of the steps point in the same direction at the end, the information about the other
+3scipy.optimize.minimize(method="trust-exact") in SciPy version 1.9.3, see [6, pp. 169–200]
+for more details. Only the successful iterations were used.
+
+GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS
+19
+0
+20
+40
+Iteration k
+0°
+20°
+40°
+60°
+80°
+100°
+120°
+140°
+160°
+180°
+Angle of sk with x-axis
+0
+5
+10
+15
+Iteration k
+Nonlinear CG
+Exact trust region
+Rosenbrock valley
+subspace at (1, 1)
+Fig. 6.4. Subspaces of the steps for nonlinear CG and exact trust region
+directions gets more and more outdated.
+In addition to the relative Frobenius norm, we also included (a relative version of)
+the Dennis–Mor´e measure from subsection 5.4 in Figure 6.3. This one measures the
+error in Ck only in the direction of the step sk. Now, one might expect that when the
+approximation is updated in one specific direction and the Dennis–Mor´e error only
+measures how good the approximation is in this one direction, the error must track
+the lower bound derived in the previous subsection quite well. Indeed, this is the
+case when the steps all lie exactly in one subspace as we could verify with manually
+generated iterates. For the exact trust region method however the step directions vary
+by about 1° in successive iterations at the end, which can be seen in Figure 6.4. Let
+s→
+k = λ1s→
+k−1 + λ2uk where uk is chosen such that s→
+k−1 and uk form an orthonormal
+basis of R2, then λ2
+1 + λ2
+2 = 1 and by multilinearity of Ck,
+(6.6)
+Ck[s→
+k ] = λ1Ck[s→
+k−1] + λ2Ck[uk].
+Therefore, in each of the last few iterations the approximation in direction s→
+k
+is
+a linear combination of the very accurate information in direction s→
+k−1 and very
+inaccurate information in direction uk. In particular, if the angle with the x-axis varies
+by about 1° we get that λ2 ≈ sin(1◦) ≈ 10−2. Combining this with the knowledge that
+the relative overall error in Ck is on the order of 1, we expect that the Dennis–Mor´e
+measure will hover around 10−2. This agrees very well with the graph in Figure 6.3 and
+shows that it is important for this method to gather accurate derivative information
+in all directions.
+7. Conclusion. We have seen in this paper that quasi-Newton updates described
+as least-change updates admit fairly straightforward generalizations to higher-order
+derivatives. Moreover, they have a closed form solution with a certain low-rank struc-
+ture to it, generalizing the rank-two characterization of regular quasi-Newton updates.
+
+20
+KARL WELZEL AND RAPHAEL A. HAUSER
+The theoretical results suggest that, as long as the directions of the steps span the
+space and stay well separated, the generated approximations converge to the true
+derivative in the limit and under suitably fast convergence of the iterates they even
+converge (in a subspace) if these assumptions are violated. This is however not the
+behaviour we see in experiments, since the problem of computing the difference be-
+tween Hessian evaluations becomes more and more ill-conditioned as the difference
+between consecutive iterates becomes smaller. These numerical limitations lead to a
+loss in accuracy: If the Hessians are computed with relative error δ we cannot expect
+the errors in the generated approximations to go below
+√
+δ. Our experiments show
+that, as long as the directions of the steps stay well separated, the method indeed
+generates accurate approximations up to the numerical limit. If the directions tend to
+fall into a subspace, the derivative information in directions orthogonal to it quickly
+gets outdated and can impact approximations as long as they do not all fall exactly
+into the subspace.
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+
diff --git a/ZdFJT4oBgHgl3EQf7i0F/content/tmp_files/load_file.txt b/ZdFJT4oBgHgl3EQf7i0F/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..75cc8e49b894ba867307bbe9287737f6d34e893e
--- /dev/null
+++ b/ZdFJT4oBgHgl3EQf7i0F/content/tmp_files/load_file.txt
@@ -0,0 +1,937 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf,len=936
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='11678v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='OC]  27 Jan 2023  GENERALIZING QUASI-NEWTON UPDATES TO HIGHER-ORDER  DERIVATIVE TENSORS∗  KARL WELZEL† AND RAPHAEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' HAUSER†  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' At the heart of all quasi-Newton methods is an update rule that enables us to gradu-  ally improve the Hessian approximation using the already available gradient evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Theoretical  results show that the global performance of optimization algorithms can be improved with higher-  order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This motivates an investigation of generalizations of quasi-Newton update rules to  obtain for example third derivatives (which are tensors) from Hessian evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Our generalization  is based on the observation that quasi-Newton updates are least-change updates satisfying the secant  equation, with different methods using different norms to measure the size of the change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We present  a full characterization for least-change updates in weighted Frobenius norms (satisfying an analogue  of the secant equation) for derivatives of arbitrary order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Moreover, we establish convergence of the  approximations to the true derivative under standard assumptions and explore the quality of the  generated approximations in numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' quasi-Newton methods, tensors, approximate derivatives, higher-order optimiza-  tion  MSC codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' 90C53, 65D25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Consider a nonconvex unconstrained optimization problem  for a function f : Rn → R  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1)  min  x∈Rn f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Recent work on worst-case global convergence rates discovered that the more de-  rivatives are provided at each iteration the fewer iterations are required to find an  approximate minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Birgin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' [2] showed that if the function is p times con-  tinuously differentiable with a Lipschitz continuous pth derivative and an oracle to  compute the first p derivatives at any point is provided, then an algorithm exists that  finds a point with ∥∇f(x)∥ ≤ ε in at most O(ε−(p+1)/p) oracle calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This result  generalized the known cases for p = 1 [20] and p = 2 [23] and was later extended by  Cartis, Gould and Toint [5], who also proved that this bound is sharp for algorithms  that minimize regularized Taylor models in each step such as the one used in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  These new insights (although largely theoretical as yet) motivate us to consider  ways of estimating higher-order derivatives from lower-order ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' A solid starting  point for formulas that estimate pth derivatives from (p − 1)st are quasi-Newton  update formulas, which estimate second derivatives from first derivatives (p = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The corresponding optimization methods use these Hessian approximations to build  approximate Taylor models, which enable rapid convergence to the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Note that this paper only considers the updating formulas and their properties  without proposing a method that uses them (for optimization or otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This  method-agnostic approach enables anyone wishing to incorporate approximate higher-  order derivatives into their algorithm to adapt the general results to their needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Pre-  vious papers concerned with approximate derivatives and higher-order models are ei-  ther using higher-order Taylor expansions to derive better conventional quasi-Newton  ∗This work is supported by the Hong Kong Innovation and Technology Commission (InnoHK  Project CIMDA)  †Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United King-  dom (welzel@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='uk, hauser@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  1  2  KARL WELZEL AND RAPHAEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' HAUSER  methods or use approximate derivatives in a higher-order optimization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' As  part of the first group, Wei, Li and Qi [29] considered a modification of the secant  equation, so that the new approximate second-order Taylor expansion interpolates  the function at the previous iterate, and show that this update predicts the curvature  information at the new iterate better, at least when evaluated in the direction of the  previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Their work was extended by Biglari, Hassan and Leong [1], who used a  fourth-order Taylor expansion to derive an even better modification, and by Enshaei  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' [10], who applied the same idea to diagonal Hessian approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  One of the very early papers on approximate higher-order models for optimization  is [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' In it, Schnabel and Chow constructed a fourth-order model that interpolates  the last few function and gradient values at every iteration and imposed a simple  low-rank structure on the third- and fourth-order tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This makes it possible to  efficiently minimize the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' A different approach was taken by Shi and Pan [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  They developed a predictor–corrector approach for the descent trajectory defined by  Newton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' First a predictor is determined using linesearch along the New-  ton direction, then a quadratic curve ϕ: R → Rn is computed that interpolates the  iterate, the predictor and their gradients (thereby approximating the descent trajec-  tory), and lastly the function is minimized along this curve to obtain the next iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  They showed that an implementation of this idea using the approximate Newton direc-  tions generated by quasi-Newton methods is able to outperform regular quasi-Newton  methods in numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Mart´ınez and Raydan [18] combined a third-order  model with a trust-region approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The third-order term is restricted to be a cer-  tain sum of n rank-one tensors which allows them to solve the model minimization  subproblem by minimizing n independent cubic polynomials in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Their  approach to computing this diagonal third-order term is similar to ours in that they  choose one which is closest to the difference of the previous two Hessian evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Most recently, Nesterov [22, 21] considered specialized algorithms for unconstrained  convex optimization based on a third-order Taylor expansion with a fourth-order reg-  ularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' He mentions that an explicit calculation of the third derivative can  be avoided since his algorithms only need the gradients of the model to solve the  subproblem, and the contribution of the tensor term to the gradients can either be  calculated through automatic differentiation or approximated with a finite-difference  formula using only first derivatives of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Our paper will be structured as follows: After introducing the tensor notation  we use in section 2, we will derive the tensor analogues of quasi-Newton updates in  section 3 and give a full characterization of these updates in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The char-  acterization includes an explicit formula and a useful recursive relationship between  successive approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Moreover, we will show that these updates exhibit a cer-  tain low-rank structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Section 5 contains results on the convergence of the approx-  imations to the exact derivative in the limit under certain conditions on the steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Lastly, in section 6 we present limited numerical experiments to verify the behaviour  predicted by the theory and discuss numerical limitations of this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' A p-tensor T of dimensions n1 × · · · × np is a multilinear map  Rn1 × · · · × Rnp → R, so that its evaluation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1)  T[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , sp],  si ∈ Rni  is linear in each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We denote the space of such p-tensors by Rn1⊗···⊗np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' If  n1 = · · · = np = n the space is denoted R⊗pn and T[s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , s] is abbreviated as T[s]p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The notation also allows to apply the tensor to q < p vectors, which then results  GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS  3  in a (p − q)-tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Moreover, we define the application of matrices W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , Wp of  appropriate dimensions to a tensor by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2)  �  T[W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , Wp]  �  [s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , sp] = T[W1s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , Wpsp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The outer product of a p1-tensor T1 with a p2-tensor T2 is defined as  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3)  (T1 ⊗ T2)[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , sp1+p2] = T1[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , sp1] · T2[sp1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , sp1+p2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  In particular, tensors of the form T = v1 ⊗ · · · ⊗ vp for vectors v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , vp ∈ Rn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  those where T[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , sp] = �n  i=1 vT  i si, are called elementary or rank-one tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  If all vectors are the same (v1 = · · · = vp = v), we abbreviate the notation above  to ⊗pv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' (Note that this notation is slightly inconsistent since 1-tensors should be  row vectors, but are represented by standard column vectors to simplify the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  This is why v[W ] = W T v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=')  For any T ∈ R⊗pn and any permutation σ ∈ Sp, let σ(T) ∈ R⊗pn be defined by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4)  σ(T)[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , sp] = T[sσ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , sσ(p)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  If σ(T) = T for all σ ∈ Sp, then T is called symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The space of all symmetric  p-tensors is denoted R⊗pn  sym .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The projection of R⊗pn onto R⊗pn  sym is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='5)  Psym(T) = 1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  �  σ∈Sp  σ(T),  see [14, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Just like matrices, tensors are fully characterized by their actions on basis vec-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This can be used to represent a p-tensor T ∈ R⊗pn as a p-dimensional array  (ti1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=',ip)1≤ij≤n,1≤j≤p where  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='6)  ti1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=',ip = T[ei1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , eip]  and  T =  n  �  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=',ip=1  ti1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=',ipei1 ⊗ · · · ⊗ eip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  There is a Frobenius inner product and a corresponding norm on these p-dimensional  arrays and by extension on R⊗pn, which we will denote by ⟨T1, T2⟩F and ∥T∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Note  that ⟨T, s1 ⊗ · · · ⊗ sp⟩F = T[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , sp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Another norm is the one induced by the  2-norm on Rn (Hackbusch [14] calls it the injective norm) which is defined by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='7)  ∥T∥2 =  max  ∥si∥2=1, 1≤i≤p|T[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , sp]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Both norms are invariant under orthogonal transformations, so that if Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , Qp ∈  Rn×n are orthogonal matrices, then T[Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , Qp] has the same Frobenius- and 2-  norm as T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' As for matrices, the 2-norm is bounded by the Frobenius norm:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='8)  ∥T∥2 =  max  ∥si∥2=1, 1≤i≤p|⟨T, s1 ⊗ · · · ⊗ sp⟩F | ≤ ∥T∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  In the last inequality we used Cauchy-Schwarz and the fact that ∥s1 ⊗ · · ·⊗ sp∥F = 1  if all si have unit 2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Using this setup we can now introduce the notation for higher order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Let f : Rn → R be a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The pth total derivative of f at x ∈ Rn is  denoted Dpf(x) and is recursively defined as the total derivative of Dp−1f where  4  KARL WELZEL AND RAPHAEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' HAUSER  D0f = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This gives a chain of linear maps which we can regard as one multilinear  map  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='9)  Dpf(x): Rn → (Rn → (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' (Rn → R))) = Rn × · · · × Rn  �  ��  �  p times  → R  making Dpf(x) a p-tensor of dimensions n×· · ·×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' By this definition the evaluation  of this p-tensor Dpf(x)[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , sp] is equal to the directional derivative of f at x  along directions s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , sp ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' For example, denoting the gradient by ∇f and the  Hessian by ∇2f we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='10)  D1f(x)[s1] = ∇f(x)T s1  and  D2f(x)[s1, s2] = sT  1 ∇2f(x)s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Moreover, Dpf(x) is symmetric, because partial derivatives commute (Schwarz’s the-  orem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The pth-order Taylor expansion of f at x evaluated at an offset s ∈ Rn can  be expressed in this notation as  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='11)  Tf,p(x, s) =  p  �  k=0  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='Dkf(x)[s]k ≈ f(x + s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We now turn to our derivation of the generalized quasi-Newton  update for higher derivatives by first introducing a few important conventional quasi-  Newton methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The most well-known update rule for quasi-Newton methods is the  BFGS method, which is often described as a rank-two update to the current Hessian  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' To motivate the generalization in this paper we take a different view  and describe BFGS and similar methods as choosing minimal updates that satisfy the  secant equation [9, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Let f ∈ C2(Rn) be twice continuously differentiable function  with gradient ∇f and Hessian ∇2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Assume that we are given some sequence of  points xk ∈ Rn for k ∈ N (possibly from minimizing f), the gradients ∇f(xk) at  each iterate and some symmetric initial Hessian approximation B0 ∈ Rn×n  sym .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Quasi-  Newton methods then update Bk at each step such that the new approximation  correctly predicts the change in gradients of the previous iteration, that is  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1)  Bk+1(xk+1 − xk) = ∇f(xk+1) − ∇f(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  This is called the secant equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We can write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1) more succinctly if we define  sk = xk+1 − xk and let �  Bk =  � 1  0 ∇2f(xk + tsk) dt be the Hessian of f averaged over  all points on the line from xk to xk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This gives the equivalent equation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2)  Bk+1sk = �  Bksk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Most quasi-Newton methods then prescribe that among all possible choices of  Bk+1 that are symmetric and satisfy the secant equation we take the one that is  closest to Bk in some norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The simplest such rule is called the Powell-symmetric-  Broyden (PSB) update and is defined as  (PSB)  Bk+1 = arg minB∈Rn×n  sym ∥B − Bk∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Bsk = �  Bksk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  It is derived from Broyden’s method [3] by adding the symmetry constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The Davidon-Fletcher-Powell (DFP) method [7, 12], even though it has been  proposed before PSB, can be understood as a way to make the PSB method scale-  invariant by choosing a weighted Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Let Wk = �  B−1/2  k  (or, in fact, any  GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS  5  nonsingular matrix with W −T  k  W −1  k  sk = �  Bksk) be the weight matrix, then the DFP  update is given by  (DFP)  Bk+1 = arg minB∈Rn×n  sym ∥W T  k (B − Bk)Wk∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Bsk = �  Bksk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  If we rescale the input to f with the nonsingular matrix A, so that ¯f(x) = f(Ax),  and also rescale the iterates using ¯xk = A−1xk, then the corresponding Hessian  approximations of ¯f determined by the DFP method satisfy ¯  Bk = AT BkA as long  as it holds for the initial choice ¯  B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Finally, the famous BFGS method named after Broyden, Fletcher, Goldfarb and  Shanno [4, 11, 13, 27], is the dual of DFP in the sense that the new approximation  minimizes the difference between inverse matrices in a weighted Frobenius norm:  (BFGS)  Bk+1 = arg minB∈Rn×n  sym ∥W −T  k  �  B−1 − B−1  k  �  W −1  k  ∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Bsk = �  Bksk  The weight matrices Wk are the same as above and in the same way they make the  method scale invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' For more on these updating rules, consult the textbook by  Dennis and Schnabel [9, Chapter 9]  Using this characterization as least-change updates allows a straightforward gen-  eralization to tensors, except for the BFGS update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Since tensors lack the concept of  an inverse tensor, (BFGS) cannot be used, and we will focus on (PSB) and (DFP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We  now need to assume that f : Rn → R is p times continuously differentiable and that as  before a sequence of points xk with a corresponding sequence of steps sk = xk+1 −xk  is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We will denote the approximations to Dpf(xk) by Ck ∈ R⊗pn  sym and the true  pth derivative averaged over all points on the line from xk to xk+1 by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3)  �Ck =  � 1  0  Dpf(xk + tsk) dt ∈ R⊗pn  sym .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  This means that, in particular, �Ck[sk] = Dp−1f(xk+1) − Dp−1f(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The pth-order  analogue of the secant equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2) is given by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4)  Ck+1[sk] = Dp−1f(xk+1) − Dp−1f(xk) = �Ck[sk]  and the generalized quasi-Newton update formula for Ck reads  (GQN)  Ck+1 = arg minC∈R⊗pn  sym ∥(C − Ck)[Wk]p∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' C[sk] = �Ck[sk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  It provides a sequence of approximations of the pth derivative of f based solely on  evaluations of the (p − 1)st derivative at the iterates xk, given some initial approxi-  mation C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Note that unlike for the DFP update we will not assume any specific choice  of weight matrices, but rather consider them to be a given sequence of nonsingu-  lar matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' In particular, that covers the generalized PSB (Wk = I) and DFP  (W −T  k  W −1  k  sk = ∇f(xk+1) − ∇f(xk)) updates, which simplify to (PSB) and (DFP)  for p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Characterization of generalized quasi-Newton updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The following  theorem provides a full characterization of the update in (GQN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Let C• ∈ R⊗pn  sym (the current approximation), �C ∈ R⊗pn  sym (the inte-  grated true derivative), a nonsingular matrix W ∈ Rn×n (the weight matrix) and a  nonzero s ∈ Rn (the step) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The following equations all have the same unique  solution C+ ∈ R⊗pn  sym (the new approximation):  6  KARL WELZEL AND RAPHAEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' HAUSER  (a) C+ = arg minC∈R⊗pn  sym ∥(C − C•)[W ]p∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' C[s] = �C[s]  (b) C+ = C• + �p  j=1(−1)j�p  j  �  (vT s)−jPsym  ��  ⊗jv  �  ⊗ (C• − �C)[s]j�  (c) C+ = C• + Psym(A ⊗ v) for a (unique) A ∈ R⊗p−1n  sym  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' C+[s] = �C[s]  (d) (C+ − �C)[W ]p = (C• − �C)[W ]p�  I − W −1ssT W −T  sT W −T W −1s  �p  where v = W −T W −1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We will first prove the result for W = I and then see how that implies  the full result for any nonsingular weight matrix W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The first characterization can be rewritten as  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1)  C+ = C• + arg minU∈R⊗pn  sym ∥U∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' U[s] = (�C − C•)[s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Let Q ∈ Rn×n be an orthogonal matrix that maps e1 to some scalar multiple of  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  This means U[Q]p has the same Frobenius norm as U and U[Q]p[ei1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , eip]  is fully determined by the equality constraint in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1) if 1 ∈ {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , ip} because of  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' On the other hand, if 1 /∈ {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , ip} there are no constraints on the  values of U[Q]p[ei1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , eip], so the unique choice that minimizes ∥U[Q]p∥F = ∥U∥F  is clearly to set all of these values to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Therefore, there is a unique solution to the  minimization problem in (a), and it is fully characterized by the fact that the update  tensor U = C+ − C• is symmetric and satisfies the following two properties:  U[s] = (�C − C•)[s]  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2a)  U[u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , up] = 0 if all ui are orthogonal to s  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2b)  We will use this as the basis to show the equivalence with all other characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  In (b) we claim that the update U has the form  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3)  p  �  j=1  (−1)j�p  j  �  ∥s∥−2j  2  Psym  ��  ⊗js  �  ⊗ (C• − �C)[s]j�  This tensor is clearly symmetric since it is a sum of symmetric tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' To show that  it satisfies property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2a) consider  p  �  j=1  (−1)j�p  j  �  ∥s∥−2j  2  Psym  ��  ⊗js  �  ⊗ (C• − �C)[s]j�  [s]  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4a)  =  p  �  j=1  (−1)j∥s∥−2j  2  ��p−1  j−1  �  Psym  ��  ⊗j−1s  �  ⊗ (C• − �C)[s]j�  ∥s∥2  2  +  �p−1  j  �  Psym  ��  ⊗js  �  ⊗ (C• − �C)[s]j+1��  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4b)  = −Psym  �  (C• − �C)[s]  �  = (�C − C•)[s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4c)  The first equality uses the fact that each summand is the symmetric projection of an  outer product of two symmetric tensors, a j-tensor and a (p − j)-tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Therefore,  there are  �p  j  �  distinct ways to orient this outer product and for  �p−1  j−1  �  of them the vector  s is applied to the j-tensor and for  �p−1  j  �  to the (p − j)-tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' A close examination  of the expression on the second line shows that it is a telescoping sum where all terms  except the ones with coefficients  �p−1  0  �  and  �p−1  p  �  cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Because  �p−1  p  �  = 0 the  GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS  7  only remaining term is the negative symmetric projection of (C• − �C)[s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Property  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2b) follows from a similar consideration to the one above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' If u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , up are all  orthogonal to s, then  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='5)  Psym  ��  ⊗js  �  ⊗ (C• − �C)[s]j�  [u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , up] = 0  for j ≥ 1 because no matter how the outer product is oriented there is always a factor  of sT ui = 0 in the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  For (c) we need to show that we can always write the update in the form Psym(A⊗  s) for some symmetric (p−1)-tensor A and that A is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Note that the expression  for the update in (b) is already of the form Psym(A ⊗ s):  p  �  j=1  (−1)j�p  j  �  ∥s∥−2j  2  Psym  ��  ⊗js  �  ⊗ (C• − �C)[s]j�  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='6a)  = Psym  \uf8eb  \uf8edPsym  \uf8eb  \uf8ed  p  �  j=1  (−1)j�p  j  �  ∥s∥−2j  2  �  ⊗j−1s  �  ⊗ (C• − �C)[s]j  \uf8f6  \uf8f8 ⊗ s  \uf8f6  \uf8f8  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='6b)  It remains to show that among all updates of the form Psym(A ⊗ s) there is only one  choice of A ∈ R⊗pn  sym such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Consider the linear map  φ: R⊗p−1n  sym  → R⊗p−1n  sym  A �→ Psym(A ⊗ s)[s]  which maps a finite-dimensional vector space to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Combining what we already  showed for (b) with the observation that the update in (b) is of the desired form, this  map is surjective (we can prescribe any (�C−C•)[s]) and so it must be bijective, which  shows uniqueness of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Lastly, (d) claims that the update can be written as  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='7)  (�C − C•) − (�C − C•)  �  I − ssT  ∥s∥2  2  �p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Clearly, property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2a) is satisfied because applying this tensor to s makes the second  term vanish, leaving only the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Moreover, for any vector u that is  orthogonal to s the matrix I − ssT  ∥s∥2  2 maps u to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This means applying the tensor  above to u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , up, all of which are orthogonal to s, will give zero because both  terms cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This is property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Now that the equivalence has been established for W = I, we consider the general  case where W ∈ Rn×n is any nonsingular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The minimization in (a)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='8)  C+ = arg minC∈R⊗pn  sym ∥(C − C•)[W ]p∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' C[s] = �C[s]  can be rewritten using D• = C•[W ]p, �D = �C[W ]p, D+ = C+[W ]p and r = W −1s as  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='9)  D+ = arg minD∈R⊗pn  sym ∥D − D•∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' D[r] = �D[r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Applying the existing characterizations and the fact that C+ = D+[W −1]p we get the  claim after some algebraic manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  8  KARL WELZEL AND RAPHAEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' HAUSER  Note that although the theorem assumes �C ∈ R⊗pn and W ∈ Rn×n to be given  the resulting C+ only depends on �C[s] ∈ R⊗p−1n  sym  and W −T W −1s ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This makes  the method applicable even when no pth derivatives exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Moreover, the equivalence between (a) and (c) shows that there is a one-to-one  correspondence between updates that exhibit a certain low-rank structure Psym(A⊗v),  where v is any vector which is not orthogonal to s, and those updates that are minimal  in some weighted Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' If we apply this result to the matrix case (p = 2)  both A and v are 1-tensors, so that Psym(A ⊗ v) is the symmetric projection of a  rank-one matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Any minimal update (such as PSB or DFP) is therefore a symmetric  matrix of rank at most two and any update matrix of type vwT + wvT is minimal in  a certain weighted Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Convergence of approximate derivates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Now that we know how the up-  dates look like, we want to show that the approximate pth derivatives converge to the  true derivative under certain assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This means that, even though we previ-  ously pointed out that the update is applicable when no pth derivatives are available,  we will assume in this section that f is p-times continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The main  tool of this section is the characterization Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1 (d) which, when applied to  (GQN), becomes  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1) (Ck+1 − �Ck)[Wk]p = (Ck − �Ck)[Wk]p[Pk]p where Pk = I − W −1  k  sksT  k W −T  k  sT  k W −T  k  W −1  k  sk  Note that Pk is the orthogonal projection onto the orthogonal complement of W −1  k  sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Convergence for pth-order polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' When Wk and �Ck are con-  stant, convergence is quite straightforward to show, and we do not need to assume  convergence of the iterates xk:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Let C0 ∈ R⊗pn  sym be given and update the approximations Ck ac-  cording to (GQN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Assume Dpf(x) = C∗ everywhere (which makes f a pth order  polynomial) and Wk = W∗ for every k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' After n steps sk such that W −1  ∗  sk are  orthogonal, we have Cn = C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Under the assumptions of the theorem, repeated application of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1) gives  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2)  (Cn − C∗)[W∗]p = (C0 − C∗)[W∗]p  �n−1  �  k=0  Pk  �p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Because W −1  0  s0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , W −1  n−1sn−1 are orthogonal and Pk are orthogonal projections  on their orthogonal complements, �n−1  k=0 Pk = 0, so that (Cn − C∗)[W∗]p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This  implies Cn = C∗ because W∗ is nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Bounded deterioration property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' For a convergence results for general  functions f we first need to establish two lemmas that will help us when xk → x∗ and  Wk → W∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The first one shows that �Ck converges to the pth derivative at x∗, which  we denote by C∗ = Dpf(x∗), and the second one gives a bound on the error term that  we incur if we replace �Ck by C∗ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Combining the two gives what Dennis and  Schnabel [9] call the bounded deterioration principle in that Ck+1 can only be slightly  worse than Ck at approximating C∗ as long as xk and xk+1 are close enough to x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  1The symmetric rank-one (SR1) update fits the latter description with v = w, but the corre-  sponding weight matrix depends on the current Hessian approximation, so it does not generalize to  higher orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS  9  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' For xk → x∗ we have �Ck → C∗ as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Moreover, if Dpf is  Lipschitz continuous with constant L, then  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3)  ∥�Ck − C∗∥2 ≤ L  2 (∥xk − x∗∥2 + ∥xk+1 − x∗∥2)  for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' By definition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3) we have  ∥�Ck − C∗∥2 = ∥  � 1  0  Dpf(xk + tsk) dt − Dpf(x∗)∥2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4a)  ≤  � 1  0  ∥Dpf(xk + tsk) − Dpf(x∗)∥2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4b)  Since Dpf is continuous the right-hand side will become arbitrarily small as xk and  xk+1 converge to x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  This shows �Ck → C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  If we additionally assume Lipschitz  continuity of Dpf, we can bound the integrand in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4b):  ∥�Ck − C∗∥2 ≤  � 1  0  L∥xk + tsk − x∗∥2 dt  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='5a)  ≤ L  � 1  0  (1 − t)∥xk − x∗∥2 + t∥xk − x∗∥2 dt  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='5b)  = L  2 (∥xk − x∗∥2 + ∥xk+1 − x∗∥2)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='5c)  This gives the second claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' If we define the error tensor Ek by  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='6)  (Ck+1 − C∗)[Wk]p = (Ck − C∗)[Wk]p[Pk]p + Ek[Wk]p  then ∥Ek∥2 ≤ (1 + κ2(Wk)p)∥�Ck − C∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Subtracting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1) from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='6) gives  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='7)  (�Ck − C∗)[Wk]p = (�Ck − C∗)[Wk]p[Pk]p + Ek[Wk]p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Multiply both sides by W −1  k  from all sides and rearrange to find  ∥Ek∥2 = ∥(�Ck − C∗) − (�Ck − C∗)[Wk]p[Pk]p[W −1  k  ]p∥2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='8a)  ≤  �  1 + ∥WkPkW −1  k  ∥p  2  �  ∥�Ck − C∗∥2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='8b)  ≤ (1 + κ2(Wk)p)∥�Ck − C∗∥2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='8c)  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' In the last step we used ∥Pk∥2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Convergence for weakly orthogonal steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Since the updates only de-  termine Ck[sk] in each step, we can never hope to recover the true pth derivative if  (asymptotically) all steps lie in a low-dimensional subspace of Rn, so we must assume  a weak orthogonality condition on the steps, namely  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='9)  �����  k0+m−1  �  k=k0  �  I − W −1  ∗  sksT  k W −T  ∗  sT  k W −T  ∗  W −1  ∗  sk  ������  2  ≤ c < 1  10  KARL WELZEL AND RAPHAEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' HAUSER  for some fixed m ∈ N, c ∈ R≥0 and all k0 ∈ N large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Here, W∗ = limk→∞ Wk  which we assume to be nonsingular as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' As Mor´e and Trangenstein [19] showed,  this assumption is equivalent to uniform linear independence of the scaled steps  W −1  ∗  sk which is in turn equivalent to the standard assumption of uniform linear  independence of the steps sk themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  This leads to the following convergence  theorem:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Let C0 ∈ R⊗pn  sym be given and update the approximations Ck accord-  ing to (GQN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Assume xk converge to x∗ ∈ Rn, Wk converge to some nonsingular  matrix W∗ ∈ Rn×n and the steps are uniformly linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Then Ck con-  verges to C∗ := Dpf(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  We need a technical lemma for this result, which shows that the projection ma-  trices Pk get closer and closer to the projection matrices that appear in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='9), which  we will abbreviate with P ∗  k = I − W −1  ∗  sksT  k W −T  ∗  sT  k W −T  ∗  W −1  ∗  sk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Let the nonsingular matrices Wk ∈ Rn×n converge to the nonsingu-  lar matrix W∗ ∈ Rn×n then  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='10)  ∥Pk − P ∗  k ∥2 → 0  and  ∥W −1  ∗  WkPkW −1  k  W∗ − P ∗  k ∥2 → 0  holds for any sequence of steps (sk)k∈N as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Without loss of generality we can assume that the steps are scaled such  that ∥W −1  ∗  sk∥2 = 1 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' That means  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='11)  ∥W −1  k  sk − W −1  ∗  sk∥2 ≤ ∥W −1  k W∗ − I∥2  �  ��  �  →0  ∥W −1  ∗  sk∥2  �  ��  �  =1  → 0  as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Since Pk projects onto the subspace that is orthogonal to W −1  k  sk and P ∗  k projects onto  the subspace that is orthogonal to W −1  ∗  sk, the difference between the two projection  matrices converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This is the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  For the second one we find that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='12)  W −1  ∗  WkPkW −1  k  W∗ − P ∗  k = (W −1  ∗  Wk − I)  �  ��  �  →0  Pk W −1  k  W∗  �  ��  �  →I  + (Pk − P ∗  k )  �  ��  �  →0  W −1  k  W∗  �  ��  �  →I  +P ∗  k (W −1  k  W∗ − I)  �  ��  �  →0  converges to 0 since the norms of Pk and P ∗  k are at most one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' It suffices to consider the case when W∗ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Otherwise,  let ¯f(x) = f(W∗x), ¯xk = W −1  ∗  xk, ¯x∗ = W −1  ∗  x∗, ¯  Wk = W −1  ∗  Wk and ¯C0 = C0[W∗]p  and update ¯Ck according to the adapted (GQN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' By construction, we then have  Ck = ¯Ck[W −1  ∗  ]p so that ¯Ck → Dp ¯f(¯x∗) implies Ck → Dpf(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Let ε > 0 be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' To prove convergence, we will show ∥Ck − C∗∥2 ≤ ε for  every k large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Consider the recurrence relation for Ck − C∗ established in  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We can rearrange it to read  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='13)  (Ck+1 − C∗) = (Ck − C∗)[WkPkW −1  k  ]p + Ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Abbreviate WkPkW −1  k  by P ′  k and note that P ′  k is no longer an orthogonal projection  but gets closer and closer to one as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' In particular, its norm approaches one  GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS  11  since ∥P ′  k∥2 ≤ κ2(Wk) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  To use the assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='9) later on we apply the  previous equality m times and get  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='14)  (Ck0+m − C∗) = (Ck0 − C∗)  �k0+m−1  �  k=k0  P ′  k  �p  +  k0+m−1  �  k=k0  Ek  �k0+m−1  �  l=k+1  P ′  l  �p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Let K1 ∈ N be such that for all k ≥ K1 we have κ2(Wk) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This means using  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3 we can bound the second term on the right-hand side of the previous  equation by  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='15)  �����  k0+m−1  �  k=k0  Ek  �k0+m−1  �  l=k+1  P ′  l  ������  2  ≤  k0+m−1  �  k=k0  2p(1 + 2p)∥�Ck − C∗∥  for all k0 ≥ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Since Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2 showed that �Ck → C∗ the bound above is smaller  than ε/2 · (1 − ( 1+c  2 )p) for k0 large enough, say k0 ≥ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Next, consider first term on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Let P ∗  k be the orthog-  onal projection matrices from the assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='9) as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Note that P ′  k ̸= P ∗  k , so  we cannot apply the assumption directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Instead, we look at the difference between  the product of P ′  k and P ∗  k and find  �����  k0+m−1  �  k=k0  P ′  k −  k0+m−1  �  k=k0  P ∗  k  �����  2  =  �����  k0+m−1  �  k=k0  P ∗  k0 · · · P ∗  k−1(P ′  k − P ∗  k )P ′  k+1 · · · P ′  k0+m−1  �����  2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='16a)  ≤  k0+m−1  �  k=k0  2m−1∥P ′  k − P ∗  k ∥2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='16b)  for k0 ≥ K1 using ∥P ′  k∥ ≤ 2 and ∥P ∗  k ∥ ≤ 1 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' For k0 large enough, say k0 ≥ K3,  the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='16) is smaller than (1 − c)/2 by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Similarly, for k0  large enough, say k0 ≥ K4, we have ∥�k0+m−1  k=k0  P ∗  k ∥ ≤ c by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Therefore,  for k0 ≥ max{K3, K4}  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='17)  �����  k0+m−1  �  k=k0  P ′  k  ����� ≤  �����  k0+m−1  �  k=k0  P ′  k −  k0+m−1  �  k=k0  P ∗  k  �����  2  +  �����  k0+m−1  �  k=k0  P ∗  k  ����� ≤ 1 + c  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  In the previous two paragraphs we have established that asymptotically for every  m steps the norm of Ck − C∗ is first multiplied by a factor that is at most slightly  larger than c and then increased by an arbitrarily small error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' In particular for  k0 ≥ K = max{K2, K3, K4} we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='18)  ∥Ck0+m − C∗∥2 ≤ ∥Ck0 − C∗∥2  �1 + c  2  �p  + ε/2 ·  �  1 −  �1 + c  2  �p�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Repeatedly applying this inequality shows that ∥Ck0+im − C∗∥2 is bounded by a  sequence (ai)i∈N which converges to ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Use this observation for all k0 ∈ {K, K +  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' , K + m − 1} to see that ∥Ck − C∗∥2 ≤ ε for all k large enough, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Let C0 ∈ R⊗pn  sym be given and update the approximations Ck ac-  cording to (GQN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Assume the steps all lie in a d-dimensional subspace spanned by  12  KARL WELZEL AND RAPHAEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' HAUSER  the columns of V ∈ Rn×d, so that the iterates are of the form xk = V ¯xk + x⊥  where V T x⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Moreover, assume that ¯xk converge to ¯x∗ ∈ Rd (or equiva-  lently xk converge to x∗ = V ¯x∗ + x⊥), Wk converge to some nonsingular matrix  W∗ ∈ Rn×n and ¯sk = ¯xk+1 − ¯xk are uniformly linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Then Ck con-  verges to C∗ := Dpf(x∗) in the subspace: Ck[V ]p → C∗[V ]p as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We will construct a sequence of ¯Ck to which we can apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4  and then show ¯Ck = Ck[V ]p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Let ¯f : Rd → R be defined by ¯f(¯x) = f(V ¯x + x⊥)  which implies  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='19)  Dp ¯f(¯x) = Dpf(V ¯x + x⊥)[V ]p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Additionally, let the weight matrices ¯  Wk be defined by ¯  Wk = (V T W −T  k  W −1  k  V )−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The operation B �→ B1/2 maps a symmetric positive semidefinite matrix B to the  unique symmetric semidefinite matrix H such that HT H = B, and is continuous  (see [17, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Here, we know that V T W −T  k  W −1  k  V is positive definite (V  is full-rank), so ¯  Wk is also positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Since the construction of ¯  Wk depends  continuously on Wk, the newly constructed weight matrices converge to a positive  definite matrix ¯  W∗ given by the same formula applied to W∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' That means if we now  define ¯C0 = C0[V ]p and apply (GQN) using ¯f, ¯xk and ¯  Wk, the resulting sequence  ¯Ck will converge to Dp ¯f(¯x∗) by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Clearly, the limit point Dp ¯f(¯x∗) = Dpf(x∗)[V ]p is already correct, it just remains  to show that Ck[V ]p = ¯Ck for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' By construction of ¯  Wk we get  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='20)  ¯  W −T  k  ¯  W −1  k  ¯sk = V T W −T  k  W −1  k  V ¯sk = V T W −T  k  W −1  k  sk  where sk = V ¯sk are steps in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1 (c) shows that Ck+1 = Ck + Psym(A ⊗  W −T W −1sk) for some (p − 1)-tensor A ∈ R⊗p−1n  sym  , where A is chosen such that  Ck+1[sk] = Dp−1f(xk+1) − Dp−1f(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This implies  Ck+1[V ]p = Ck[V ]p + Psym(A ⊗ W −T W −1sk)[V ]p  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='21a)  = Ck[V ]p + Psym(A[V ]p−1 ⊗ V T W −T W −1sk)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='21b)  = Ck[V ]p + Psym(A[V ]p−1 ⊗ ¯  W −T  k  ¯  W −1  k  ¯sk)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='21c)  and Ck+1[V ]p[¯sk] = Dp−1 ¯f(¯xk+1) − Dp−1 ¯f(¯xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Again by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1 (c) ¯Ck and  Ck[V ]p are updated in the same way, so inductively they define the same sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Generalized Dennis–Mor´e condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Dennis and Mor´e [8] showed that  if optimization methods choose their iterates using xk+1 = xk − B−1  k ∇f(xk) and  converge to x∗ then this convergence is Q-superlinear if and only if  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='22)  lim  k→∞  ∥(Bk − ∇2f(x∗))sk∥F  ∥sk∥2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  This equation is called the Dennis–Mor´e condition and is weaker than convergence of  Bk to ∇2f(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Instead, it suffices when Bk converges in the step directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  We are not concerned with convergence rates for any particular optimization  method in this paper, but it seems natural to ask whether the approximations Ck  satisfy a generalized Dennis–Mor´e condition  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='23)  lim  k→∞  ∥(Ck − Dpf(x∗))[sk]∥F  ∥sk∥2  = 0  when they are updated according to (GQN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS  13  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Let C0 ∈ R⊗pn  sym be given and update the approximations Ck ac-  cording to (GQN) where the function has a Lipschitz continuous pth derivative Dpf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Assume xk converge to x∗ ∈ Rn and Wk converge to some nonsingular matrix  W∗ ∈ Rn×n fast enough such that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='24)  �  k≥0  ∥xk − x∗∥2 < ∞  and  �  k≥0  ∥Wk − W∗∥2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Then the generalized Dennis–Mor´e condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='23) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' As in the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4 we assume W∗ = I without loss of gen-  erality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Since the condition number is locally Lipschitz continuous around I, the  assumption �  k≥0∥Wk − I∥2 < ∞ also implies Cκ := �  k≥0(κ2(Wk) − 1) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  As a first step we need to use the bounded deterioration principle to show that  ∥Ck − C∗∥2 stays bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' For this we again consider the m-step recursive formula  established in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='14):  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='25)  (Cm − C∗) = (C0 − C∗)  �m−1  �  k=0  P ′  k  �p  +  m−1  �  k=0  Ek  � m−1  �  l=k+1  P ′  l  �p  where P ′  k = WkPkW −1  k  and Ek is defined in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Clearly,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='26)  ln  ������  m−1  �  k=0  P ′  k  �����  2  �  ≤  m−1  �  k=0  ln(∥P ′  k∥2) ≤  m−1  �  k=0  ln(κ2(Wk)) ≤  m−1  �  k=0  (κ2(Wk) − 1) ≤ Cκ  is uniformly bounded for all m, which means the same is true for ∥�m−1  k=0 P ′  k∥2 itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Therefore,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='27)  ∥Cm − C∗∥2 ≤ exp(Cκ)p  �  ∥C0 − C∗∥2 +  m−1  �  k=0  ∥Ek∥2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Because Dpf is Lipschitz continuous and κ2(Wk) stays bounded, Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3  show that there is a constant CE < ∞ such that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='28)  ∥Ek∥2 ≤ CE/2(∥xk − x∗∥ + ∥xk+1 − x∗∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  This implies �m−1  k=0 ∥Ek∥2 ≤ CE  �m  k=0∥xk − x∗∥2 is uniformly bounded for all m and  therefore ∥Cm − C∗∥2 is as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The next step is crucial for this proof and similar to the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2  in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We note that in the Frobenius norm for any tensor T ∈ R⊗pn and any nonzero  vector v ∈ Rn we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='29)  ∥T∥2  F =  ����T  �vvT  vT v  �����  2  F  +  ����T  �  I − vvT  vT v  �����  2  F  = ∥T[v]∥2  F  ∥v∥2  2  +  ����T  �  I − vvT  vT v  �����  2  F  because the matrices in brackets are orthogonal projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We can apply this to the  14  KARL WELZEL AND RAPHAEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' HAUSER  case where I − vvT /(vT v) is Pk to get  ∥(Ck − C∗)[Wk]p[Pk]p∥  2  F  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='30a)  ≤ ∥(Ck − C∗)[Wk]p[Pk]∥2  F  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='30b)  = ∥(Ck − C∗)[Wk]p∥2  F −  ��(Ck − C∗)[Wk]p�  W −1  k  sk  ���2  F  ∥W −1  k  sk∥2  2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='30c)  = ∥(Ck − C∗)[Wk]p∥2  F −  ���(Ck − C∗)[sk][Wk]p−1���  2  F  ∥W −1  k  sk∥2  2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='30d)  ≤ ∥(Ck − C∗)[Wk]p∥2  F − ∥(Ck − C∗)[sk]∥2  F  ∥sk∥2  2  ∥W −1  k ∥−2p  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='30e)  Adding Ek[Wk]p into the Frobenius norm on the left-hand side gives  ∥(Ck+1 − C∗)[Wk]p∥2  F = ∥(Ck − C∗)[Wk]p[Pk]p + Ek[Wk]∥2  F  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='31a)  = ∥(Ck − C∗)[Wk]p[Pk]p∥2  F + ∥Ek[Wk]p∥2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='31b)  The inner product term ⟨(Ck − C∗)[Wk]p[Pk]p, Ek[Wk]p⟩F is missing in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='31b)  because it is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' To show that, note that we can rewrite (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='7) from the proof of  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3 as  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='32)  (�Ck − Ek − C∗)[Wk]p = (�Ck − C∗)[Wk]p[Pk]p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Using the equivalence between Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1 (d) and (c) the error tensor can be written  explicitly as −Ek = Psym(A ⊗ W −T  k  W −1  k  sk) for some (p − 1)-tensor A and  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='33)  Ek[Wk]p = −Psym(A[Wk]p−1 ⊗ W −1  k  sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  In other words, Ek[Wk]p can be expressed as a sum of outer products between W −1  k  sk  and some (p−1)-tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Any inner product of such a tensor with (Ck − C∗)[Wk]p[Pk]p  must be zero as Pk maps W −1  k  sk to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='30) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='31) and rearranging gives  ∥(Ck − C∗)[sk]∥2  F  ∥sk∥2  2  ≤ ∥W −1  k  ∥2p  2  �  ∥(Ck − C∗)[Wk]p∥2  F  − ∥(Ck+1 − C∗)[Wk]p∥2  F + ∥Ek[Wk]p∥2  F  �  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='34a)  ≤ ∥(Ck − C∗)∥2  Fκ2(Wk)2p − ∥(Ck+1 − C∗)∥2  F + ∥Ek∥2  Fκ2(Wk)2p  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='34b)  Lastly, we wish to sum up both sides of the previous inequality over all k ≥ 0 and  show that the right-hand side stays bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This immediately gives the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' A  few technicalities are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We already showed that Ck − C∗ stays bounded, so let  C∆ < ∞ be a constant such that ∥Ck−C∗∥F ≤ C∆ for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' As established above,  �  k≥0(κ2(Wk) − 1) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This also implies that �  k≥0  �  κ2(Wk)2p − 1  �  = Cκ,2p < ∞  for some constant Cκ,2p since κ2(Wk)2p − 1 ≤ 4p(κ2(Wk) − 1) for κ2(Wk) small  enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS  15  Consider the Ck − C∗ terms on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='34) first:  m−1  �  k=0  �  ∥(Ck − C∗)∥2  F κ2(Wk)2p − ∥(Ck+1 − C∗)∥2  F  �  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='35a)  = ∥C0 − C∗∥2  F κ2(W0)2p  �  ��  �  constant  +  m−1  �  k=1  �  κ2(Wk)2p − 1  �  �  ��  �  =Cκ,2p  ∥Ck − C∗∥2  F  �  ��  �  ≤C2  ∆  − ∥Cm − C∗∥2  F  �  ��  �  ≥0  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='35b)  is uniformly bounded for all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The same is true for the Ek term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Because the  Frobenius norm and the 2-norm for p-tensors are both norms on finite-dimensional  vector spaces they are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Moreover, κ2(Wk) stays bounded, so for some  constant C  m−1  �  k=0  ∥Ek∥2  F κ2(Wk)2p ≤ C  m−1  �  k=0  ∥Ek∥2  2 ≤ C  �m−1  �  k=0  ∥Ek∥2  �2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='36a)  holds, and the term is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Therefore,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='37)  �  k≥0  ∥(Ck − C∗)[sk]∥2  F  ∥sk∥2  2  < ∞  and  ∥(Ck − C∗)[sk]∥F  ∥sk∥2  → 0 as k → ∞  as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' While the previous sections investigated the the-  oretical properties of the generalized quasi-Newton updates, we now turn to numerical  experiments to understand how quickly convergence of the approximations sets in and  how the algorithm behaves for different kinds of iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This is not supposed to be  a comprehensive treatment of the numerical performance of the algorithm, but rather  give an idea of its general behaviour by considering a small toy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The implementation was done in Python using NumPy [15] and uses the explicit  formula in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1 (b) at its core:  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1)  Ck+1 = Ck +  p  �  j=1  (−1)j�p  j  ��  vT  k sk  �−jPsym  ��  ⊗jvk  �  ⊗ (Ck[sk] − Dk)[sk]j−1�  where vk = W −T  k  W −1  k  sk and Dk = Dp−1f(xk+1) − Dp−1f(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  This makes it  particularly easy to implement the analogues of (PSB) (where vk = sk) and (DFP)  (where vk = (∇f(xk+1) − ∇f(xk))/∥sk∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' To increase legibility and since these two  choices produce roughly similar approximations, only the PSB variant will be shown  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  We chose to use the two-dimensional Rosenbrock function f(x, y) = (1 − x)2 +  100(y − x2)2 as our objective function because it is a simple, yet widely used test  function with nonconstant third derivative  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2)  D3f(x, y) =  ��  −2400x  −400  −400  0  �  �  −400  0  0  0  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The iterates xk are generated by different minimization algorithms starting at (0, 0)  and converging to the global minimum of f at x∗ = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Lastly, the initial approx-  imation C0 is just the zero 3-tensor in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  16  KARL WELZEL AND RAPHAEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' HAUSER  0  10  20  30  40  Iteration k  10−14  10−12  10−10  10−8  10−6  10−4  10−2  100  ∥Ck − D3f(xk)∥F /∥D3f(xk)∥F  ∥xk − x∗∥2/∥x∗∥2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Convergence of iterates and approximations for nonlinear CG  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Numerical limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' In the first experiment, a nonlinear CG method2  was used to minimize the Rosenbrock function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Although nonlinear CG methods  with restarts can achieve superlinear local convergence, this implementation does not  include restarts, and we observe roughly linear convergence of xk to x∗ in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The relative error of each Ck is not measured with respect to C∗ = D3f(1, 1) here  but instead with respect to the true third derivative at each iterate D3f(xk), since  this is the more relevant metric in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' From the convergence theorems in the  previous section we expect that this quantity will also converge to zero, since both  Ck and D3f(xk) converge to C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Unfortunately, although at first this seems to be true and the two error curves  roughly coincide, from iteration 27 onwards the error in Ck increases quite consid-  erably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The issue, as it turns out, stems from rounding errors in finite precision  arithmetic, which we did not consider in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Specifically the computation of  Dk becomes more ill-conditioned the smaller the step sk is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  In each iteration, the current approximation Ck moves closer to the integrated  derivative �Ck, so we cannot expect Ck to approximate D3f(xk) better than �Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Of  course, the only part of �Ck which is used is its component in the direction of sk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Dk/∥sk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' In exact arithmetic the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2 also shows that  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3)  Dk  ∥sk∥2  = �Ck[s→  k ] = D3f(xk)[s→  k ] + ∆Dk  with  ∥∆Dk∥2 ≤ L  2 ∥sk∥2  where s→  k is the normed step sk/∥sk∥2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' the unit norm vector pointing in the same  direction as sk, and L is the (local) Lipschitz constant of Dpf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Now, let �Dk be the computed Dk = Dp−1f(xk+1)−Dp−1f(xk) under the influence  of rounding errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  As Higham [16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  9] explains, if we subtract two numbers  2scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='minimize(method="CG") in SciPy version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3, implementing the Polak–  Ribi`ere variant of nonlinear CG [24, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' 122]  GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS  17  0  10  20  30  40  Iteration k  10−14  10−12  10−10  10−8  10−6  10−4  10−2  100  ∥Ck − D3f(xk)∥F /∥D3f(xk)∥F  ∥xk − x∗∥2/∥x∗∥2  max{∥sk−1∥2, εmach/∥sk−1∥2}  √εmach  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Convergence of iterates and approximations for nonlinear CG (extended)  ˆa = a(1 + ∆a) and ˆb = b(1 + ∆b) from each other, and assume the relative errors ∆a  and ∆b are bounded by δ, the absolute error in the result is bounded by  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4)  |−a∆a + b∆b| ≤ δ(|a| + |b|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  The a and b in our case are the entries of Dp−1f(xk+1) and Dp−1f(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' They are  computed with the exact formulas, but stored in finite precision, so the best possible  error bound δ is the machine precision εmach ≈ 10−16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This gives  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='5)  �Dk  ∥sk∥2  =  Dk  ∥sk∥2  + ∆�Dk = D3f(xk)[s→  k ] + ∆Dk + ∆�Dk  where ∥∆�Dk∥F ≤  √  2εmach  �  ∥Dp−1f(xk+1)∥F + ∥Dp−1f(xk)∥F  �  /∥sk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='5) shows that there are two sources of error, one from using the  secant equation and one from calculating Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The former is proportional to ∥sk∥2  whereas the latter is proportional to 1/∥sk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  This leads to the V-shaped graph  in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' If we assume that Dp−1f, Dpf and L have roughly the same scale,  we can estimate that the lowest possible relative error in �Dk/∥sk∥2 (compared to  D3f(xk)[sk/∥sk∥2]) is √εmach and is achieved when ∥sk∥2 ≈ √εmach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This analysis  is analogous to one for numerical differentiation schemes where the same lower bound  is derived, see for example [25, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' One notable exception to the rule occurs  when Dp−1f(x∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' In that case ∥∆�Dk∥F stays close to machine precision and  the approximations get better and better as sk converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This is one of the  reasons that regular quasi-Newton methods (p = 2) work very well for optimization  algorithms as they converge to stationary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  In Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2 one can see that indeed the best relative error achieved is roughly  √εmach and that the maximum of ∥sk−1∥2 and εmach/∥sk−1∥2 is a pretty good proxy  for a lower bound on the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  18  KARL WELZEL AND RAPHAEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' HAUSER  0  2  4  6  8  10  12  14  16  Iteration k  10−14  10−12  10−10  10−8  10−6  10−4  10−2  100  ∥Ck − D3f(xk)∥F /∥D3f(xk)∥F  ∥(Ck − D3f(xk))[s→  k ]∥F /∥D3f(xk)∥F  ∥xk − x∗∥2/∥x∗∥2  max{∥sk−1∥2, εmach/∥sk−1∥2}  √εmach  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Convergence of iterates and approximations for exact trust region  Note that these numerical issues cannot be overcome with a different imple-  mentation but are inherent in this approach of extracting third-order information  from successive evaluations of second-order derivatives, since computing Dk is an  ill-conditioned problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' A practical way to avoid losing accuracy in the last few it-  erations would be to employ a heuristic that skips updating Ck when the expected  size of errors ∆�Dk exceeds the size of the update or simply when ∥sk∥2 becomes too  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Convergence in a subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' In the second experiment, we used a trust  region approach with exact Hessian evaluations to generate the iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3  As pre-  dicted by the theory for these methods, we observe local quadratic convergence to the  minimizer (and much fewer iterations in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' The dotted line shows that there  are very few iterations in which accurate information about the third derivatives can  be obtained, but even then the relative error in Ck is several orders of magnitude  larger than the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  A key difference in the two experiments is how the directions of the steps are  distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4 plots the angle of each sk with the x-axis, normalized between  0° and 180° so that opposite directions coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Whereas the steps generated by  the nonlinear CG algorithm cover multiple well-separated directions during the main  part of the algorithm, the steps generated by the trust region method tend to fall into  a one-dimensional subspace, especially towards the end when convergence happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  This directly explains why the relative Frobenius error stays high and even increases  towards the end in the second experiment: All the information we can extract from  the (averaged) true derivative �Ck is its evaluation in the direction sk and since most  of the steps point in the same direction at the end, the information about the other  3scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='minimize(method="trust-exact") in SciPy version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3, see [6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' 169–200]  for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Only the successful iterations were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  GENERALIZING QUASI-NEWTON UPDATES TO HIGHER ORDERS  19  0  20  40  Iteration k  0°  20°  40°  60°  80°  100°  120°  140°  160°  180°  Angle of sk with x-axis  0  5  10  15  Iteration k  Nonlinear CG  Exact trust region  Rosenbrock valley  subspace at (1, 1)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Subspaces of the steps for nonlinear CG and exact trust region  directions gets more and more outdated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  In addition to the relative Frobenius norm, we also included (a relative version of)  the Dennis–Mor´e measure from subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4 in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This one measures the  error in Ck only in the direction of the step sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Now, one might expect that when the  approximation is updated in one specific direction and the Dennis–Mor´e error only  measures how good the approximation is in this one direction, the error must track  the lower bound derived in the previous subsection quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Indeed, this is the  case when the steps all lie exactly in one subspace as we could verify with manually  generated iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' For the exact trust region method however the step directions vary  by about 1° in successive iterations at the end, which can be seen in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Let  s→  k = λ1s→  k−1 + λ2uk where uk is chosen such that s→  k−1 and uk form an orthonormal  basis of R2, then λ2  1 + λ2  2 = 1 and by multilinearity of Ck,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='6)  Ck[s→  k ] = λ1Ck[s→  k−1] + λ2Ck[uk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  Therefore, in each of the last few iterations the approximation in direction s→  k  is  a linear combination of the very accurate information in direction s→  k−1 and very  inaccurate information in direction uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' In particular, if the angle with the x-axis varies  by about 1° we get that λ2 ≈ sin(1◦) ≈ 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Combining this with the knowledge that  the relative overall error in Ck is on the order of 1, we expect that the Dennis–Mor´e  measure will hover around 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This agrees very well with the graph in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='3 and  shows that it is important for this method to gather accurate derivative information  in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' We have seen in this paper that quasi-Newton updates described  as least-change updates admit fairly straightforward generalizations to higher-order  derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Moreover, they have a closed form solution with a certain low-rank struc-  ture to it, generalizing the rank-two characterization of regular quasi-Newton updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content='  20  KARL WELZEL AND RAPHAEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' HAUSER  The theoretical results suggest that, as long as the directions of the steps span the  space and stay well separated, the generated approximations converge to the true  derivative in the limit and under suitably fast convergence of the iterates they even  converge (in a subspace) if these assumptions are violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' This is however not the  behaviour we see in experiments, since the problem of computing the difference be-  tween Hessian evaluations becomes more and more ill-conditioned as the difference  between consecutive iterates becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' These numerical limitations lead to a  loss in accuracy: If the Hessians are computed with relative error δ we cannot expect  the errors in the generated approximations to go below  √  δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' Our experiments show  that, as long as the directions of the steps stay well separated, the method indeed  generates accurate approximations up to the numerical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
+page_content=' If the directions tend to  fall into a subspace, the derivative information in directions orthogonal to it quickly  gets outdated and can impact approximations as long as they do not all fall exactly  into the subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFJT4oBgHgl3EQf7i0F/content/2301.11678v1.pdf'}
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+Full control of the libration potential in rotational levitodynamics
+J. A. Zielińska,1, ∗ F. van der Laan,1 A. Norrman,1, 2 R. Reimann,1, 3 L. Novotny,1 and M. Frimmer1
+1Photonics Laboratory, ETH Zürich, CH-8093 Zürich, Switzerland
+2Center for Photonics Sciences, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
+3Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE
+Control of the potential energy and free evolution lie at the heart of levitodynamics as key re-
+quirements for sensing, wave function expansion, and mechanical squeezing protocols.
+Here, we
+experimentally demonstrate full control over the optical potential governing the librational degrees
+of freedom of a levitated anisotropic nanoparticle. This control is achieved by introducing the degree
+of polarization as a new tool for rotational levitodynamics. We demonstrate the free rotation of a
+levitated anisotropic scatterer around its short axis and and we use the rotational degrees of freedom
+to probe the local spin of a strongly focused laser beam.
+Introduction.—
+Levitodynamics is the science of con-
+trolling the motion of levitated mesoscopic objects [1].
+The field has received growing attention in the last
+decade as a platform for force, torque, and electric field
+sensing [2]. Next to the translational degrees of freedom,
+the rotational dynamics of levitated anisotropic bodies
+offer particularly promising opportunities. More specifi-
+cally, new functionalities demonstrated for optically levi-
+tated rotors include controllable diffusion [3], gyroscopic
+stabilization [4], spinning with GHz rotation rates [5–8],
+and the realization of rotational “washboard potentials”
+by carefully trading off conservative and non-conservative
+torques in elliptically polarized fields [3, 9].
+A particularly enticing prospect is to harness levitated
+rotors as ultra-sensitive torque sensors [7], in applica-
+tions ranging from photonic torque microscopy [10–15],
+to seismology [16, 17] and space-based alignment proce-
+dures [18]. Another use case are tests of quantum co-
+herence at macroscopic scales [19, 20]. With librational
+degrees of freedom currently on track to reach the quan-
+tum regime [21–24], control over the depth and inversion
+of the potential will enable the generation of large delo-
+calized orientational states [25, 26] and the preparation
+of mechanical squeezed states [27].
+Therefore, to realize the full promise of levitated ro-
+tors, a scheme is required to release a librator from the
+optical potential pinning its orientation, allowing it to
+freely evolve. In this state, the system becomes an opti-
+cally suspended gyroscope that is extremely sensitive to
+DC torques, in full analogy to previously developed DC
+force sensing schemes [28]. The open question is how to
+deactivate the optical potential used to trap the levitated
+object’s orientation while keeping the trapping potential
+for its center-of-mass (COM) motion fully intact.
+In this Letter we demonstrate full control over the con-
+servative libration potential of an optically levitated par-
+ticle. Our scheme makes use of the degree of polarization
+of the trapping field. We experimentally realize near-zero
+libration frequencies up to the point where the libration
+signal vanishes, giving way to thermally driven free evo-
+lution of the levitated rotor. Additionally, for particles
+with cylindrical symmetry (dumbbells), we observe the
+signature of the transverse spin of light locally present in
+a strongly focused trapping beam.
+Key concept.—
+Consider an anisotropic dipolar point
+scatterer of polarizability α = diag(α1, α2, α3) in the
+body frame (spanned by unit vectors e1, e2, e3), with
+α3 > α2 > α1, as illustrated in Fig. 1. The orientation of
+the particle with respect to the lab frame is described by
+the three Euler angles Φ, Θ, and Ψ (see Supplement [29]).
+In a field linearly polarized along x in the lab frame, the
+particle will align with its axis of largest polarizability e3
+to the polarization axis x (Φ = Θ = 0) while it can freely
+rotate by any angle Ψ around its long axis e3.
+Small
+deviations of the long axis e3 from the polarization axis
+represent libration modes, i.e., harmonic oscillator de-
+grees of freedom, described by the angles Φ and Θ.
+Let us now consider an unpolarized electric field, whose
+field vector remains in the xy-plane. Here, the particle
+will “lie flat” in the polarization plane, i.e., align with its
+axis of smallest polarizability e1 along the z axis. Devia-
+tions from this alignment, i.e., tilts out of the polarization
+plane, again represent two libration modes described by
+the angles Θ and Ψ. At the same time, the particle can
+freely rotate by any angle Φ, as the field vector has no
+preferred direction in the xy-plane. Accordingly, both in
+a linearly and in an unpolarized field, one angular de-
+gree of freedom is free. Importantly, in the unpolarized
+case, the free rotation is measurable by available detec-
+Figure 1.
+The orientation of an anisotropic particle’s body
+frame (given by e1, e2, e3) relative to the lab frame (x, y, z) is
+described by the three angles Φ, Θ, and Ψ.
+arXiv:2301.04536v1  [physics.optics]  11 Jan 2023
+
+2
+tion schemes [21, 22] and therefore highly attractive for
+torque sensing applications. In the following, we experi-
+mentally investigate the dynamics of a levitated rotor as
+it is transitioned from a linearly polarized to an unpolar-
+ized trapping field.
+Experiment.—
+At the heart of our experimental
+setup, illustrated in Fig. 2(a), is an optical trap with vari-
+able degree of polarization (DOP), formed by focusing a
+trapping beam with a lens (NA=0.8) inside a vacuum
+chamber (pressure 1.5 mbar). To vary the DOP, a laser
+beam (wavelength 1550 nm) is split into two components
+with orthogonal linear polarization, which are then fre-
+quency shifted with acousto-optic modulators (AOMs)
+by ±80 MHz, respectively. The frequency-shifted polar-
+ization components are subsequently recombined on a po-
+larizing beamsplitter to form the trapping beam (power
+450 mW), which propagates along the z direction. Spher-
+ical silica nanoparticles (nominal diameter 143 nm) are
+loaded into the trap with a nebulizer. The dynamics of
+the trapped object are detected using forward-scattered
+light.
+The COM motion is recorded using a quadrant
+photodiode, and the libration signal using a standard ho-
+PBS
+PBS
+COM 
+99:1
++80 MHz
+AOM 2
+AOM 1
+-80 MHz
+Laser
+(a)
+(b)
+(c)
+1.5 mbar
+QWP
+ND 
+PBS
+HWP
+LD
+z
+y
+x
+Figure 2.
+(a) Simplified schematic of the experimental setup.
+The two polarization components of a laser beam are sepa-
+rated and each frequency-shifted by ±80 MHz, respectively,
+with an acousto-optic modulator (AOM). The components’
+amplitudes Ex and Ey are varied by adjusting the driving
+powers of the AOMs.
+After recombining the polarization
+components on a polarizing beamsplitter (PBS), the beam
+is focused in a vacuum chamber with a high-NA lens to form
+an optical trap with variable degree of polarization. The li-
+bration signal is detected in the forward direction using a
+combination of a quarter-wave plate (QWP), half-wave plate
+(HWP), a neutral density filter (ND), a PBS, and a balanced
+detector (LD). (b) Power spectral density (PSD) of a particle
+cluster in a linearly polarized trap. (c) PSD of a dumbbell in
+a linearly polarized trap.
+modyne detection scheme [22].
+In this work, we focus on two distinct classes of
+anisotropic particles, identified by their characteristic li-
+bration spectra shown in Figs. 2(b) and (c). The first
+class are “clusters”, that is, objects composed of more
+than two particles, described by a fully anisotropic po-
+larizability tensor (as the particle symbolically depicted
+in Fig. 1). The cluster spectrum shown in Fig. 2(b) ex-
+hibits two modes at 390 kHz and at 425 kHz, respec-
+tively, which we associate with the libration modes de-
+scribed by the angles Θ and Φ from Fig. 1. The second
+class of anisotropic particles are dumbbells (cylindrically
+symmetric objects composed of two spherical particles
+in touching contact), characterized by a sharp libration
+peak flanked by broad shoulders, as shown in Fig. 2(c).
+This spectrum originates from two libration modes that
+are coupled by the thermally driven spinning around the
+symmetry axis [22, 30, 31].
+Degree
+of
+polarization.—
+Having
+introduced
+the
+spectra for linearly polarized light we now turn to fields
+of variable DOP. In our setup, the tweezer field be-
+fore the trapping lens reads E = (Exeiωxt, Eyeiωyt, 0)T ,
+where the angular frequency difference ∆ω = ωx − ωy =
+2π ×160 MHz is kept constant, while the real-valued am-
+plitudes Ex and Ey can be controlled with the AOMs
+(see Supplement for details [29]). The instantaneous po-
+larization state of the trapping beam (before focusing) is
+described via the four Stokes parameters [32]
+S0 = E2
+x + E2
+y,
+(1a)
+S1 = E2
+x − E2
+y,
+(1b)
+S2 = 2ExEy cos (∆ωt),
+(1c)
+S3 = −2ExEy sin (∆ωt).
+(1d)
+The DOP is defined as
+P =
+�
+⟨S1⟩2 + ⟨S2⟩2 + ⟨S3⟩2/⟨S0⟩,
+(2)
+where ⟨·⟩ denotes the time average [33]. Since the op-
+tical modulation frequency ∆ω is more than two orders
+of magnitude larger than the librational dynamics, the
+cosine and the sine terms average out and the DOP sim-
+plifies to P = |s1|, where s1 = S1/S0. The upper bound,
+P = 1, corresponds to fully linearly polarized light, while
+the lower bound, P = 0, denotes unpolarized light. The
+intermediate regime, 0 < P < 1, describes partial polar-
+ization.
+Results.—
+Let us discuss our experimental observa-
+tions for a cluster trapped in a beam with variable DOP.
+In Fig. 3(a), we show in false color the power spectral
+density (PSD) of the libration signal as a function of
+frequency and P. For linearly polarized trapping light
+(P = 1), we observe the spectrum from Fig. 2(b), with
+a feature composed of two closely spaced peaks near
+400 kHz.
+As the DOP is reduced (P < 1), the two
+peaks split and their frequencies decrease. Remarkably,
+
+3
+(a)
+(b)
+Figure 3. PSDs measured by the libration detector as a function of the DOP and the normalized Stokes parameter s1 of the
+trapping beam. Each subfigure consists of 100 PSDs, where the first one (s1 = −1) corresponds to y-polarized trapping light
+and the last one (s1 = 1) to x-polarized trapping light. In the case of s1 = 0 the trapping beam is effectively unpolarized
+and the frequency corresponding to libration in the xy plane tends to zero. Frequencies corresponding to translational COM
+motion are visible as horizontal lines in the range between 40 and 150 kHz. (a) Cluster (non-rotationally symmetric particle).
+Red lines show theoretical prediction calculated from Eqs. (3a) and (3b) using only libration frequencies measured at the linear
+polarization setting. (b) Dumbbell (cylindrically symmetric particle). Black solid lines are precession frequencies calculated
+from the theoretical model including spinning of the dumbbell along its long axis [29], with ωs/2π (proportional to the spinning
+rate) shown as dashed black line.
+for unpolarized light (P = 0) the frequency of one of the
+modes vanishes, while the frequency of the second mode
+approaches 300 kHz.
+Our experimental observations for a trapped dumbbell,
+shown in Fig. 3(b), strikingly differ from that of a cluster.
+The single peak at 500 kHz (surrounded by broad shoul-
+ders) observed in linearly polarized light (P = 1), see
+Fig. 2(c), splits in two as the DOP is reduced (P < 1). In
+contrast to the cluster, the dumbbell exhibits one mode
+that shifts to higher frequencies and settles at 680 kHz
+for unpolarized light (P = 0), while the second mode
+frequency tends towards zero, where its signal strength
+vanishes.
+Model.—
+To understand our observations, we model
+the orientational dynamics of an anisotropic dipolar scat-
+terer in a field of variable DOP. Let the scatterer be
+characterized by its polarizability α = diag(α1, α2, α3)
+and its tensor of inertia I = diag(I1, I2, I3), which are
+both diagonal in the intrinsic body frame spanned by
+e1, e2, e3. We furthermore assume α1 ≤ α2 ≤ α3 and
+I1 ≥ I2 ≥ I3.
+We calculate the potential energy of a
+fully anisotropic scatterer in a field of variable DOP as a
+function of the orientation angles Φ, Θ, and Ψ and iden-
+tify the global energy minimum (see Supplement [29]).
+Small deviations of the orientation angles Φ, Θ, and Ψ
+from their equilibrium values resemble, to first order, har-
+monic oscillator degrees of freedom, whose characteristic
+libration frequencies are given by
+Ω1 = A1
+√
+P,
+(3a)
+Ω2 = A2
+�
+1 + P
+2
+,
+(3b)
+Ω3 = A3
+�
+1 − P
+2
+,
+(3c)
+respectively, where Ai
+= [(|αj − αk|S0)/2Ii]1/2 and
+{i, j, k} are permutations of {1, 2, 3}.
+Equations (3a)–(3c) indicate that we can directly con-
+trol the librational potential governing the orientation of
+the rotor via the DOP. Although our detection is only
+sensitive to libration in the xy plane [21], the coupling
+between the different libration modes [30] is responsible
+for the second libration mode in the spectrum. To com-
+pare our theoretical prediction with our measurement,
+we plot the calculated values for the libration frequen-
+cies Ω1 and Ω2 from Eqs. (3a) and (3b) as red lines in
+the measurement of the trapped cluster in Fig. 3(a). The
+required parameters A1 and A2 are defined by the libra-
+tion frequencies extracted at P = 1. The theoretical lines
+trace the observed libration frequencies remarkably well,
+demonstrating that our model correctly captures the ro-
+tational dynamics and providing strong support for our
+initial assumption that the trapped object is a cluster
+without symmetry. We stress that in a field with zero
+
+P
+1.0
+0.5
+0.0
+0.5
+1.0P
+1.0
+0.5
+0.0
+0.5
+1.04
+DOP the cluster’s libration frequency Ω1 vanishes.
+In
+other words, the orientation angle Φ undergoes free evo-
+lution.
+Let us turn our attention to the dynamics of the
+trapped dumbbells. Inspection of Eqs. (3c) shows that
+for an object of cylindrical symmetry A3 and therefore
+also Ω3 vanish. This observation intuitively makes sense,
+since such a scatterer can always freely rotate around
+its long axis. However, for dumbbells, the libration fre-
+quencies as a function of P, experimentally observed in
+Fig. 3(b), deviate significantly from those predicted by
+Eqs. (3a)–(3c). As has been pointed out before [30, 31],
+the spinning of the dumbbell at a stationary rate ˙Ψ0
+around its axis of symmetry couples the libration modes
+with frequencies Ω1 and Ω2 into precession modes with
+frequencies ΩA and ΩB according to
+(ΩA − ΩB)2 = ω2
+s + (Ω1 − Ω2)2,
+(4)
+with the coupling rate ωs = µ ˙Ψ0 and the inertial cou-
+pling constant µ = (I1 − I3)/I1. We interpret the salient
+libration features in our data in Fig. 3(b) as the preces-
+sion frequencies ΩA and ΩB of the dumbbell, and fit their
+functional dependence with Eq. (4), where Ω1 and Ω2 are
+in turn given by Eqs. (3a) and (3b). The fit [black solid
+lines in Fig. 3(b)] describes our experimental observa-
+tion very well and yields a fitted coupling rate ωs, shown
+as the dashed black line in Fig. 3(b) [29]. We conclude
+that the rate of spinning around the long axis in our ex-
+periment reaches ˙Ψ0 = 2π × 200 kHz in the regime of
+unpolarized light P = 0, where we used the dumbbells
+length-to-diameter ratio of L/D ≈ 1.8 [3] to estimate the
+inertial coupling factor as µ ≈ 0.375. We will explain the
+origin of this spinning motion in the next section.
+Discussion.—
+Our results in Fig. 3 demonstrate that
+the DOP of the trapping field allows us to control the li-
+bration frequencies of the optically levitated particle. In
+particular, we stress the fact that for a cluster in a field
+with vanishing DOP, the libration frequency Ω1 vanishes.
+In other words, the cluster becomes a free rotor regarding
+its orientation angle around the optical z axis, while its
+two axes of largest moment of inertia (I2 and I3) are har-
+monically trapped in the xy plane of polarization. The
+situation is analogous for the dumbbell, whose long axis
+is harmonically trapped with characteristic frequency ΩB
+in the focal xy plane in unpolarized light, while the ori-
+entation of the long axis undergoes free evolution within
+this plane. Thus, the DOP allows us, for the first time,
+to tune the angular motion of a levitated object from
+librations of several hundered kHz all the way to free
+evolution. This demonstration is the main result of this
+paper.
+Let us now provide an explanation for the torque that
+drives the trapped dumbbell into spinning motion around
+its long axis. We note that this torque must lie in the
+focal plane (which is the plane the dumbbell’s long axis
+is pinned to). Strongly focused fields can indeed carry
+transverse spin angular momentum [35, 36], which gives
+rise to a torque when transferred to a particle. In Fig. 4,
+we illustrate the transverse part of the spin vector [37] in
+the focal plane of a strongly focused y polarized Gaus-
+sian beam.
+The spin is depicted as arrows whose di-
+rection (length) indicates the spin’s orientation (magni-
+tude). The spin points predominantly along the positive
+(negative) x direction in the range y < 0 (y > 0). To
+understand how a dumbbell can be exposed to the trans-
+verse spin, we consider a trap composed of not only a
+y polarized beam but also of an additional strong x po-
+larized beam, used for trapping the particle’s COM (the
+intensity of which is illustrated as a colormap in Fig. 4).
+The x polarized trapping beam aligns the particle’s long
+axis along the x direction. If we displace the x polar-
+ized beam along the y direction then the transverse spin
+of the y polarized beam will spin the dumbbell along
+its long axis, as experimentally observed. Even though
+the torque along the dumbbell’s long axis is very weak,
+the effect is visible since the dumbbell is free to rotate
+along its long axis. We can thus explain the observed
+spinning motion of the dumbbell as a signature of the
+transverse spin angular momentum of light in a strongly
+focused field, together with an inevitably imperfect align-
+ment between the two cross-polarized beams forming our
+trap with tunable DOP. Effectively the trapped dumbbell
+locally senses the spin of an additional light field. Note
+I(x=0,y)
+negative
+spin 
+positive
+spin 
+Figure 4.
+Illustration of the spatial mismatch between the
+trapping beam components leading to a dumbbell being
+trapped in a region of nonzero transverse spin. A vector plot
+shows the transverse spin pattern in the xy plane generated
+by y polarized strongly focused Gaussian beam [34]. Simul-
+taneously, we show the dumbbell (to scale) trapped in the
+intensity maximum of a stronger x polarized Gaussian beam
+(contour plot) displaced by 100 nm from y = 0. Beam size
+and lens parameters correspond to the experiment. On the
+right we plot the intensity profile I(x = 0, y), indicating re-
+gions with spin pointing in negative and positive x direction.
+
+5
+that, in contrast to dumbbells, clusters are not driven
+into rotation along their long axis by a weak transverse
+spin of light, since A3 ̸= 0 and the angle of rotation
+around the long axis is restrained for P < 1 [see Eq. (3c)].
+Finally, let us comment on the limitations of our vari-
+able DOP potential control scheme.
+Throughout this
+work, we have only considered the mean polarization of
+the beams, neglecting their polarization oscillations at
+the frequency ∆ω [see Eqs. (1c) and (1d)]. In analogy
+to Paul traps operating at RF frequencies, the oscillat-
+ing polarization will give rise to an additional potential
+term [29] and a small amplitude micromotion at ∆ω. The
+frequencies associated with the additional potential term
+are of the order of magnitude of Bi ≈ A2
+i /∆ω [38]. In
+our experiment the correction to the libration frequencies
+caused by the oscillating polarization is negligible. Note
+that 1/Bi limits the maximum free evolution time for a
+given value of ∆ω.
+Moreover, in the current implementation ∆ω is re-
+stricted by the AOM bandwidth, but this limit can be
+readily removed by using two laser sources with differ-
+ent wavelengths (without the need for relative frequency
+stabilization).
+Increasing the free evolution time limit
+to 1 s requires 3 nm difference between the wavelengths
+of the two polarization components (assuming a cen-
+ter wavelength of 1550 nm). We also note that exper-
+iments involving librational potential switching [25, 27]
+via DOP control can be implemented on timescales of a
+few nanoseconds (only limited by AOM rise time).
+Conclusions.—
+We have demonstrated for the first
+time the complete tunability of the librational frequencies
+of optically levitated clusters of silica nanoparticles. This
+tunability is accomplished by the DOP and is indepen-
+dent of the COM trapping potential. Our work is impor-
+tant for the development of high-performance nanoscale
+gyroscopes and for the study of macroscopic rotational
+quantum physics [23, 39]. Furthermore, we have exper-
+imentally confirmed that symmetric rotors can serve as
+a precise tool for sensing torques not only perpendicu-
+lar, but also parallel to the long axis of the rotor. This
+feature, together with the high control over libration de-
+grees of freedom, may enable the full characterization of
+three-dimensional Stokes parameters [10, 40–42].
+The authors would like to thank A. Militaru, O.
+Romero-Isart, C. Gonzalez-Ballestero and all trappers in
+the Photonics Laboratory for fruitful discussions. This
+research was supported by the European Union’s Hori-
+zon 2020 research and innovation programme under grant
+agreement No. [863132] (IQLev), as well as ETH Grant
+No.
+ETH-47 20-2.
+A.N. thanks the Jane and Aatos
+Erkko Foundation (Finland) for funding.
+∗ jzielinska@eth.ch
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+Supplemental Material
+S1. VARIABLE DOP TRAPPING BEAM PREPARATION
+This section describes the preparation of the variable DOP trapping beam. We provide experimental details of the
+most important aspects, such as ensuring the spatial overlap of the two orthogonally-polarized components of the
+tweezer beam and the calibration of the DOP in the trapping region.
+The tweezer beam generation system is depicted in Fig. S1(a). Before entering the vacuum chamber, the trapping
+beam, propagating along the z axis, is split into two components with equal power but orthogonal linear polarizations.
+Each of these constituent beams is then frequency shifted by an AOM (y-polarized beam is shifted by −80 MHz by
+AOM 1, and x-polarized beam is shifted by +80 MHz by AOM 2). Subsequently, the beams are spatially overlapped
+and recombined on a polarizing beam-splitter (PBS). Both beams are first focused into their respective AOMs using
+one lens before the split (f = 200 mm), and then collimated using another lens (f = 500 mm) after the recombination.
+The spatial mode overlap of the two beams is optimised by maximising the visibility of interference at the difference
+frequency ∆ω = 160 MHz, recorded on the auxiliary detector (PD in Fig. S1(a)). The beams are overlapped by
+means of two steering mirrors, and the beam sizes are matched using a corner-cube (CC) mounted on a translation
+stage. The CC allows us to adjust the path length difference between the x- and y-polarized components between
+focusing (f = 200 mm) and collimating (f = 500 mm) lenses. The maximum achieved interference visibility is 85%.
+The aforementioned system allows us to control the relative contributions of x- and y-polarized light to the trapping
+beam via controlling the driving power of the AOMs. For example, if AOM 1 is driven at maximum power, and AOM 2
+receives no drive, the tweezer is y polarized, whereas when both AOM 1 and AOM 2 are driven so that they provide
+the same diffraction efficiency, the tweezer is composed of x and y polarizations in equal measure.
+In the remainder of this section we will describe the calibration of the contributions of x- and y-polarized light to
+the intensity in the trapping volume, which directly determine the DOP in the trap. In order to avoid systematic
+errors introduced by spatial mode mismatch of the x- and y-polarized beams, we perform this calibration using the
+COM motion of the trapped particle.
+The relationship between the focal intensity and the AOM driving power is determined for each beam separately
+with the help of the transverse COM frequencies describing the particle motion in the focal plane along x and y
+(denoted as fx and fy respectively). Let us denote the average transverse COM frequency as fav = 1
+2(fx + fy). We
+use the average transverse COM frequency squared f 2
+av as a proxy for the focal intensity measurement.
+The measured dependence of f 2
+av on the driving power of each AOM (together with a quadratic fit) is depicted
+in Fig. S1(b). We verify the calibration by performing the scan of the degree of polarization shown in Fig. S1(c).
+As desired, we find that fav remains constant when s1 is varied. Note that the average transverse COM frequency
+fav does not decrease for s1 = 0 (when 50% of focal intensity is x-polarized and 50% is y-polarized), even though
+the calibration is performed for each polarization component separately. This shows that the volumes of the traps
+generated by the two beams overlap well and their trap depths add together to form the final trapping potential.
+The DOP calibration procedure is potentially affected by the fact that in our experiment the trapping beam has a
+slightly elliptical shape introduced by the AOMs. For a tweezer formed by a strongly focused circular Gaussian input
+beam, fx and fy are not the same [34]. We quantify this effect using ϵ = (fx − fy)/(fx + fy). Clearly, in the case of
+circular Gaussian beam, the value of |ϵ| should not depend on whether the tweezer is x- or y-polarized. Therefore,
+different values of |ϵ| for x and y tweezer polarization (see Fig. S1(c) at points s1 = ±1) suggest that the beam cross
+section is elliptical.
+In order to quantify the effect of the trapping beam ellipticity on f 2
+av, we have simulated the focal field produced by
+the trapping lens [34]. The beam waists along x and y before focusing are used as free parameters. The obtained focal
+intensity cross sections along x and y for both x and y polarized tweezer light are shown in Fig. S1(d) and reproduce
+well the observed values of ϵ for an x- and y-polarized tweezer (ϵ = 0.057 and ϵ = −0.116 respectively). We find that
+the values of the average transverse COM frequency fav for both polarizations differs by less than 1%. Hence, we
+conclude that f 2
+av offers a good focal intensity estimate in our experiment.
+S2. LIBRATIONAL POTENTIALS
+In this section we derive the potential governing the libration of an anisotropic particle. The potential is induced
+by the electric field of the trapping beam which is propagating along z and is composed of two frequency shifted
+1
+
+(a)
+(b)
+(c)
+(d)
+f=200
+f=500
+PBS
+PBS
+HWP
++80MHz
+AOM 2
+AOM 1
+-80MHz
+PD 
+HWP
+to trap
+Laser
+CC
+z
+y
+x
+Figure S1.
+(a) Experimental diagram of the beam preparation stage.
+The laser beam is split into two parts, which are
+frequency-shifted by ±80 MHz (shown by violet and orange). The two parts are subsequently recombined and sent to the trap.
+Abbreviations: half-wave plate (HWP), polarizing beam splitter (PBS), photo-diode (PD), corner cube (CC). (b) The average
+COM transverse frequency squared f 2
+av as a function of AOM driving power for both AOMs. The data points are shown together
+with quadratic fits (solid lines). (c) The power spectral density (PSD) of COM motion as a function of DOP of the trapping
+beam. This subfigure consists of 100 PSDs, where the first one (s1 = −1) corresponds to y-polarized trapping light and the
+last one (s1 = 1) to x-polarized trapping light. The PSD is measured by the quadrant photo detector (see Fig. 2(a), both QPD
+channels are summed here). COM transverse frequencies fx and fy are visible in the 120 to 150 kHz region. (d) Simulated
+intensity profiles along x and y directions for a strongly focused elliptical beam. Note that the cross-section of the focal spot
+is more circular for y-polarised tweezer beam.
+components (x- and y-polarized) of different amplitudes:
+E = (Exeiωxt, Eyeiωyt, 0)T .
+(S1)
+The x and y field components acquire a phase difference that grows linearly in time in proportion to the frequency
+∆ω = ωx − ωy. Since the initial phase difference is not important to the dynamics, we assume that the amplitudes
+Ex and Ey are real.
+We assume that both the moment of inertia and the polarizability tensor can be simultaneously diagonalized in the
+principal axes reference frame of the particle. We refer to this frame of reference as “particle frame” and represent
+the principal axes as (e1, e2, e3). We denote α = diag(α1, α2, α3) and I = diag(I1, I2, I3) as the static polarizability
+and inertia tensors of the object in the intrinsic body frame, respectively, assuming α1 ≤ α2 ≤ α3 and I1 ≥ I2 ≥ I3.
+We refer to e3 (the principal axis with the largest polarizability) as the “long axis” of the object.
+In order to describe the orientation of the particle frame with respect to the laboratory frame (x, y, z), we use the
+intrinsic x-convention of Euler angles denoted as φ, θ and ψ (see [43] and §35 in [44]). The Euler angles φ and θ
+2
+
+describe the orientation of the long axis of the rotor (the angle measured in the experiment is φ, which corresponds to
+the orientation of the long axis in the xy plane). In order to transform a vector from laboratory to particle frame we
+first rotate it by the angle φ around z, then by θ around e1 and finally by ψ around e3. The transformation matrix
+corresponding to these three rotation operations reads [43]
+R =
+�
+�
+cos ψ cos φ − cos θ sin ψ sin φ
+cos θ sin ψ cos φ + cos ψ sin φ
+sin θ sin ψ
+− cos θ cos ψ sin φ − sin ψ cos φ
+cos θ cos ψ cos φ − sin ψ sin φ
+sin θ cos ψ
+sin θ sin φ
+− sin θ cos φ
+cos θ
+�
+� .
+(S2)
+The dipole moment induced in a trapped anisotropic particle, expressed in the laboratory frame of reference, yields:
+p = R−1αRE,
+(S3)
+where ⃗E is also expressed in the laboratory reference frame. Since in general ⃗p and ⃗E are not parallel, the potential
+energy Utot associated with the orientation of the particle (after averaging over optical frequencies) yields:
+Utot = −1
+4Re (p · E∗) .
+(S4)
+If we consider the electric field as in Eq. (S1), the potential Utot can be written as a sum of two terms
+Utot = U0 + U1 cos (∆ωt),
+(S5)
+with
+U0 =S0
+8
+�
+−(α1 + α2) + sin2 θ (1 − s1 cos 2φ)
+�
+α1 sin2 ψ + α2 cos2 ψ − α3
+�
++ s1 (α1 − α2) (cos θ sin 2ψ sin 2φ − cos 2ψ cos 2φ)
+�
+,
+(S6a)
+U1 = − S0
+�
+1 − s2
+1
+8
+�
+sin2 θ sin 2φ (α1 sin2 ψ + α2 cos2 ψ − α3)
++ (α1 − α2)(cos θ sin 2ψ cos 2φ + cos 2ψ sin 2φ)
+�
+,
+(S6b)
+where S0 and s1 = S1/S0 are defined in Eqs. (1a) and (1b). The oscillating term U1 cos(∆ωt) describes the interaction
+of a dipole moment induced by the field oscillating at ωx with the field oscillating at ωy (and vice versa).
+The fast oscillating term U1 cos(∆ωt) will give rise to a small amplitude micromotion at ∆ω. This “fast” micromotion
+will in turn have an effect on the “slow” librational dynamics, which can be described as an additional, constant-in-time
+effective potential term U ′
+1 [44, 45], yielding
+U ′
+1 =
+1
+2(∆ω)2
+�
+i,k
+A−1
+ik ∂iU1∂kU1 ,
+(S7)
+where the indices i and k run through θ, φ and ψ and Aik are matrix elements of the quadratic form describing the
+kinetic energy T, such that
+T = 1
+2( ˙θ, ˙φ, ˙ψ)A( ˙θ, ˙φ, ˙ψ)T.
+(S8)
+The matrix A depends on the Euler angles and the inertial moment according to
+A =
+�
+�
+I1 cos2 ψ + I2 sin2 ψ
+(I1 − I2) sin θ sin ψ cos ψ
+0
+(I1 − I2) sin θ sin ψ cos ψ
+(I1 sin2 ψ + I2 cos2 ψ) sin2 θ + I3 cos2 θ
+I3 cos θ
+0
+I3 cos θ
+I3
+�
+� .
+(S9)
+As an example, let us find the explicit expressions for U0 and U ′
+1 in the case of a rotor with cylindrical symmetry,
+such as a dumbbell (α1 = α2, I1 = I2):
+U0 = A2
+1
+4 I1(s1 cos 2φ − 1) sin2 θ ,
+(S10a)
+U ′
+1 = A4
+1I1(1 − s2
+1) sin2 θ
+8(∆ω)2
+�
+I1 sin2 θ cos2 2φ + (I1 sin2 θ + I3 cos2 θ) cos2 θ sin2 2φ
+�
+(I1 sin2 θ + I3 cos2 θ)
+,
+(S10b)
+3
+
+where the parameter A1 =
+�
+(α3−α1)S0
+2I1
+is equal to the libration frequency in a linearly polarized trap. Equations (S10a)
+and (S10b) indicate that the residual potential U ′
+1 is shallower than U0 by approximately a factor of (A1/∆ω)2. Note
+that for our experimental parameters, we have (A1/∆ω)2 ≈ 10−5.
+Let us now consider how both potential terms affect the dynamics of the orientation of the dumbbell in the xy
+plane described by the angle φ. We can expect the contribution from U ′
+1 to be negligible, except when s1 ≈ 0 (the
+trap is unpolarized) and U0 is independent of φ. Therefore, in this case, U ′
+1 is the dominant potential term governing
+the dynamics of φ. The minima of U ′
+1 occur at φ = nπ/4 with n ∈ {1, 2, 3, 4}, which means that our symmetric rotor
+will become diagonally oriented in the xy plane.
+However, in our experiment we have U ′
+1 ≪ kBT, where kB is the Boltzmann constant and T is the temperature.
+Therefore, even when the tweezer is unpolarized, the effect of the residual potential U ′
+1 is completely obscured by the
+interaction with the environment (background gas).
+S3. TORQUES AND LIBRATION FREQUENCIES FOR ASYMMETRIC ROTOR
+In this section we use the potential derived in the previous section to calculate restoring torques acting on a rotor
+trapped in a beam with an arbitrary DOP. We derive the libration frequencies as functions of DOP.
+We denote particle-frame torque components as Ki and ki =
+Ki
+Ii .
+The full expressions for torques due to the
+time-independent potential U0 [see Eq. (S6a)] read
+k1 = kN cos ψ + (kz − k3 cos θ) sin ψ
+sin θ
+= A2
+1
+2 sin θ [(1 − s1 cos 2φ) cos θ cos ψ + s1 sin ψ sin 2φ] ,
+(S11a)
+k2 = −kN sin ψ + (kz − k3 cos θ) cos ψ
+sin θ
+= −A2
+2
+2 sin θ[(1 − s1 cos 2φ) cos θ sin ψ − s1 cos ψ sin 2φ] ,
+(S11b)
+k3 = −A2
+3
+2
+�
+(1 − s1 cos 2φ) sin2 θ sin ψ cos ψ + s1(cos θ cos 2ψ sin 2φ + sin 2ψ cos 2φ)
+�
+,
+(S11c)
+where k3 = −∂U0/∂ψ, kN = −∂U0/∂θ, kz = −∂U0/∂φ. Additionally we have Ai = [(|αj − αk|S0)/2Ii]1/2 where
+{i, j, k} are permutations of {1, 2, 3}.
+To gain an intuitive understanding, let us analyze in detail the case s1 = 1, i.e., the tweezer is linearly polarized
+along x. The object is trapped in a potential minimum (along x), for which θ = φ = π
+2 , and ψ is unrestrained (free).
+This means that the rotor aligns itself with its long axis to the polarization axis, while it can freely spin around it. Let
+us now move beyond purely linear polarization. As soon as s1 < 1, a potential minimum appears at ψ = π
+2 and the
+angle ψ also becomes trapped. Colloquially speaking, the rotor can now minimize its potential energy by “lying flat”
+in the polarization plane. Having discussed the orientation with minimal potential energy, let us consider deviations
+from that orientation and the associated dynamics, which corresponds to librational motion for small angles. To this
+end, we introduce the libration angles describing the motion around the equilibrium position as Θ = θ − π
+2 , Φ = φ− π
+2 ,
+and Ψ = ψ− π
+2 . Without loss of generality, it is convenient to assume a tweezer field that is predominantly x-polarized,
+such that the resulting torque components can be expressed in the laboratory frame. To first order in the Euler angles,
+these restoring torques acting on the angles Θ, Φ, and Ψ read
+kz ≈ −s1A2
+1Φ,
+(S12a)
+ky ≈ −s1 + 1
+2
+A2
+2Θ,
+(S12b)
+kx ≈ −1 − s1
+2
+A2
+3Ψ.
+(S12c)
+The libration frequencies associated with these restoring torques read
+Ω1 =
+√
+PA1,
+(S13a)
+Ω2 =
+�
+P + 1
+2
+A2,
+(S13b)
+Ω3 =
+�
+1 − P
+2
+A3.
+(S13c)
+In the above expressions we have replaced s1 with P so that Eqs. (S13a)–(S13c) are also valid around s1 = −1. In
+the case of s1 ≈ −1 the tweezer is almost y-polarized, and the potential minimum occurs for a different particle
+4
+
+orientation. Therefore, the angles θ, φ and ψ librate around different equilibrium positions (denoting the potential
+minimum) for different polarization states of the tweezer field. However, the libration frequencies are, to linear order,
+always given by Eq. (S13).
+S4. EQUATIONS OF MOTION FOR A SYMMETRIC ROTOR
+In this section, we analyze the motion of a symmetric rotor (I1 = I2, α1 = α2) under restoring torques given by
+Eqs. (S11a)–(S11c) and an additional small spinning torque. Since the parameter A3 vanishes for the symmetric rotor,
+Eq. (S11c) implies that there is no restoring torque pointing along the object’s long axis. Therefore, for symmetric
+rotors the angle ψ is free for any value of s1.
+Let us now explore a scenario in which the symmetric rotor is performing a small amplitude libration around the x
+axis (s1 ≈ 1) and an additional constant torque is present. We assume that the additional torque is much smaller than
+the restoring torques along y and z [see Eqs. (S12a) and (S12b)]. Therefore, only the additional torque component
+pointing along x will have an effect on the dynamics, causing the rotor to spin around its long axis (which points long
+x). We denote the additional torque component along x as kext. Let us write the rotational equations of motion [44]
+(Euler’s equations), including both the restoring torques and kext for a symmetric rotor performing small amplitude
+libration around the x axis :
+¨Φ = −µ ˙Θ ˙Ψ − Ω2
+1Φ,
+(S14a)
+¨Θ = µ ˙Ψ ˙Φ − Ω2
+2Θ,
+(S14b)
+¨Ψ = kext,
+(S14c)
+where µ = I1−I3
+I1
+. Provided that kext is larger than fluctuating (thermal) torques, the dumbbell will start to spin
+consistently around its long axis. The spinning rate will increase until it reaches the friction-dependent stationary
+value denoted as ˙Ψ0. Note that Eqs. (S14a) and (S14b) include coupling between angular degrees of freedom, which
+is an inherent feature of rotational mechanics. Fast spinning, i.e., large ˙Ψ causes strong coupling between the Φ and
+Θ libration modes which become hybrid precession modes. Similarly to Ref. [30] we use the following simple model
+¨Φ = −ωs ˙Θ − Ω2
+1Φ,
+(S15a)
+¨Θ = ωs ˙Φ − Ω2
+2Θ,
+(S15b)
+where ωs = µ ˙Ψ0. We will henceforth refer to ωs as the rotational coupling rate. The eigenfunctions of Eqs. (S15a)-
+(S15b) (precession modes) oscillate at eigenfrequencies ΩA and ΩB, which read
+ΩA =
+�
+Ω2
+1 + Ω2
+2 + ω2s −
+�
+(Ω2
+1 + Ω2
+2 + ω2s )2 − 4Ω2
+1Ω2
+2
+√
+2
+,
+(S16a)
+ΩB =
+�
+Ω2
+1 + Ω2
+2 + ω2s +
+�
+(Ω2
+1 + Ω2
+2 + ω2s )2 − 4Ω2
+1Ω2
+2
+√
+2
+.
+(S16b)
+We notice that the precession frequencies ΩA and ΩB satisfy the following relation:
+(ΩA − ΩB)2 = ω2
+s + (Ω1 − Ω2)2.
+(S17)
+This relation illustrates the signature of the spinning along the long axis, which can be described as an additional
+splitting between the eigenfrequencies visible in the libration spectrum. We have used Eq. (S17) to reconstruct the
+value of the rotational coupling ωs (as function of s1) from the measured precession mode splitting (ΩA − ΩB) and
+the libration mode splitting (Ω1 −Ω2) predicted from Eqs. (S13a) and (S13b). The value of A1 used in this procedure
+was measured at P = 0. The obtained rotational coupling rate is shown in Fig. 3(b) as a dashed black line.
+Note that ωs depends only on the difference between observed peak frequencies in the libration spectrum. We
+have calculated ωs from the data in Fig. 3(b) and used it to find the precession mode frequencies from Eqs. (S16a)
+and (S16b).
+We find that the resulting precession frequencies (as functions of s1) fit the behavior of the modes
+observed in the libration spectrum well, which confirms the spinning hypothesis.
+5
+
+Lastly, we have only discussed the effect of the additional small torque on a symmetric rotor (dumbbell). We note
+that non-symmetric rotors (clusters), when exposed to the same additional torque, will not spin. This is due to the
+fact that for P < 1 all degrees of freedom describing cluster orientation are trapped in a potential minimum, such
+that they librate and do not spin, unless the additional torque exceeds the restoring torque.
+6
+
diff --git a/btE3T4oBgHgl3EQfdgrZ/content/tmp_files/load_file.txt b/btE3T4oBgHgl3EQfdgrZ/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6b9d0c5616556154475364f2d93c923770db7eb5
--- /dev/null
+++ b/btE3T4oBgHgl3EQfdgrZ/content/tmp_files/load_file.txt
@@ -0,0 +1,747 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf,len=746
+page_content='Full control of the libration potential in rotational levitodynamics  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Zielińska,1, ∗ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' van der Laan,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Norrman,1, 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Reimann,1, 3 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Novotny,1 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Frimmer1  1Photonics Laboratory, ETH Zürich, CH-8093 Zürich, Switzerland  2Center for Photonics Sciences, University of Eastern Finland, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Box 111, FI-80101 Joensuu, Finland  3Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE  Control of the potential energy and free evolution lie at the heart of levitodynamics as key re-  quirements for sensing, wave function expansion, and mechanical squeezing protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Here, we  experimentally demonstrate full control over the optical potential governing the librational degrees  of freedom of a levitated anisotropic nanoparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' This control is achieved by introducing the degree  of polarization as a new tool for rotational levitodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We demonstrate the free rotation of a  levitated anisotropic scatterer around its short axis and and we use the rotational degrees of freedom  to probe the local spin of a strongly focused laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='—  Levitodynamics is the science of con-  trolling the motion of levitated mesoscopic objects [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The field has received growing attention in the last  decade as a platform for force, torque, and electric field  sensing [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Next to the translational degrees of freedom,  the rotational dynamics of levitated anisotropic bodies  offer particularly promising opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' More specifi-  cally, new functionalities demonstrated for optically levi-  tated rotors include controllable diffusion [3], gyroscopic  stabilization [4], spinning with GHz rotation rates [5–8],  and the realization of rotational “washboard potentials”  by carefully trading off conservative and non-conservative  torques in elliptically polarized fields [3, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  A particularly enticing prospect is to harness levitated  rotors as ultra-sensitive torque sensors [7], in applica-  tions ranging from photonic torque microscopy [10–15],  to seismology [16, 17] and space-based alignment proce-  dures [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Another use case are tests of quantum co-  herence at macroscopic scales [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' With librational  degrees of freedom currently on track to reach the quan-  tum regime [21–24], control over the depth and inversion  of the potential will enable the generation of large delo-  calized orientational states [25, 26] and the preparation  of mechanical squeezed states [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Therefore, to realize the full promise of levitated ro-  tors, a scheme is required to release a librator from the  optical potential pinning its orientation, allowing it to  freely evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' In this state, the system becomes an opti-  cally suspended gyroscope that is extremely sensitive to  DC torques, in full analogy to previously developed DC  force sensing schemes [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The open question is how to  deactivate the optical potential used to trap the levitated  object’s orientation while keeping the trapping potential  for its center-of-mass (COM) motion fully intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  In this Letter we demonstrate full control over the con-  servative libration potential of an optically levitated par-  ticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Our scheme makes use of the degree of polarization  of the trapping field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We experimentally realize near-zero  libration frequencies up to the point where the libration  signal vanishes, giving way to thermally driven free evo-  lution of the levitated rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Additionally, for particles  with cylindrical symmetry (dumbbells), we observe the  signature of the transverse spin of light locally present in  a strongly focused trapping beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Key concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='—  Consider an anisotropic dipolar point  scatterer of polarizability α = diag(α1, α2, α3) in the  body frame (spanned by unit vectors e1, e2, e3), with  α3 > α2 > α1, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The orientation of  the particle with respect to the lab frame is described by  the three Euler angles Φ, Θ, and Ψ (see Supplement [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  In a field linearly polarized along x in the lab frame, the  particle will align with its axis of largest polarizability e3  to the polarization axis x (Φ = Θ = 0) while it can freely  rotate by any angle Ψ around its long axis e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Small  deviations of the long axis e3 from the polarization axis  represent libration modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=', harmonic oscillator de-  grees of freedom, described by the angles Φ and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Let us now consider an unpolarized electric field, whose  field vector remains in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Here, the particle  will “lie flat” in the polarization plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=', align with its  axis of smallest polarizability e1 along the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Devia-  tions from this alignment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=', tilts out of the polarization  plane, again represent two libration modes described by  the angles Θ and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' At the same time, the particle can  freely rotate by any angle Φ, as the field vector has no  preferred direction in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Accordingly, both in  a linearly and in an unpolarized field, one angular de-  gree of freedom is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Importantly, in the unpolarized  case, the free rotation is measurable by available detec-  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The orientation of an anisotropic particle’s body  frame (given by e1, e2, e3) relative to the lab frame (x, y, z) is  described by the three angles Φ, Θ, and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='04536v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='optics]  11 Jan 2023  2  tion schemes [21, 22] and therefore highly attractive for  torque sensing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' In the following, we experi-  mentally investigate the dynamics of a levitated rotor as  it is transitioned from a linearly polarized to an unpolar-  ized trapping field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='—  At the heart of our experimental  setup, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 2(a), is an optical trap with vari-  able degree of polarization (DOP), formed by focusing a  trapping beam with a lens (NA=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='8) inside a vacuum  chamber (pressure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='5 mbar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' To vary the DOP, a laser  beam (wavelength 1550 nm) is split into two components  with orthogonal linear polarization, which are then fre-  quency shifted with acousto-optic modulators (AOMs)  by ±80 MHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The frequency-shifted polar-  ization components are subsequently recombined on a po-  larizing beamsplitter to form the trapping beam (power  450 mW), which propagates along the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Spher-  ical silica nanoparticles (nominal diameter 143 nm) are  loaded into the trap with a nebulizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The dynamics of  the trapped object are detected using forward-scattered  light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The COM motion is recorded using a quadrant  photodiode, and the libration signal using a standard ho-  PBS  PBS  COM  99:1  +80 MHz  AOM 2  AOM 1  80 MHz  Laser  (a)  (b)  (c)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='5 mbar  QWP  ND  PBS  HWP  LD  z  y  x  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (a) Simplified schematic of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The two polarization components of a laser beam are sepa-  rated and each frequency-shifted by ±80 MHz, respectively,  with an acousto-optic modulator (AOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The components’  amplitudes Ex and Ey are varied by adjusting the driving  powers of the AOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  After recombining the polarization  components on a polarizing beamsplitter (PBS), the beam  is focused in a vacuum chamber with a high-NA lens to form  an optical trap with variable degree of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The li-  bration signal is detected in the forward direction using a  combination of a quarter-wave plate (QWP), half-wave plate  (HWP), a neutral density filter (ND), a PBS, and a balanced  detector (LD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (b) Power spectral density (PSD) of a particle  cluster in a linearly polarized trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (c) PSD of a dumbbell in  a linearly polarized trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  modyne detection scheme [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  In this work, we focus on two distinct classes of  anisotropic particles, identified by their characteristic li-  bration spectra shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 2(b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The first  class are “clusters”, that is, objects composed of more  than two particles, described by a fully anisotropic po-  larizability tensor (as the particle symbolically depicted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The cluster spectrum shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 2(b) ex-  hibits two modes at 390 kHz and at 425 kHz, respec-  tively, which we associate with the libration modes de-  scribed by the angles Θ and Φ from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The second  class of anisotropic particles are dumbbells (cylindrically  symmetric objects composed of two spherical particles  in touching contact), characterized by a sharp libration  peak flanked by broad shoulders, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  This spectrum originates from two libration modes that  are coupled by the thermally driven spinning around the  symmetry axis [22, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Degree  of  polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='—  Having  introduced  the  spectra for linearly polarized light we now turn to fields  of variable DOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' In our setup, the tweezer field be-  fore the trapping lens reads E = (Exeiωxt, Eyeiωyt, 0)T ,  where the angular frequency difference ∆ω = ωx − ωy =  2π ×160 MHz is kept constant, while the real-valued am-  plitudes Ex and Ey can be controlled with the AOMs  (see Supplement for details [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The instantaneous po-  larization state of the trapping beam (before focusing) is  described via the four Stokes parameters [32]  S0 = E2  x + E2  y,  (1a)  S1 = E2  x − E2  y,  (1b)  S2 = 2ExEy cos (∆ωt),  (1c)  S3 = −2ExEy sin (∆ωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (1d)  The DOP is defined as  P =  �  ⟨S1⟩2 + ⟨S2⟩2 + ⟨S3⟩2/⟨S0⟩,  (2)  where ⟨·⟩ denotes the time average [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Since the op-  tical modulation frequency ∆ω is more than two orders  of magnitude larger than the librational dynamics, the  cosine and the sine terms average out and the DOP sim-  plifies to P = |s1|, where s1 = S1/S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The upper bound,  P = 1, corresponds to fully linearly polarized light, while  the lower bound, P = 0, denotes unpolarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The  intermediate regime, 0 < P < 1, describes partial polar-  ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='—  Let us discuss our experimental observa-  tions for a cluster trapped in a beam with variable DOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 3(a), we show in false color the power spectral  density (PSD) of the libration signal as a function of  frequency and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' For linearly polarized trapping light  (P = 1), we observe the spectrum from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 2(b), with  a feature composed of two closely spaced peaks near  400 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  As the DOP is reduced (P < 1), the two  peaks split and their frequencies decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Remarkably,  3  (a)  (b)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' PSDs measured by the libration detector as a function of the DOP and the normalized Stokes parameter s1 of the  trapping beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Each subfigure consists of 100 PSDs, where the first one (s1 = −1) corresponds to y-polarized trapping light  and the last one (s1 = 1) to x-polarized trapping light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' In the case of s1 = 0 the trapping beam is effectively unpolarized  and the frequency corresponding to libration in the xy plane tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Frequencies corresponding to translational COM  motion are visible as horizontal lines in the range between 40 and 150 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (a) Cluster (non-rotationally symmetric particle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Red lines show theoretical prediction calculated from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (3a) and (3b) using only libration frequencies measured at the linear  polarization setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (b) Dumbbell (cylindrically symmetric particle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Black solid lines are precession frequencies calculated  from the theoretical model including spinning of the dumbbell along its long axis [29], with ωs/2π (proportional to the spinning  rate) shown as dashed black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  for unpolarized light (P = 0) the frequency of one of the  modes vanishes, while the frequency of the second mode  approaches 300 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Our experimental observations for a trapped dumbbell,  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 3(b), strikingly differ from that of a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The single peak at 500 kHz (surrounded by broad shoul-  ders) observed in linearly polarized light (P = 1), see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 2(c), splits in two as the DOP is reduced (P < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' In  contrast to the cluster, the dumbbell exhibits one mode  that shifts to higher frequencies and settles at 680 kHz  for unpolarized light (P = 0), while the second mode  frequency tends towards zero, where its signal strength  vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='—  To understand our observations, we model  the orientational dynamics of an anisotropic dipolar scat-  terer in a field of variable DOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Let the scatterer be  characterized by its polarizability α = diag(α1, α2, α3)  and its tensor of inertia I = diag(I1, I2, I3), which are  both diagonal in the intrinsic body frame spanned by  e1, e2, e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We furthermore assume α1 ≤ α2 ≤ α3 and  I1 ≥ I2 ≥ I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  We calculate the potential energy of a  fully anisotropic scatterer in a field of variable DOP as a  function of the orientation angles Φ, Θ, and Ψ and iden-  tify the global energy minimum (see Supplement [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Small deviations of the orientation angles Φ, Θ, and Ψ  from their equilibrium values resemble, to first order, har-  monic oscillator degrees of freedom, whose characteristic  libration frequencies are given by  Ω1 = A1  √  P,  (3a)  Ω2 = A2  �  1 + P  2  ,  (3b)  Ω3 = A3  �  1 − P  2  ,  (3c)  respectively, where Ai  = [(|αj − αk|S0)/2Ii]1/2 and  {i, j, k} are permutations of {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Equations (3a)–(3c) indicate that we can directly con-  trol the librational potential governing the orientation of  the rotor via the DOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Although our detection is only  sensitive to libration in the xy plane [21], the coupling  between the different libration modes [30] is responsible  for the second libration mode in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' To com-  pare our theoretical prediction with our measurement,  we plot the calculated values for the libration frequen-  cies Ω1 and Ω2 from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (3a) and (3b) as red lines in  the measurement of the trapped cluster in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The  required parameters A1 and A2 are defined by the libra-  tion frequencies extracted at P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The theoretical lines  trace the observed libration frequencies remarkably well,  demonstrating that our model correctly captures the ro-  tational dynamics and providing strong support for our  initial assumption that the trapped object is a cluster  without symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We stress that in a field with zero  P  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='0P  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='04  DOP the cluster’s libration frequency Ω1 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  In  other words, the orientation angle Φ undergoes free evo-  lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Let us turn our attention to the dynamics of the  trapped dumbbells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Inspection of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (3c) shows that  for an object of cylindrical symmetry A3 and therefore  also Ω3 vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' This observation intuitively makes sense,  since such a scatterer can always freely rotate around  its long axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' However, for dumbbells, the libration fre-  quencies as a function of P, experimentally observed in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 3(b), deviate significantly from those predicted by  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (3a)–(3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' As has been pointed out before [30, 31],  the spinning of the dumbbell at a stationary rate ˙Ψ0  around its axis of symmetry couples the libration modes  with frequencies Ω1 and Ω2 into precession modes with  frequencies ΩA and ΩB according to  (ΩA − ΩB)2 = ω2  s + (Ω1 − Ω2)2,  (4)  with the coupling rate ωs = µ ˙Ψ0 and the inertial cou-  pling constant µ = (I1 − I3)/I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We interpret the salient  libration features in our data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 3(b) as the preces-  sion frequencies ΩA and ΩB of the dumbbell, and fit their  functional dependence with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (4), where Ω1 and Ω2 are  in turn given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (3a) and (3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The fit [black solid  lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 3(b)] describes our experimental observa-  tion very well and yields a fitted coupling rate ωs, shown  as the dashed black line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 3(b) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We conclude  that the rate of spinning around the long axis in our ex-  periment reaches ˙Ψ0 = 2π × 200 kHz in the regime of  unpolarized light P = 0, where we used the dumbbells  length-to-diameter ratio of L/D ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='8 [3] to estimate the  inertial coupling factor as µ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We will explain the  origin of this spinning motion in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='—  Our results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 3 demonstrate that  the DOP of the trapping field allows us to control the li-  bration frequencies of the optically levitated particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' In  particular, we stress the fact that for a cluster in a field  with vanishing DOP, the libration frequency Ω1 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  In other words, the cluster becomes a free rotor regarding  its orientation angle around the optical z axis, while its  two axes of largest moment of inertia (I2 and I3) are har-  monically trapped in the xy plane of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The  situation is analogous for the dumbbell, whose long axis  is harmonically trapped with characteristic frequency ΩB  in the focal xy plane in unpolarized light, while the ori-  entation of the long axis undergoes free evolution within  this plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Thus, the DOP allows us, for the first time,  to tune the angular motion of a levitated object from  librations of several hundered kHz all the way to free  evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' This demonstration is the main result of this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Let us now provide an explanation for the torque that  drives the trapped dumbbell into spinning motion around  its long axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We note that this torque must lie in the  focal plane (which is the plane the dumbbell’s long axis  is pinned to).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Strongly focused fields can indeed carry  transverse spin angular momentum [35, 36], which gives  rise to a torque when transferred to a particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 4,  we illustrate the transverse part of the spin vector [37] in  the focal plane of a strongly focused y polarized Gaus-  sian beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The spin is depicted as arrows whose di-  rection (length) indicates the spin’s orientation (magni-  tude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The spin points predominantly along the positive  (negative) x direction in the range y < 0 (y > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' To  understand how a dumbbell can be exposed to the trans-  verse spin, we consider a trap composed of not only a  y polarized beam but also of an additional strong x po-  larized beam, used for trapping the particle’s COM (the  intensity of which is illustrated as a colormap in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The x polarized trapping beam aligns the particle’s long  axis along the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' If we displace the x polar-  ized beam along the y direction then the transverse spin  of the y polarized beam will spin the dumbbell along  its long axis, as experimentally observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Even though  the torque along the dumbbell’s long axis is very weak,  the effect is visible since the dumbbell is free to rotate  along its long axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We can thus explain the observed  spinning motion of the dumbbell as a signature of the  transverse spin angular momentum of light in a strongly  focused field, together with an inevitably imperfect align-  ment between the two cross-polarized beams forming our  trap with tunable DOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Effectively the trapped dumbbell  locally senses the spin of an additional light field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Note  I(x=0,y)  negative  spin  positive  spin  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Illustration of the spatial mismatch between the  trapping beam components leading to a dumbbell being  trapped in a region of nonzero transverse spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' A vector plot  shows the transverse spin pattern in the xy plane generated  by y polarized strongly focused Gaussian beam [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Simul-  taneously, we show the dumbbell (to scale) trapped in the  intensity maximum of a stronger x polarized Gaussian beam  (contour plot) displaced by 100 nm from y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Beam size  and lens parameters correspond to the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' On the  right we plot the intensity profile I(x = 0, y), indicating re-  gions with spin pointing in negative and positive x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  5  that, in contrast to dumbbells, clusters are not driven  into rotation along their long axis by a weak transverse  spin of light, since A3 ̸= 0 and the angle of rotation  around the long axis is restrained for P < 1 [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (3c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Finally, let us comment on the limitations of our vari-  able DOP potential control scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Throughout this  work, we have only considered the mean polarization of  the beams, neglecting their polarization oscillations at  the frequency ∆ω [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (1c) and (1d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' In analogy  to Paul traps operating at RF frequencies, the oscillat-  ing polarization will give rise to an additional potential  term [29] and a small amplitude micromotion at ∆ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The  frequencies associated with the additional potential term  are of the order of magnitude of Bi ≈ A2  i /∆ω [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' In  our experiment the correction to the libration frequencies  caused by the oscillating polarization is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Note  that 1/Bi limits the maximum free evolution time for a  given value of ∆ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Moreover, in the current implementation ∆ω is re-  stricted by the AOM bandwidth, but this limit can be  readily removed by using two laser sources with differ-  ent wavelengths (without the need for relative frequency  stabilization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Increasing the free evolution time limit  to 1 s requires 3 nm difference between the wavelengths  of the two polarization components (assuming a cen-  ter wavelength of 1550 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We also note that exper-  iments involving librational potential switching [25, 27]  via DOP control can be implemented on timescales of a  few nanoseconds (only limited by AOM rise time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='—  We have demonstrated for the first  time the complete tunability of the librational frequencies  of optically levitated clusters of silica nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' This  tunability is accomplished by the DOP and is indepen-  dent of the COM trapping potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Our work is impor-  tant for the development of high-performance nanoscale  gyroscopes and for the study of macroscopic rotational  quantum physics [23, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Furthermore, we have exper-  imentally confirmed that symmetric rotors can serve as  a precise tool for sensing torques not only perpendicu-  lar, but also parallel to the long axis of the rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' This  feature, together with the high control over libration de-  grees of freedom, may enable the full characterization of  three-dimensional Stokes parameters [10, 40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The authors would like to thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Militaru, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Romero-Isart, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Gonzalez-Ballestero and all trappers in  the Photonics Laboratory for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' This  research was supported by the European Union’s Hori-  zon 2020 research and innovation programme under grant  agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' [863132] (IQLev), as well as ETH Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  ETH-47 20-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' thanks the Jane and Aatos  Erkko Foundation (Finland) for funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  ∗ jzielinska@eth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='ch  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Gonzalez-Ballestero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Aspelmeyer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Novotny,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Quidant, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Romero-Isart, Science 374, eabg3027  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rademacher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Millen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Li, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Tech-  nol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 9, 227 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Bellando, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Kleine, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Amarouchene, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Perrin, and  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Louyer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 129, 023602 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Kuhn, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Stickler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Kosloff, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Patolsky, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Horn-  berger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Arndt, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Millen, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 8, 1670  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Reimann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Doderer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Hebestreit, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Diehl,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Frimmer,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Windey,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Tebbenjohanns,  and  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Novotny, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 121, 033602 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Ahn, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Bang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Deng, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Hoang,  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Han, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Ma,  and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Li, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 121,  033603 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Ahn, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Bang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Ju, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Gao, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Li, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 15, 89 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [8] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' van der Laan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Reimann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Militaru, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Tebbenjo-  hanns, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Windey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Frimmer, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Novotny, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' A 102, 013505 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Kuhn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Kosloff, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Stickler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Patolsky, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Horn-  berger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Arndt, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Millen, Optica 4, 356 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Irrera,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Magazzù,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Artoni,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Simpson,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Hanna, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Jones, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Priolo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Gucciardi, and  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Maragò, Nano Letters 16, 4181 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [11] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Svak, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Brzobohatý, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Šiler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Jákl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Kaňka,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Zemánek, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Simpson, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 9, 5453  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [12] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Arita, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Simpson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Zemánek, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Dholakia,  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content='  [13] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Liang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Yan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Yao, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Lei, Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content='  [14] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content='  Simpson,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Zemánek,  and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Dholakia, arXiv preprint  arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Hu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Kingsley-Smith, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Xu,  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Millen, arXiv  preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Guattari, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' de Toldi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Garcia, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Mimoun,  in International Conference on Space Optics — ICSO  2018, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 11180, International Society for Optics and  Photonics (SPIE, 2019) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 1118080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Braun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Ripepe, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Igel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Volcanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Geotherm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Kong, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Stickler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Papendell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Kuhn, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Schrinski,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Millen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Arndt, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Hornberger, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Schrinski, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Hornberger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Militaru, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Norrman, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' van der  Laan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Frimmer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Tebbenjohanns, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Frimmer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 127,  023601 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [26] Once the ground state is reached, we estimate that co-  herent spatial expansion by a factor of 102 is possible  even in the presence of backaction-induced heating rates  reported in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Janszky and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Adam, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' A 46, 6091 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [28] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Hebestreit, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Frimmer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Reimann, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Novotny,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 121, 063602 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [29] See Supplemental material for the detailed description of  the experimental setup, derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (3) and further  discussion of spinning along the long axis driven by the  transverse spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [30] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Seberson and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Robicheaux, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' A 99, 013821  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Bang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Seberson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Ju, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Ahn, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Gao,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Robicheaux,  and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Li, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 2, 043054  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Gil and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Ossikovski, Polarized Light and the  Mueller Matrix Approach, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (CRC Press, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Shevchenko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Roussey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Friberg, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Setälä,  Optica 4, 64 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Novotny and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Hecht, Principles of Nano-Optics, 2nd  ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (Cambridge University Press, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Eismann,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Nicholls,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Roth,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Alonso,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Banzer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rodríguez-Fortuño, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Zayats,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Nori,  and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Bliokh, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Dong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Cai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Norrman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Friberg,  and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Setälä, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Norrman,  and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+page_content=' Norrman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Friberg, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Setälä, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 47, 2566 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [43] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Weisstein, From MathWorld–A Wolfram Web Re-  source .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [44] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Landau and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Lifshitz, Mechanics, Third Edi-  tion: Volume 1 (Course of Theoretical Physics), 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (Butterworth-Heinemann, 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  [45] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rahav, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Gilary, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Fishman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' A 68,  013820 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Supplemental Material  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' VARIABLE DOP TRAPPING BEAM PREPARATION  This section describes the preparation of the variable DOP trapping beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We provide experimental details of the  most important aspects, such as ensuring the spatial overlap of the two orthogonally-polarized components of the  tweezer beam and the calibration of the DOP in the trapping region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The tweezer beam generation system is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' S1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Before entering the vacuum chamber, the trapping  beam, propagating along the z axis, is split into two components with equal power but orthogonal linear polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Each of these constituent beams is then frequency shifted by an AOM (y-polarized beam is shifted by −80 MHz by  AOM 1, and x-polarized beam is shifted by +80 MHz by AOM 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Subsequently, the beams are spatially overlapped  and recombined on a polarizing beam-splitter (PBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Both beams are first focused into their respective AOMs using  one lens before the split (f = 200 mm), and then collimated using another lens (f = 500 mm) after the recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The spatial mode overlap of the two beams is optimised by maximising the visibility of interference at the difference  frequency ∆ω = 160 MHz, recorded on the auxiliary detector (PD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' S1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The beams are overlapped by  means of two steering mirrors, and the beam sizes are matched using a corner-cube (CC) mounted on a translation  stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The CC allows us to adjust the path length difference between the x- and y-polarized components between  focusing (f = 200 mm) and collimating (f = 500 mm) lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The maximum achieved interference visibility is 85%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The aforementioned system allows us to control the relative contributions of x- and y-polarized light to the trapping  beam via controlling the driving power of the AOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' For example, if AOM 1 is driven at maximum power, and AOM 2  receives no drive, the tweezer is y polarized, whereas when both AOM 1 and AOM 2 are driven so that they provide  the same diffraction efficiency, the tweezer is composed of x and y polarizations in equal measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  In the remainder of this section we will describe the calibration of the contributions of x- and y-polarized light to  the intensity in the trapping volume, which directly determine the DOP in the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' In order to avoid systematic  errors introduced by spatial mode mismatch of the x- and y-polarized beams, we perform this calibration using the  COM motion of the trapped particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The relationship between the focal intensity and the AOM driving power is determined for each beam separately  with the help of the transverse COM frequencies describing the particle motion in the focal plane along x and y  (denoted as fx and fy respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Let us denote the average transverse COM frequency as fav = 1  2(fx + fy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We  use the average transverse COM frequency squared f 2  av as a proxy for the focal intensity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The measured dependence of f 2  av on the driving power of each AOM (together with a quadratic fit) is depicted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' S1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We verify the calibration by performing the scan of the degree of polarization shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' S1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  As desired, we find that fav remains constant when s1 is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Note that the average transverse COM frequency  fav does not decrease for s1 = 0 (when 50% of focal intensity is x-polarized and 50% is y-polarized), even though  the calibration is performed for each polarization component separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' This shows that the volumes of the traps  generated by the two beams overlap well and their trap depths add together to form the final trapping potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The DOP calibration procedure is potentially affected by the fact that in our experiment the trapping beam has a  slightly elliptical shape introduced by the AOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' For a tweezer formed by a strongly focused circular Gaussian input  beam, fx and fy are not the same [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We quantify this effect using ϵ = (fx − fy)/(fx + fy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Clearly, in the case of  circular Gaussian beam, the value of |ϵ| should not depend on whether the tweezer is x- or y-polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Therefore,  different values of |ϵ| for x and y tweezer polarization (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' S1(c) at points s1 = ±1) suggest that the beam cross  section is elliptical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  In order to quantify the effect of the trapping beam ellipticity on f 2  av, we have simulated the focal field produced by  the trapping lens [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The beam waists along x and y before focusing are used as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The obtained focal  intensity cross sections along x and y for both x and y polarized tweezer light are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' S1(d) and reproduce  well the observed values of ϵ for an x- and y-polarized tweezer (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='057 and ϵ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='116 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We find that  the values of the average transverse COM frequency fav for both polarizations differs by less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Hence, we  conclude that f 2  av offers a good focal intensity estimate in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' LIBRATIONAL POTENTIALS  In this section we derive the potential governing the libration of an anisotropic particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The potential is induced  by the electric field of the trapping beam which is propagating along z and is composed of two frequency shifted  1  (a)  (b)  (c)  (d)  f=200  f=500  PBS  PBS  HWP  +80MHz  AOM 2  AOM 1  80MHz  PD  HWP  to trap  Laser  CC  z  y  x  Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (a) Experimental diagram of the beam preparation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The laser beam is split into two parts, which are  frequency-shifted by ±80 MHz (shown by violet and orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The two parts are subsequently recombined and sent to the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Abbreviations: half-wave plate (HWP), polarizing beam splitter (PBS), photo-diode (PD), corner cube (CC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (b) The average  COM transverse frequency squared f 2  av as a function of AOM driving power for both AOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The data points are shown together  with quadratic fits (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (c) The power spectral density (PSD) of COM motion as a function of DOP of the trapping  beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' This subfigure consists of 100 PSDs, where the first one (s1 = −1) corresponds to y-polarized trapping light and the  last one (s1 = 1) to x-polarized trapping light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The PSD is measured by the quadrant photo detector (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 2(a), both QPD  channels are summed here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' COM transverse frequencies fx and fy are visible in the 120 to 150 kHz region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (d) Simulated  intensity profiles along x and y directions for a strongly focused elliptical beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Note that the cross-section of the focal spot  is more circular for y-polarised tweezer beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  components (x- and y-polarized) of different amplitudes:  E = (Exeiωxt, Eyeiωyt, 0)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (S1)  The x and y field components acquire a phase difference that grows linearly in time in proportion to the frequency  ∆ω = ωx − ωy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Since the initial phase difference is not important to the dynamics, we assume that the amplitudes  Ex and Ey are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  We assume that both the moment of inertia and the polarizability tensor can be simultaneously diagonalized in the  principal axes reference frame of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We refer to this frame of reference as “particle frame” and represent  the principal axes as (e1, e2, e3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We denote α = diag(α1, α2, α3) and I = diag(I1, I2, I3) as the static polarizability  and inertia tensors of the object in the intrinsic body frame, respectively, assuming α1 ≤ α2 ≤ α3 and I1 ≥ I2 ≥ I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  We refer to e3 (the principal axis with the largest polarizability) as the “long axis” of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  In order to describe the orientation of the particle frame with respect to the laboratory frame (x, y, z), we use the  intrinsic x-convention of Euler angles denoted as φ, θ and ψ (see [43] and §35 in [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The Euler angles φ and θ  2  describe the orientation of the long axis of the rotor (the angle measured in the experiment is φ, which corresponds to  the orientation of the long axis in the xy plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' In order to transform a vector from laboratory to particle frame we  first rotate it by the angle φ around z, then by θ around e1 and finally by ψ around e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The transformation matrix  corresponding to these three rotation operations reads [43]  R =  �  �  cos ψ cos φ − cos θ sin ψ sin φ  cos θ sin ψ cos φ + cos ψ sin φ  sin θ sin ψ  − cos θ cos ψ sin φ − sin ψ cos φ  cos θ cos ψ cos φ − sin ψ sin φ  sin θ cos ψ  sin θ sin φ  − sin θ cos φ  cos θ  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (S2)  The dipole moment induced in a trapped anisotropic particle, expressed in the laboratory frame of reference, yields:  p = R−1αRE,  (S3)  where ⃗E is also expressed in the laboratory reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Since in general ⃗p and ⃗E are not parallel, the potential  energy Utot associated with the orientation of the particle (after averaging over optical frequencies) yields:  Utot = −1  4Re (p · E∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (S4)  If we consider the electric field as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (S1), the potential Utot can be written as a sum of two terms  Utot = U0 + U1 cos (∆ωt),  (S5)  with  U0 =S0  8  �  −(α1 + α2) + sin2 θ (1 − s1 cos 2φ)  �  α1 sin2 ψ + α2 cos2 ψ − α3  �  + s1 (α1 − α2) (cos θ sin 2ψ sin 2φ − cos 2ψ cos 2φ)  �  ,  (S6a)  U1 = − S0  �  1 − s2  1  8  �  sin2 θ sin 2φ (α1 sin2 ψ + α2 cos2 ψ − α3)  + (α1 − α2)(cos θ sin 2ψ cos 2φ + cos 2ψ sin 2φ)  �  ,  (S6b)  where S0 and s1 = S1/S0 are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (1a) and (1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The oscillating term U1 cos(∆ωt) describes the interaction  of a dipole moment induced by the field oscillating at ωx with the field oscillating at ωy (and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The fast oscillating term U1 cos(∆ωt) will give rise to a small amplitude micromotion at ∆ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' This “fast” micromotion  will in turn have an effect on the “slow” librational dynamics, which can be described as an additional, constant-in-time  effective potential term U ′  1 [44, 45], yielding  U ′  1 =  1  2(∆ω)2  �  i,k  A−1  ik ∂iU1∂kU1 ,  (S7)  where the indices i and k run through θ, φ and ψ and Aik are matrix elements of the quadratic form describing the  kinetic energy T, such that  T = 1  2( ˙θ, ˙φ, ˙ψ)A( ˙θ, ˙φ, ˙ψ)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (S8)  The matrix A depends on the Euler angles and the inertial moment according to  A =  �  �  I1 cos2 ψ + I2 sin2 ψ  (I1 − I2) sin θ sin ψ cos ψ  0  (I1 − I2) sin θ sin ψ cos ψ  (I1 sin2 ψ + I2 cos2 ψ) sin2 θ + I3 cos2 θ  I3 cos θ  0  I3 cos θ  I3  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (S9)  As an example, let us find the explicit expressions for U0 and U ′  1 in the case of a rotor with cylindrical symmetry,  such as a dumbbell (α1 = α2, I1 = I2):  U0 = A2  1  4 I1(s1 cos 2φ − 1) sin2 θ ,  (S10a)  U ′  1 = A4  1I1(1 − s2  1) sin2 θ  8(∆ω)2  �  I1 sin2 θ cos2 2φ + (I1 sin2 θ + I3 cos2 θ) cos2 θ sin2 2φ  �  (I1 sin2 θ + I3 cos2 θ)  ,  (S10b)  3  where the parameter A1 =  �  (α3−α1)S0  2I1  is equal to the libration frequency in a linearly polarized trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Equations (S10a)  and (S10b) indicate that the residual potential U ′  1 is shallower than U0 by approximately a factor of (A1/∆ω)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Note  that for our experimental parameters, we have (A1/∆ω)2 ≈ 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Let us now consider how both potential terms affect the dynamics of the orientation of the dumbbell in the xy  plane described by the angle φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We can expect the contribution from U ′  1 to be negligible, except when s1 ≈ 0 (the  trap is unpolarized) and U0 is independent of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Therefore, in this case, U ′  1 is the dominant potential term governing  the dynamics of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The minima of U ′  1 occur at φ = nπ/4 with n ∈ {1, 2, 3, 4}, which means that our symmetric rotor  will become diagonally oriented in the xy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  However, in our experiment we have U ′  1 ≪ kBT, where kB is the Boltzmann constant and T is the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Therefore, even when the tweezer is unpolarized, the effect of the residual potential U ′  1 is completely obscured by the  interaction with the environment (background gas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' TORQUES AND LIBRATION FREQUENCIES FOR ASYMMETRIC ROTOR  In this section we use the potential derived in the previous section to calculate restoring torques acting on a rotor  trapped in a beam with an arbitrary DOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We derive the libration frequencies as functions of DOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  We denote particle-frame torque components as Ki and ki =  Ki  Ii .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  The full expressions for torques due to the  time-independent potential U0 [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (S6a)] read  k1 = kN cos ψ + (kz − k3 cos θ) sin ψ  sin θ  = A2  1  2 sin θ [(1 − s1 cos 2φ) cos θ cos ψ + s1 sin ψ sin 2φ] ,  (S11a)  k2 = −kN sin ψ + (kz − k3 cos θ) cos ψ  sin θ  = −A2  2  2 sin θ[(1 − s1 cos 2φ) cos θ sin ψ − s1 cos ψ sin 2φ] ,  (S11b)  k3 = −A2  3  2  �  (1 − s1 cos 2φ) sin2 θ sin ψ cos ψ + s1(cos θ cos 2ψ sin 2φ + sin 2ψ cos 2φ)  �  ,  (S11c)  where k3 = −∂U0/∂ψ, kN = −∂U0/∂θ, kz = −∂U0/∂φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Additionally we have Ai = [(|αj − αk|S0)/2Ii]1/2 where  {i, j, k} are permutations of {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  To gain an intuitive understanding, let us analyze in detail the case s1 = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=', the tweezer is linearly polarized  along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The object is trapped in a potential minimum (along x), for which θ = φ = π  2 , and ψ is unrestrained (free).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  This means that the rotor aligns itself with its long axis to the polarization axis, while it can freely spin around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Let  us now move beyond purely linear polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' As soon as s1 < 1, a potential minimum appears at ψ = π  2 and the  angle ψ also becomes trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Colloquially speaking, the rotor can now minimize its potential energy by “lying flat”  in the polarization plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Having discussed the orientation with minimal potential energy, let us consider deviations  from that orientation and the associated dynamics, which corresponds to librational motion for small angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' To this  end, we introduce the libration angles describing the motion around the equilibrium position as Θ = θ − π  2 , Φ = φ− π  2 ,  and Ψ = ψ− π  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Without loss of generality, it is convenient to assume a tweezer field that is predominantly x-polarized,  such that the resulting torque components can be expressed in the laboratory frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' To first order in the Euler angles,  these restoring torques acting on the angles Θ, Φ, and Ψ read  kz ≈ −s1A2  1Φ,  (S12a)  ky ≈ −s1 + 1  2  A2  2Θ,  (S12b)  kx ≈ −1 − s1  2  A2  3Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (S12c)  The libration frequencies associated with these restoring torques read  Ω1 =  √  PA1,  (S13a)  Ω2 =  �  P + 1  2  A2,  (S13b)  Ω3 =  �  1 − P  2  A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (S13c)  In the above expressions we have replaced s1 with P so that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (S13a)–(S13c) are also valid around s1 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' In  the case of s1 ≈ −1 the tweezer is almost y-polarized, and the potential minimum occurs for a different particle  4  orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Therefore, the angles θ, φ and ψ librate around different equilibrium positions (denoting the potential  minimum) for different polarization states of the tweezer field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' However, the libration frequencies are, to linear order,  always given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (S13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' EQUATIONS OF MOTION FOR A SYMMETRIC ROTOR  In this section, we analyze the motion of a symmetric rotor (I1 = I2, α1 = α2) under restoring torques given by  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (S11a)–(S11c) and an additional small spinning torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Since the parameter A3 vanishes for the symmetric rotor,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (S11c) implies that there is no restoring torque pointing along the object’s long axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Therefore, for symmetric  rotors the angle ψ is free for any value of s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Let us now explore a scenario in which the symmetric rotor is performing a small amplitude libration around the x  axis (s1 ≈ 1) and an additional constant torque is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We assume that the additional torque is much smaller than  the restoring torques along y and z [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (S12a) and (S12b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Therefore, only the additional torque component  pointing along x will have an effect on the dynamics, causing the rotor to spin around its long axis (which points long  x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We denote the additional torque component along x as kext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Let us write the rotational equations of motion [44]  (Euler’s equations), including both the restoring torques and kext for a symmetric rotor performing small amplitude  libration around the x axis :  ¨Φ = −µ ˙Θ ˙Ψ − Ω2  1Φ,  (S14a)  ¨Θ = µ ˙Ψ ˙Φ − Ω2  2Θ,  (S14b)  ¨Ψ = kext,  (S14c)  where µ = I1−I3  I1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Provided that kext is larger than fluctuating (thermal) torques, the dumbbell will start to spin  consistently around its long axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The spinning rate will increase until it reaches the friction-dependent stationary  value denoted as ˙Ψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Note that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (S14a) and (S14b) include coupling between angular degrees of freedom, which  is an inherent feature of rotational mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Fast spinning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=', large ˙Ψ causes strong coupling between the Φ and  Θ libration modes which become hybrid precession modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' Similarly to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' [30] we use the following simple model  ¨Φ = −ωs ˙Θ − Ω2  1Φ,  (S15a)  ¨Θ = ωs ˙Φ − Ω2  2Θ,  (S15b)  where ωs = µ ˙Ψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We will henceforth refer to ωs as the rotational coupling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The eigenfunctions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (S15a)-  (S15b) (precession modes) oscillate at eigenfrequencies ΩA and ΩB, which read  ΩA =  �  Ω2  1 + Ω2  2 + ω2s −  �  (Ω2  1 + Ω2  2 + ω2s )2 − 4Ω2  1Ω2  2  √  2  ,  (S16a)  ΩB =  �  Ω2  1 + Ω2  2 + ω2s +  �  (Ω2  1 + Ω2  2 + ω2s )2 − 4Ω2  1Ω2  2  √  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (S16b)  We notice that the precession frequencies ΩA and ΩB satisfy the following relation:  (ΩA − ΩB)2 = ω2  s + (Ω1 − Ω2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  (S17)  This relation illustrates the signature of the spinning along the long axis, which can be described as an additional  splitting between the eigenfrequencies visible in the libration spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (S17) to reconstruct the  value of the rotational coupling ωs (as function of s1) from the measured precession mode splitting (ΩA − ΩB) and  the libration mode splitting (Ω1 −Ω2) predicted from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (S13a) and (S13b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The value of A1 used in this procedure  was measured at P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' The obtained rotational coupling rate is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 3(b) as a dashed black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  Note that ωs depends only on the difference between observed peak frequencies in the libration spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We  have calculated ωs from the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' 3(b) and used it to find the precession mode frequencies from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' (S16a)  and (S16b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  We find that the resulting precession frequencies (as functions of s1) fit the behavior of the modes  observed in the libration spectrum well, which confirms the spinning hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  5  Lastly, we have only discussed the effect of the additional small torque on a symmetric rotor (dumbbell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' We note  that non-symmetric rotors (clusters), when exposed to the same additional torque, will not spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content=' This is due to the  fact that for P < 1 all degrees of freedom describing cluster orientation are trapped in a potential minimum, such  that they librate and do not spin, unless the additional torque exceeds the restoring torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
+page_content='  6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE3T4oBgHgl3EQfdgrZ/content/2301.04536v1.pdf'}
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+arXiv:2301.02278v1  [cond-mat.mtrl-sci]  5 Jan 2023
+Computational characterization of novel nanostructured materials: Case study of
+NiCl2
+Elizaveta B. Kalika,1 Alexey V. Verkhovtsev,2, ∗ Mikhail M. Maslov,3 Konstantin P. Katin,3 and Andrey V. Solov’yov2
+1Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Moscow Region, 141700, Russia
+2MBN Research Center, Altenhöferallee 3, 60438 Frankfurt am Main, Germany
+3Department of Condensed Matter Physics, National Research Nuclear University “MEPhI”,
+Kashirskoe Shosse 31, Moscow, 115409, Russian Federation
+A computational multiscale approach combining dispersion-corrected density functional theory
+(DFT) and long-timescale classical molecular dynamics is employed to characterize the geometrical,
+mechanical and thermal properties of a recently proposed two-dimensional (2D) transition metal
+dichalcogenide, NiCl2. A classical interatomic force field is proposed whose parameters are derived
+from the results of DFT calculations. The developed force field is used to study the mechanical
+response, thermal stability and melting of a NiCl2 monolayer. It is found that the NiCl2 sheet is
+thermally stable up to the melting point temperature of 1315 K. At higher temperatures structural
+degradation of the system is observed, which involves several subsequent structural transformations,
+namely the formation of a highly porous 2D sheet, 1D nanowires, and nanodroplets. The compu-
+tational methodology presented through an illustrative case study of NiCl2 can be utilized for the
+computational characterization of other novel 2D materials, including recently synthesized NiO2,
+NiS2 and NiSe2.
+I.
+INTRODUCTION
+Nickel-based nanomaterials are considered promising
+for sensors, adsorbents, and catalysts. Recently synthe-
+sized Ni-based composites have proven their effectiveness
+as sensors for various organic and biological molecules
+[1–3].
+Nickel nanoparticles possess catalytic activity
+for many reactions, including the synthesis of primary
+amines and hydrogen oxidation [4, 5]. Nickel-based two-
+dimensional (2D) materials can be used for the efficient
+generation of molecular oxygen [6, 7], electrocatalytic wa-
+ter splitting [8] and glucose oxidation [9]. In addition,
+recently synthesized pentagonal 2D sheets of nickel di-
+azenide (NiN2) have tunable direct band gap and may
+serve as a precursor for pentagonal 2D materials [10].
+Nickel-doped graphene materials are also widely used
+for many applications. In particular, they provide ex-
+cellent sensing properties [11–13] and can effectively cat-
+alyze such technologically important reactions as hydro-
+gen production from water [14] and ethanol steam [15],
+oxygen reduction under alkaline conditions [16], and car-
+bon monoxide reduction [17]. Another important appli-
+cation of Ni-doped graphene is hydrogen storage. Due to
+3d electrons and the spillover effect, nickel can hold hy-
+drogen molecules [18–20]. At the same time, graphene is
+not the only 2D material that can be doped with nickel.
+Less common 2D materials, such as silicene, germanene
+and MoS2, have also been used as a substrate for nickel
+atoms and nanoparticles [21–23].
+The aforementioned properties of nickel-doped 2D ma-
+terials are based on a combination of the mechanical char-
+acteristics of a 2D sheet with the adsorption and cat-
+alytic properties of nickel. A further development of this
+∗ verkhovtsev@mbnexplorer.com
+idea is that nickel can be not an alloying impurity but
+the basis for a 2D material. Pure nickel is a metal and
+cannot exist in the form of the 2D allotrope. However,
+an extensive computational search for layered crystals
+and related 2D materials has recently predicted the ex-
+istence of several 2D nickel-based materials of the com-
+position NiX2 (X = O, Cl, Br, I, S, Se) [24]. Some of
+these materials have already been synthesized. Technolo-
+gies developed for widespread 2D MoS2/MoSe2 materials
+[25] and their analogs based on niobium [26], titanium
+[27] and tungsten [28] proved to be useful for synthesiz-
+ing NiX2 monolayers. Large-area NiO2 monolayers sep-
+arated by lanthanum atoms have been observed using
+scanning tunneling microscopy [29]. Recent experiments
+have proven the feasibility of chemical vapor deposition
+synthesis of nanometer layers of NiCl2 [30], NiS2 [31] and
+NiSe2 [32]. Moreover, NiCl2 films up to four layers thick
+have been recently synthesized [30]. In all these mate-
+rials, nickel atoms are organized in a regular 2D lattice
+so that the materials possess enhanced adsorption and
+catalytic properties.
+Currently, there is limited experimental information
+on the properties of such 2D materials since only a few
+laboratories have synthesized them in limited quantities.
+Atomistic computer simulations can serve as an alterna-
+tive approach to characterize the structure of such novel
+materials and explore their properties.
+This paper presents the results of a computational
+characterization of a novel 2D material, NiCl2, by means
+of a multiscale modeling approach. NiCl2 has been cho-
+sen as an illustrative and experimentally relevant repre-
+sentative of the Ni-based 2D materials family. A combi-
+nation of quantum and classical approaches into a unified
+multiscale methodology permits a comprehensive investi-
+gation of the structure, mechanical properties, and ther-
+mal stability of NiCl2. Density-functional theory (DFT)
+calculations provide reference data on geometrical char-
+
+2
+acteristics of the material. On this basis, a new classi-
+cal interatomic potential is developed and benchmarked
+against the results of quantum-mechanical calculations.
+The validated potential is used for atomistic modeling of
+mechanical deformations and thermal stability of NiCl2
+by means of the advanced software packages MBN Ex-
+plorer [33] and MBN Studio [34].
+Through an illustrative case study of NiCl2 we present
+a general methodology that can be utilized for the com-
+putational characterization of other novel materials, in-
+cluding the recently synthesized NiO2 [29], NiS2 [31] and
+NiSe2 [32]. To the best of our knowledge, NiCl2 has not
+been studied computationally except for a recent study
+[35], which investigated structural defects in NiCl2 and
+similar materials and confirmed their stability in the en-
+vironment by means of DFT calculations.
+The paper is organized as follows. Section II describes
+the key aspects of theoretical methods utilized in DFT
+calculations and corresponding classical simulations. A
+particular focus is made on the procedure to determine
+the classical force field parameters for NiCl2. This follows
+with the discussion of the obtained results in Section III.
+The accuracy of the developed force field is evaluated
+in Section III A through the analysis of structural and
+energetic parameters of NiCl2. Section III B is devoted to
+the analysis of mechanical properties of NiCl2 by means
+of DFT and classical energy minimization calculations
+using the developed force field. In Section III C the force
+field is utilized to study the thermal stability of NiCl2
+and evaluate its melting temperature by means of long-
+scale MD simulations. Finally, in Section IV we draw the
+conclusion from this work and give an outlook for further
+developments in this research direction.
+II.
+COMPUTATIONAL METHODOLOGY
+A.
+DFT calculations
+DFT calculations of the structural properties of a
+NiCl2 sheet have been performed using the Quantum
+ESPRESSO 6.5 software package [36, 37].
+The plane-
+wave basis set for valence electron states, generalized
+gradient approximation (GGA) in the Perdew-Burke-
+Ernzerhof (PBE) functional form for the exchange-
+correlation energy [38], and projector-augmented-wave
+(PAW) pseudopotentials [39, 40] for core-electron inter-
+actions were used to perform the calculations. We have
+employed the kinetic energy cut-off for wave functions of
+100 Ry (∼1360 eV) and kinetic energy cut-off for charge
+density and potential of 600 Ry (∼8160 eV) with checking
+convergence of energy and charge to increase the simu-
+lation accuracy. In addition, the van der Waals inter-
+actions have been taken into account through the D3
+Grimme (DFT-D3) dispersion corrections [41].
+DFT-
+D3 approach possesses improved accuracy due to the use
+of environment-dependent dispersion coefficients and the
+inclusion of a three-body component to the dispersion
+FIG. 1. Structure of a NiCl2 supercell. Ni and Cl atoms are
+shown in blue and green colors, respectively.
+correction energy term.
+The interlayer distance between separate NiCl2 sheets
+has been set equal to 30 Å, which provides sufficient space
+separation to avoid nonphysical interactions. Thus, opti-
+mization of the lattice parameters along the axis perpen-
+dicular to the NiCl2 sheet plane was unnecessary. The
+atomic equilibrium positions have been obtained by the
+total minimization of the supercell using the calculated
+forces and stress on the atoms. The convergence criterion
+of self-consistent calculations for ionic relaxations was set
+to 10−10 eV between two consecutive steps. The geom-
+etry optimization of the unstressed NiCl2 sheet was car-
+ried out without symmetry constraints until the Hellman-
+Feynman forces acting on the atoms became smaller than
+10−4 hartree/bohr. Such criteria ensure that the abso-
+lute value of stress is less than 0.01 kbar. The parameters
+of the supercell have also been optimized. The first Bril-
+louin zone integrations have been performed by using the
+Monkhorst-Pack k-point sampling scheme [42] with the
+6 × 6 × 1 mesh grid together with the Methfessel-Paxton
+smearing [43] with the smearing width of 0.02 Ry.
+The NiCl2 sheet is represented by a hexagonal Ni9Cl18
+periodic cell containing 3 × 3 elementary NiCl2 cells, see
+Fig. 1. The hexagonal symmetry corresponds to the re-
+sults obtained earlier for this monolayer [24]. After op-
+timization, we obtained the lattice constant a = 3a0 =
+10.328 Å, where a0 is the parameter of the NiCl2 prim-
+itive unit cell, while the evaluated Ni–Cl bond length
+was equal to 2.377 Å. Additional DFT-based geometry
+optimization calculations have been performed using the
+Gaussian software package [44] employing three differ-
+ent exchange-correlation functionals, namely LDA, PBE0
+
+3
+and HSE06.
+B.
+Classical MD simulations
+Classical geometry optimization calculations and MD
+simulations have been performed by means of MBN Ex-
+plorer [33] – a software package for advanced multiscale
+modeling of complex molecular structure and dynam-
+ics.
+MBN Explorer permits to simulate a wide range
+of Meso-Bio-Nano (MBN) systems, including nanosys-
+tems [33, 45–47]; nanostructured materials [48–51]; com-
+posite and hybrid materials [52–55] with sizes ranging
+from atomic to mesoscopic. The dedicated MBN Studio
+toolkit [34] has been utilized to create the systems, pre-
+pare all necessary input files, and analyze and visualize
+simulation outputs.
+C.
+Determination of the classical force field for
+NiCl2
+The first part of this study has been devoted to the
+determination of parameters of a classical force field for
+NiCl2.
+The geometry of the 3 × 3 NiCl2 cell obtained after
+DFT optimization was taken as an initial geometry for
+the calculations using the classical force field. The NiCl2
+sheet has been simulated by assigning the NiCl2 super-
+cell to a simulation box which is replicated in space using
+periodic boundary conditions. First, several MD simu-
+lations have been carried out with the trial force field
+parameters fitted for transitional metal dichalides NiF2
+[56], ZnCl2 [57], AlCl3 and FeCl3 [58]. The force field
+employed in these simulations has been constructed as a
+sum of the short-term exponential, long-term r−6 power
+and the Coulomb potentials. Such force fields have been
+commonly utilized in MD simulations of various inorganic
+and ionic crystals. The simulation box size was set ac-
+cording to the lattice size optimized by DFT. The NiCl2
+system was unstable with any of the parameters; it was
+also noticeable that the sheet would bend using most of
+the given force field potentials.
+As the NiCl2 sheet would bend with all of the above
+mentioned force fields, the simulation box size has been
+gradually increased in two dimensions. It has been found
+that the NiCl2 sheet remains flat when the simulation
+box size is increased by 4%.
+As such, the simulation
+box size used for all further simulations has been set to
+53.32 Å × 92.59 Å × 30.00 Å.
+Next, a new set of classical force field parameters for
+the NiCl2 system has been determined in the following
+way. The Nelder-Mead algorithm [59] has been utilized
+for the optimization of the force field parameters. The al-
+gorithm is simplex-based, meaning that in n-dimensional
+space it maintains a set of n + 1 points called simplex.
+After calculating the value of the function at each point
+of the simplex, it extrapolates the behavior of the func-
+tion to find a new test point and replaces the test points
+with the worst result for a new one. The process repeat-
+edly continues until the result converges to the tolerance
+value or reaches the maximum number of iterations. The
+Nelder-Mead algorithm was implemented in Python via
+scipy.optimize.minimize library [60]. This algorithm was
+used to fit the geometrical parameters of NiCl2 calculated
+by MBN Explorer [33] to the ones obtained through the
+DFT-based optimization; details are given below in Sec-
+tion III A.
+NiCl2 is an ionic crystal with partial charges on nickel
+and chlorine atoms. In order to describe more accurately
+the mutual polarization interaction between atoms of the
+system, the r−6 potential has been replaced with a r−4
+power potential. The resulting total interaction potential
+between atoms i and j of the system is given by the
+following expression:
+U(rij) = Aij exp(−αijrij) + Cij
+r4
+ij
++
+qiqj
+4πε0rij
+(1)
+The following six parameters of the force field, Eq. (1),
+have been optimized using the Nelder-Mead algorithm:
+ANi−Cl, αNi−Cl, ACl−Cl, αCl−Cl, C and q. Partial charges
+on Ni and Cl atoms have been set to ensure that the
+system is electrically neutral, |qNi| = 2|qCl|. A 7 Å cut-
+off was applied for the exponential and power terms of
+the potential. The Coulomb potential was calculated us-
+ing the Ewald summation method with precision of 10−5
+and the cutoff of 12 Å. MD simulations with simulation
+time of 30 ps and time step of 1 fs, Langevin thermostat
+temperature of 300 K and damping time of 0.1 ps were
+performed for each generated set of parameters.
+The following geometrical parameters of NiCl2 have
+been compared to the results of DFT calculations: (i) Ni–
+Cl bond length (lNi−Cl); (ii) Ni–Ni bond length (lNi−Ni);
+(iii) Cl–Cl bond length (lCl−Cl); (iv) Cl–Ni–Cl angle
+(ϕCl−Ni−Cl), denoted hereafter as ϕ for brevity. In addi-
+tion, the fifth parameter, namely the system’s energy as
+a function of Ni–Cl interplanar distance has also been an-
+alyzed. The distance between the Ni and Cl planes has
+been gradually varied and single point energy calcula-
+tions were carried out using MBN Explorer for different
+Ni–Cl interplanar distances. The resulting dependence
+of system’s energy on the Ni-Cl interplanar distance has
+been fitted by a quadratic function and its minimum, d,
+was found. The value of d obtained from classical force
+field calculations was compared to the results of DFT
+calculations.
+Then, the standard deviation
+σ =
+�
+�
+�
+��
+i
+�li − lDFT
+i
+lDFT
+i
+�2
++
+�ϕ − ϕDFT
+ϕDFT
+�2
++
+�d − dDFT
+dDFT
+�2
+(2)
+has been calculated and minimized using the Nelder-
+Mead algorithm. Here the summation has been carried
+out over different covalent bonds, namely Ni–Ni, Ni–Cl
+and Cl–Cl. Each round of optimization consisted of 30
+
+4
+FIG. 2. Evolution of parameters of the force field, Eq. (1), as a function of the number of global optimization iterations carried
+out using the Nelder-Mead optimization algorithm.
+iterations and the parameters resulting from each opti-
+mization round were used as the starting parameters for
+the following one. In this study, 430 subsequent itera-
+tions have been carried out to determine the parameters
+of the force field, Eq. (1).
+The evolution of the force field parameters in the
+course of the parameter optimization procedure is shown
+in Fig. 2. The convergence is due to the decrease of σ
+value, Eq. (2). The final values of the force field param-
+eters and partial charges derived from the comparison
+of NiCl2 geometrical parameters to the DFT results are
+listed in Table I. These values have been employed in all
+the simulations described below in Section III.
+III.
+RESULTS AND DISCUSSION
+A.
+Benchmarking the accuracy of the constructed
+force field
+NiCl2 is a novel material for which very little infor-
+mation has been obtained both experimentally and com-
+putationally. Hence, it is important to evaluate the ac-
+curacy of the developed classical force field in order to
+make computational predictions of structural and dy-
+namical properties of NiCl2.
+Table II lists the values
+of Ni–Cl, Ni–Ni and Cl–Cl bond lengths and Cl–Ni–Cl
+angles obtained via different optimization methods: clas-
+sical force field optimization (column labeled ‘FF’) and
+DFT optimization calculations using different exchange-
+correlation functionals. In addition to DFT-based cal-
+culations described in Sect. II A, several complementary
+calculations have been carried out using the Gaussian
+software package [44] to evaluate the optimal geometri-
+cal parameters of NiCl2 with three different exchange-
+correlation functionals (PBE0, HSE06 and LDA).
+The results listed in Table II indicate that the Ni–
+Cl bond length determined from classical structure opti-
+mization calculations is close to the values obtained by
+means of DFT with a relative discrepancy of 6%. The
+equilibrium Ni–Ni bond length is longer than the values
+obtained by DFT by about 5%. At the same time, the
+equilibrium Cl–Cl bond length is shorter than the values
+obtained by DFT, with a relative deviation of about 20%.
+A smaller Ni–Cl interplanar distance results in smaller
+values of the Cl–Ni–Cl angles. The discrepancy might be
+attributed to limitations of the Nelder-Mead optimiza-
+tion algorithm exploited in this study and a limited num-
+ber of optimization iterations.
+Overall, the classical force field. Eq. (1), with the pa-
+rameters listed in Table I describes with a reasonable ac-
+curacy the geometrical parameters of a 2D NiCl2 sheet.
+The range of discrepancies between the results of classical
+force field based and DFT-based geometry optimization
+calculations is similar to that obtained in earlier studies
+of transition-metal clusters and bulk materials [47, 48].
+B.
+Mechanical properties of NiCl2
+In this study, mechanical properties of a NiCl2 sheet
+have been studied by both DFT and classical force field
+calculations. DFT-based calculations have used as input
+the supercell described above in Section II A. The uni-
+form stretching of the NiCl2 sheet has been simulated by
+simultaneously increasing the lattice parameters of the
+
+5.0-
+-0.60-
+4.9 -
+-0.65.
+11000
+(eV)
+-0.70-
+4.8
+(eV
+Ni-CI
+10500
+4.7.
+C
+α
+-0.75
+A
+4.6.
+-0.80
+10000
+4.5
+-0.85
+1550-
+-0.78
+3.8-
+-0.80
+1500-
+3.7
+(eV)
+-0.82
+1
+A
+1450-
+-0.84
+3.6-
+b
+-0.86-
+1400-
+3.5-
+-0.88 -
+-0.90.
+1350-
+3.4 -
+400
+0
+100
+200
+300
+400
+0
+100
+200
+300
+0
+100
+200
+300
+400
+Number of optimization iterations
+Number of optimization iterations
+Number of optimization iterations5
+TABLE I. Parameters of the force field, Eq. (1), derived in this study for the description of a 2D NiCl2 material.
+Aij (eV)
+αij (Å−1)
+C (eV Å4)
+q, |e|
+Ni–Cl
+10804.52
+4.73
+Ni
++1.718
+Cl–Cl
+1465.99
+3.55
+−0.805
+Cl
+−0.859
+TABLE II. Comparison of geometrical parameters of a NiCl2 2D sheet after geometry optimization using the classical force
+field, Eq. (1) (column labeled ‘FF’), and DFT with different exchange-correlation functionals.
+Optimization method
+FF
+DFT
+PBE PBE0 HSE06 LDA
+average
+Ni–Cl, Å
+2.24 ± 0.02 2.38
+2.39
+2.40
+2.34 2.38 ± 0.03
+Ni–Ni, Å
+3.57 ± 0.03 3.44
+3.40
+3.38
+3.33 3.39 ± 0.05
+Cl–Cl, Å
+2.68 ± 0.02 3.28
+3.41
+3.38
+3.33 3.35 ± 0.06
+Cl–Ni–Cl, deg
+73.5 ± 0.8
+87.2
+91.0
+89.8
+88.9
+89.2 ± 1.6
+supercell along the x− and y−directions (parallel to the
+NiCl2 plane) and further optimizing atomic positions in
+the supercell.
+To estimate elasticity of the sheet, the latter was bi-
+axially stretched so that the lattice constant a increased
+according to the relation
+a = a0(1 + ε) ,
+(3)
+where a0 is the equilibrium lattice constant and ε is the
+relative deformation. The deformation energy ∆E has
+been calculated as follows:
+∆E = E − E0 ,
+(4)
+where E is the total energy of the deformed system and
+E0 is the energy of the initial (unperturbed) system. As
+the unit cell for DFT- and classical force field optimiza-
+tion calculations contained different numbers of atoms,
+the calculated deformation energy value was divided by
+the number of atoms in the respective unit cell.
+Figure 3 shows that the dependence of the NiCl2 sheet
+deformation energy on ε is well described by a quadratic
+function. The threshold for the elastic deformation was
+found to be 20 % and 18% for DFT-based and classical
+force field based optimization calculations, respectively.
+Note that graphene possesses similar elastic properties
+[61].
+For 2D materials,
+the common definition of the
+Young’s modulus is not applicable, since the thickness of
+a 2D sheet cannot be determined unambiguously. There-
+fore, 2D materials should preferably be characterized by
+the 2D Young’s modulus E2D, which does not depend
+on the sheet thickness [62].
+The value of E2D can be
+evaluated using the formula
+E2D =
+1
+2S0
+(∆E)′′ ,
+(5)
+where S0 is the area of the relaxed unit cell and (∆E)′′
+is the second derivative of the deformation energy with
+respect to biaxial deformation ε. The value of E2D =
+FIG. 3. Deformation energy ∆E of a biaxially stretched NiCl2
+sheet as a function of relative deformation ε.
+32.4 N/m has been obtained in this study through the
+DFT calculations of a NiCl2 sheet. The classical force
+field, Eq. (1), predicts a very close value of the 2D
+Young’s modulus, E2D = 34.9 N/m.
+To estimate the common 3D Young’s modulus, the ob-
+tained value E2D should be divided by the effective sheet
+thickness. According to the results presented in Table II
+the average NiCl2 sheet thickness (that is the average
+Cl–Cl bond length obtained from the DFT calculations
+with different exchange-correlation functionals), is equal
+to 3.35 Å. Using this value one obtains the Young’s mod-
+ulus equal to 97 GPa. This value is close to the value of
+107 GPa obtained for NiCl2 via DFT calculations in a re-
+cent study [35]. Note that the Young’s modulus reported
+for graphene is about 10 times higher [63]. 2D elastic
+modulus of monolayers with similar structures, MoS2 and
+WS2, are about 170 N/m [64]. Therefore, the results of
+calculations carried out in this study predict that NiCl2
+is a less rigid material compared to other well-studied 2D
+sheets, and its Young’s modulus is close to that of GaP
+monolayer [62]. Such a behavior is a consequence of the
+
+1.2
+口
+FF optimization
+DFT optimization
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.00
+0.05
+0.10
+0.15
+0.20
+Relative deformation ε6
+corrugated structure of 2D NiCl2 and high elasticity of
+the Ni–Cl bonds.
+C.
+Thermal stability of NiCl2
+The interatomic force field described in Section II C has
+been utilized to study thermal properties of a 2D NiCl2
+sheet and evaluate its melting temperature.
+This has
+been done by carrying out a series of constant tempera-
+ture MD simulations. The system initially equilibrated
+at 300 K was heated gradually by either 50 degrees (in
+the temperature range from 300 to 2000 K) or 100 degrees
+(in the range from 2000 to 5000 K) over 200 ps and then
+equilibrated over 10 ns at each temperature. A smaller
+temperature increment of 20 K has been considered in
+the temperature range from 1300 to 1500 K to evaluate
+the melting temperature of NiCl2 more accurately. The
+first 4 ns of the simulated trajectories were excluded from
+the analysis. The remaining part of each trajectory was
+used to evaluate the average value of the system’s to-
+tal energy at a given temperature. The resulting caloric
+curve, that is the dependence of the time-averaged total
+energy of the system on temperature, ⟨Etot⟩(T ), is shown
+in Fig. 4(a) by symbols. Figure 4(b) shows the temper-
+ature dependence of heat capacity at constant volume
+Cv, defined as a derivative of the internal energy of the
+system over temperature:
+Cv =
+�∂E
+∂T
+�
+v
+.
+(6)
+The melting process in a macroscopic system occurs at a
+specific temperature under fixed external pressure. This
+process reveals itself as a first-order phase transition via a
+spike in the heat capacity of the system at the transition
+temperature.
+At the initial phase of NiCl2 heating, the system’s to-
+tal energy grows linearly with temperature, see the inset
+in Fig. 4(a). At temperatures above ∼1200 K the NiCl2
+sheet transforms into a pre-melted state when the crys-
+talline structure is strongly deformed, but the 2D sheet
+still maintains its integrity (see the simulation snapshot
+in Fig. 7(b)). A slight increase in the heat capacity as a
+function of temperature also indicates the onset of sys-
+tem’s pre-melting (see Fig. 4(b)).
+The calculated heat capacity curve has a sharp maxi-
+mum at T ≈ 1315 K, indicating the melting temperature
+of NiCl2. This value is considerably lower than the melt-
+ing temperature for bulk nickel, Tm(Ni) = 1728 K. On
+the other hand, the evaluated value of T ≈ 1315 K is
+of the same order of magnitude as the melting temper-
+ature of a well-studied 2D material tungsten disulfide,
+Tm(WS2) ∼ 1520 K [65]. The evalualted melting temper-
+ature of NiCl2 also lies within the melting temperature
+range determined computationally for different 2D metal
+monoxides and monochlorides such as BeO, MgO, LiCl,
+and NaCl [66].
+FIG. 4.
+Caloric curve for NiCl2 (panel (a)) and the cor-
+responding heat capacity Cv as a function of temperature
+(panel (b)). The peak in the Cv(T ) dependence at T ≈ 1315 K
+indicates the melting temperature of NiCl2.
+At temperatures above the melting point, the system’s
+total energy ⟨Etot⟩ continues to grow linearly with T un-
+til the onset of another phase transition takes place at
+T ∼ 3200 K. In contrast to the melting phase transi-
+tion at Tm ≈ 1315 K, this high-temperature transition
+takes place over a broad temperature range from approx-
+imately 3200 to 4400 K, as indicated by a broad peak
+in the Cv(T ) dependence. This transition corresponds
+to the multifragmentation of NiCl2 into separate NiCl2
+molecular fragments.
+The phase transitions corresponding to the peaks in
+the heat capacity curve can be visualized by analyzing
+the radial distribution function (RDF) of atoms in the
+system.
+The RDF characterizes the number of atoms
+located at certain radial distance r from a reference atom.
+The RDF for Ni–Ni and Cl–Cl atomic pairs, evaluated
+at different temperatures of the system, are plotted in
+Figures 5(a) and 5(b), respectively. A rapid change of
+the RDFs between 1310 and 1320 K indicates the melting
+phase transition.
+It is of particular interest to explore, on the atomistic
+level, the phase transition from the crystalline NiCl2 to
+
+-8600
+-7000
+-8800
+Total energy (eV)
+-9000
+500
+1000
+1500
+-8000
+(a)
+-9000
+(b)
+4
+Heat capacity (eV/K)
+2
+0
+1000
+2000
+3000
+4000
+5000
+Temperature (K)7
+FIG. 5. Radial distribution function for nickel (panel (a)) and clorine (panel (b)) atoms of NiCl2 at different temperatures of
+the system. A rapid change of the RDFs between 1310 and 1320 K indicates the melting phase transition.
+FIG. 6.
+Variation of the system’s total energy, Etot, as a
+function of simulation time t. Colored curves show the Etot(t)
+dependencies for different temperatures, namely T = 1310 K
+(just below the phase transition temperature), T = 1320 K (in
+the phase transition region), as well as T = 1360 K and 1380 K
+(above the phase transition temperature). Labels (1) to (5)
+correspond to different time instants for the curve at T =
+1320 K. These instants correspond to the system’s snapshots
+shown in panels (c)–(g) of Fig. 7.
+its molten state. Figure 6 shows the variation of Etot as
+a function of simulation time t. Colored curves show the
+Etot(t) dependencies for different temperatures, namely
+T = 1310 K (just below the phase transition tempera-
+ture), T = 1320 K (in the phase transition region), as
+well as T = 1360 K and 1380 K (above the phase transi-
+tion temperature).
+At T = 1310 K (black curve in Fig. 6) the total energy
+of NiCl2 fluctuates around the average value of about
+−8800 eV over the 10 ns long simulation, which indi-
+cates that the system remains in the crystalline state.
+However, when the temperature increases to 1320 K the
+system’s energy stays nearly constant within the first
+∼3.7 ns of the simulated trajectory, which follows by
+a rapid rise of Etot(t) within ∼0.4 ns.
+This jump in
+the total energy is attributed to the formation of holes
+in the NiCl2 sheet as shown in Fig. 7(c,d). After that
+the system’s total energy decreases as the system relaxes
+into a highly porous 2D structure (see Fig. 7(e)). This
+structure is metastable, and it evolves quickly into an ar-
+ray of quasi-1D structures, where denser regions are con-
+nected by thin NiCl2 links (see Fig. 7(f)). This structure
+is also metastable, and it eventually relaxes into spheri-
+cal droplets with a diameter of ∼ 3.5nm (see Fig. 7(g)).
+Interestingly, as the system’s temperature increases, ran-
+dom evaporation of single NiCl2 units from the droplets
+and the fast diffusion of Ni and Cl atoms occurs. As a re-
+sult of these processes, the droplets eventually merge into
+a 1D NiCl2 nanowire which is thermally stable at tem-
+peratures below ∼3000 K within the simulation times
+considered in this study. This behavior, however, may
+be attributed to relative small sizes of the system and
+the simulation box, which enable the interaction of Ni
+and Cl atoms with their periodic images across the simu-
+lation box boundaries. A detailed analysis of system’s
+size effects on its thermal properties is an interesting
+topic which however goes beyond the scope of the present
+study. At higher temperatures, the NiCl2 nanowires dis-
+integrate and the multifragmentation process takes place,
+which is seen from the change of the ⟨Etot⟩(T ) slope
+shown in Fig. 4(a).
+IV.
+CONCLUSIONS
+Two-dimensional (2D) materials possess properties
+that are technologically relevant and differ from prop-
+erties of the corresponding bulk counterparts. Monolay-
+ers of layered crystals are of particular interest because
+they can be synthesized with a well-known top-down ap-
+
+-8550
+1310K
+2
+1320 K
+-8600
+1360K
+1380 K
+Total energy (eV
+-8650
+(3)
+-8700
+-8750
+-8800
+-8850
+0
+2
+6
+8
+Simulation time (ns)20
+10
+(b)
+CI-CI
+(a)
+Ni-Ni
+300 K
+300 K
+600 K
+600 K
+Radial distribution function
+8
+1000K
+1000 K
+15
+1310 K
+1310 K
+1320 K
+1320 K
+6-
+4500K
+4500 K
+10-
+4-
+5.
+0
+2
+8
+2
+4
+6
+10
+6
+8
+10
+Interatomic distance (A)
+Interatomic distance (A)8
+FIG. 7. Snapshots of the NiCl2 system at different temperatures T as indicated. Nickel and chlorine atoms are shown in blue
+and green colors, respectively. Panels (c) to (g) illustrate the system’s structure at different time instants corresponding to
+labels (1)–(5) in Fig. 6.
+proach. Extensive experimental characterization of such
+materials is expensive and time-consuming, so computa-
+tional modeling may provide valuable support for exper-
+imental research.
+In this study we have reported a detailed computer
+simulation of the recently proposed novel 2D material
+NiCl2. The combination of a rigorous DFT approach and
+a classical approach based on an interatomic force field
+provides insight into the mechanical and thermal prop-
+erties of this material. It has been found that the 2D
+NiCl2 sheet is mechanically stable at the relative defor-
+mation up to 18−20%, similar to graphene. At the same
+time, the obtained Young’s modulus for NiCl2 is 5 − 10
+times lower than the Young’s modulus of other commonly
+studied 2D materials such as graphene, MoS2 and WS2.
+It has been found that the NiCl2 sheet is thermally sta-
+ble up to the melting point temperature of 1315 K. The
+evaluated melting point for NiCl2 is of the same order
+of magnitude as the melting temperature of WS2 and
+lies within the melting temperature range determined
+computationally for different 2D metal monoxides and
+monochlorides, such as BeO, MgO, LiCl, and NaCl. At
+temperatures above the melting point structural degra-
+dation of NiCl2 has been observed, which involves several
+subsequent structural transformations, namely the for-
+mation of a highly porous 2D sheet, 1D nanowires, and
+nanodroplets.
+The classical force field developed in this study for sim-
+ulating the 2D NiCl2 material describes reasonably accu-
+rate the geometrical parameters of NiCl2, determined on
+the basis of DFT calculations. The force field parameters
+for NiCl2 presented in this study might be useful for fur-
+ther studies of more complex characteristics of this ma-
+terial, such as thermal conductivity, defects and phonon
+spectrum. The general methodology presented through
+an illustrative case study of NiCl2 can be utilized for the
+computational characterization of other novel 2D materi-
+als, including recently synthesized NiO2, NiS2 and NiSe2
+materials.
+ACKNOWLEDGEMENTS
+The authors are grateful for the support of this work
+by the Deutsche Forschungsgemeinschaft provided within
+the collaborative project “Hydrogen adsorption on novel
+two-dimensional materials: ab initio and molecular dy-
+namics study” (project number 452691275).
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index 0000000000000000000000000000000000000000..339fa0fab55c2fe1189ed9379fb7352e6540b880
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf,len=1227
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='02278v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='mtrl-sci]  5 Jan 2023  Computational characterization of novel nanostructured materials: Case study of  NiCl2  Elizaveta B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Kalika,1 Alexey V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Verkhovtsev,2, ∗ Mikhail M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Maslov,3 Konstantin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Katin,3 and Andrey V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Solov’yov2  1Moscow Institute of Physics and Technology, Institutskiy per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Dolgoprudny,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Moscow Region,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 141700,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Russia  2MBN Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Altenhöferallee 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 60438 Frankfurt am Main,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Germany  3Department of Condensed Matter Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' National Research Nuclear University “MEPhI”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Kashirskoe Shosse 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 115409,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Russian Federation  A computational multiscale approach combining dispersion-corrected density functional theory  (DFT) and long-timescale classical molecular dynamics is employed to characterize the geometrical,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  mechanical and thermal properties of a recently proposed two-dimensional (2D) transition metal  dichalcogenide,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' NiCl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' A classical interatomic force field is proposed whose parameters are derived  from the results of DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The developed force field is used to study the mechanical  response, thermal stability and melting of a NiCl2 monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' It is found that the NiCl2 sheet is  thermally stable up to the melting point temperature of 1315 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' At higher temperatures structural  degradation of the system is observed, which involves several subsequent structural transformations,  namely the formation of a highly porous 2D sheet, 1D nanowires, and nanodroplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The compu-  tational methodology presented through an illustrative case study of NiCl2 can be utilized for the  computational characterization of other novel 2D materials, including recently synthesized NiO2,  NiS2 and NiSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  INTRODUCTION  Nickel-based nanomaterials are considered promising  for sensors, adsorbents, and catalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Recently synthe-  sized Ni-based composites have proven their effectiveness  as sensors for various organic and biological molecules  [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Nickel nanoparticles possess catalytic activity  for many reactions, including the synthesis of primary  amines and hydrogen oxidation [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Nickel-based two-  dimensional (2D) materials can be used for the efficient  generation of molecular oxygen [6, 7], electrocatalytic wa-  ter splitting [8] and glucose oxidation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' In addition,  recently synthesized pentagonal 2D sheets of nickel di-  azenide (NiN2) have tunable direct band gap and may  serve as a precursor for pentagonal 2D materials [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Nickel-doped graphene materials are also widely used  for many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' In particular, they provide ex-  cellent sensing properties [11–13] and can effectively cat-  alyze such technologically important reactions as hydro-  gen production from water [14] and ethanol steam [15],  oxygen reduction under alkaline conditions [16], and car-  bon monoxide reduction [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Another important appli-  cation of Ni-doped graphene is hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Due to  3d electrons and the spillover effect, nickel can hold hy-  drogen molecules [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' At the same time, graphene is  not the only 2D material that can be doped with nickel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Less common 2D materials, such as silicene, germanene  and MoS2, have also been used as a substrate for nickel  atoms and nanoparticles [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The aforementioned properties of nickel-doped 2D ma-  terials are based on a combination of the mechanical char-  acteristics of a 2D sheet with the adsorption and cat-  alytic properties of nickel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' A further development of this  ∗ verkhovtsev@mbnexplorer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='com  idea is that nickel can be not an alloying impurity but  the basis for a 2D material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Pure nickel is a metal and  cannot exist in the form of the 2D allotrope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' However,  an extensive computational search for layered crystals  and related 2D materials has recently predicted the ex-  istence of several 2D nickel-based materials of the com-  position NiX2 (X = O, Cl, Br, I, S, Se) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Some of  these materials have already been synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Technolo-  gies developed for widespread 2D MoS2/MoSe2 materials  [25] and their analogs based on niobium [26], titanium  [27] and tungsten [28] proved to be useful for synthesiz-  ing NiX2 monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Large-area NiO2 monolayers sep-  arated by lanthanum atoms have been observed using  scanning tunneling microscopy [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Recent experiments  have proven the feasibility of chemical vapor deposition  synthesis of nanometer layers of NiCl2 [30], NiS2 [31] and  NiSe2 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Moreover, NiCl2 films up to four layers thick  have been recently synthesized [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' In all these mate-  rials, nickel atoms are organized in a regular 2D lattice  so that the materials possess enhanced adsorption and  catalytic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Currently, there is limited experimental information  on the properties of such 2D materials since only a few  laboratories have synthesized them in limited quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Atomistic computer simulations can serve as an alterna-  tive approach to characterize the structure of such novel  materials and explore their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  This paper presents the results of a computational  characterization of a novel 2D material, NiCl2, by means  of a multiscale modeling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' NiCl2 has been cho-  sen as an illustrative and experimentally relevant repre-  sentative of the Ni-based 2D materials family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' A combi-  nation of quantum and classical approaches into a unified  multiscale methodology permits a comprehensive investi-  gation of the structure, mechanical properties, and ther-  mal stability of NiCl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Density-functional theory (DFT)  calculations provide reference data on geometrical char-  2  acteristics of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' On this basis, a new classi-  cal interatomic potential is developed and benchmarked  against the results of quantum-mechanical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The validated potential is used for atomistic modeling of  mechanical deformations and thermal stability of NiCl2  by means of the advanced software packages MBN Ex-  plorer [33] and MBN Studio [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Through an illustrative case study of NiCl2 we present  a general methodology that can be utilized for the com-  putational characterization of other novel materials, in-  cluding the recently synthesized NiO2 [29], NiS2 [31] and  NiSe2 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' To the best of our knowledge, NiCl2 has not  been studied computationally except for a recent study  [35], which investigated structural defects in NiCl2 and  similar materials and confirmed their stability in the en-  vironment by means of DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Section II describes  the key aspects of theoretical methods utilized in DFT  calculations and corresponding classical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' A  particular focus is made on the procedure to determine  the classical force field parameters for NiCl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' This follows  with the discussion of the obtained results in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The accuracy of the developed force field is evaluated  in Section III A through the analysis of structural and  energetic parameters of NiCl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Section III B is devoted to  the analysis of mechanical properties of NiCl2 by means  of DFT and classical energy minimization calculations  using the developed force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' In Section III C the force  field is utilized to study the thermal stability of NiCl2  and evaluate its melting temperature by means of long-  scale MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Finally, in Section IV we draw the  conclusion from this work and give an outlook for further  developments in this research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  COMPUTATIONAL METHODOLOGY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  DFT calculations  DFT calculations of the structural properties of a  NiCl2 sheet have been performed using the Quantum  ESPRESSO 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='5 software package [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The plane-  wave basis set for valence electron states, generalized  gradient approximation (GGA) in the Perdew-Burke-  Ernzerhof (PBE) functional form for the exchange-  correlation energy [38], and projector-augmented-wave  (PAW) pseudopotentials [39, 40] for core-electron inter-  actions were used to perform the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' We have  employed the kinetic energy cut-off for wave functions of  100 Ry (∼1360 eV) and kinetic energy cut-off for charge  density and potential of 600 Ry (∼8160 eV) with checking  convergence of energy and charge to increase the simu-  lation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' In addition, the van der Waals inter-  actions have been taken into account through the D3  Grimme (DFT-D3) dispersion corrections [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  DFT-  D3 approach possesses improved accuracy due to the use  of environment-dependent dispersion coefficients and the  inclusion of a three-body component to the dispersion  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Structure of a NiCl2 supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Ni and Cl atoms are  shown in blue and green colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  correction energy term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The interlayer distance between separate NiCl2 sheets  has been set equal to 30 Å, which provides sufficient space  separation to avoid nonphysical interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Thus, opti-  mization of the lattice parameters along the axis perpen-  dicular to the NiCl2 sheet plane was unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The  atomic equilibrium positions have been obtained by the  total minimization of the supercell using the calculated  forces and stress on the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The convergence criterion  of self-consistent calculations for ionic relaxations was set  to 10−10 eV between two consecutive steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The geom-  etry optimization of the unstressed NiCl2 sheet was car-  ried out without symmetry constraints until the Hellman-  Feynman forces acting on the atoms became smaller than  10−4 hartree/bohr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Such criteria ensure that the abso-  lute value of stress is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='01 kbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The parameters  of the supercell have also been optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The first Bril-  louin zone integrations have been performed by using the  Monkhorst-Pack k-point sampling scheme [42] with the  6 × 6 × 1 mesh grid together with the Methfessel-Paxton  smearing [43] with the smearing width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='02 Ry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The NiCl2 sheet is represented by a hexagonal Ni9Cl18  periodic cell containing 3 × 3 elementary NiCl2 cells, see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The hexagonal symmetry corresponds to the re-  sults obtained earlier for this monolayer [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' After op-  timization, we obtained the lattice constant a = 3a0 =  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='328 Å, where a0 is the parameter of the NiCl2 prim-  itive unit cell, while the evaluated Ni–Cl bond length  was equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='377 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Additional DFT-based geometry  optimization calculations have been performed using the  Gaussian software package [44] employing three differ-  ent exchange-correlation functionals, namely LDA, PBE0  3  and HSE06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Classical MD simulations  Classical geometry optimization calculations and MD  simulations have been performed by means of MBN Ex-  plorer [33] – a software package for advanced multiscale  modeling of complex molecular structure and dynam-  ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  MBN Explorer permits to simulate a wide range  of Meso-Bio-Nano (MBN) systems, including nanosys-  tems [33, 45–47];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' nanostructured materials [48–51];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' com-  posite and hybrid materials [52–55] with sizes ranging  from atomic to mesoscopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The dedicated MBN Studio  toolkit [34] has been utilized to create the systems, pre-  pare all necessary input files, and analyze and visualize  simulation outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Determination of the classical force field for  NiCl2  The first part of this study has been devoted to the  determination of parameters of a classical force field for  NiCl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The geometry of the 3 × 3 NiCl2 cell obtained after  DFT optimization was taken as an initial geometry for  the calculations using the classical force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The NiCl2  sheet has been simulated by assigning the NiCl2 super-  cell to a simulation box which is replicated in space using  periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' First, several MD simu-  lations have been carried out with the trial force field  parameters fitted for transitional metal dichalides NiF2  [56], ZnCl2 [57], AlCl3 and FeCl3 [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The force field  employed in these simulations has been constructed as a  sum of the short-term exponential, long-term r−6 power  and the Coulomb potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Such force fields have been  commonly utilized in MD simulations of various inorganic  and ionic crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The simulation box size was set ac-  cording to the lattice size optimized by DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The NiCl2  system was unstable with any of the parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' it was  also noticeable that the sheet would bend using most of  the given force field potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  As the NiCl2 sheet would bend with all of the above  mentioned force fields, the simulation box size has been  gradually increased in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' It has been found  that the NiCl2 sheet remains flat when the simulation  box size is increased by 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  As such, the simulation  box size used for all further simulations has been set to  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='32 Å × 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='59 Å × 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='00 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Next, a new set of classical force field parameters for  the NiCl2 system has been determined in the following  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The Nelder-Mead algorithm [59] has been utilized  for the optimization of the force field parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The al-  gorithm is simplex-based, meaning that in n-dimensional  space it maintains a set of n + 1 points called simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  After calculating the value of the function at each point  of the simplex, it extrapolates the behavior of the func-  tion to find a new test point and replaces the test points  with the worst result for a new one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The process repeat-  edly continues until the result converges to the tolerance  value or reaches the maximum number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The  Nelder-Mead algorithm was implemented in Python via  scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='minimize library [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' This algorithm was  used to fit the geometrical parameters of NiCl2 calculated  by MBN Explorer [33] to the ones obtained through the  DFT-based optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' details are given below in Sec-  tion III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  NiCl2 is an ionic crystal with partial charges on nickel  and chlorine atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' In order to describe more accurately  the mutual polarization interaction between atoms of the  system, the r−6 potential has been replaced with a r−4  power potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The resulting total interaction potential  between atoms i and j of the system is given by the  following expression:  U(rij) = Aij exp(−αijrij) + Cij  r4  ij  +  qiqj  4πε0rij  (1)  The following six parameters of the force field, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' (1),  have been optimized using the Nelder-Mead algorithm:  ANi−Cl, αNi−Cl, ACl−Cl, αCl−Cl, C and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Partial charges  on Ni and Cl atoms have been set to ensure that the  system is electrically neutral, |qNi| = 2|qCl|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' A 7 Å cut-  off was applied for the exponential and power terms of  the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The Coulomb potential was calculated us-  ing the Ewald summation method with precision of 10−5  and the cutoff of 12 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' MD simulations with simulation  time of 30 ps and time step of 1 fs, Langevin thermostat  temperature of 300 K and damping time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='1 ps were  performed for each generated set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The following geometrical parameters of NiCl2 have  been compared to the results of DFT calculations: (i) Ni–  Cl bond length (lNi−Cl);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' (ii) Ni–Ni bond length (lNi−Ni);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  (iii) Cl–Cl bond length (lCl−Cl);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' (iv) Cl–Ni–Cl angle  (ϕCl−Ni−Cl), denoted hereafter as ϕ for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' In addi-  tion, the fifth parameter, namely the system’s energy as  a function of Ni–Cl interplanar distance has also been an-  alyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The distance between the Ni and Cl planes has  been gradually varied and single point energy calcula-  tions were carried out using MBN Explorer for different  Ni–Cl interplanar distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The resulting dependence  of system’s energy on the Ni-Cl interplanar distance has  been fitted by a quadratic function and its minimum, d,  was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The value of d obtained from classical force  field calculations was compared to the results of DFT  calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Then, the standard deviation  σ =  �  �  �  ��  i  �li − lDFT  i  lDFT  i  �2  +  �ϕ − ϕDFT  ϕDFT  �2  +  �d − dDFT  dDFT  �2  (2)  has been calculated and minimized using the Nelder-  Mead algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Here the summation has been carried  out over different covalent bonds, namely Ni–Ni, Ni–Cl  and Cl–Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Each round of optimization consisted of 30  4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Evolution of parameters of the force field, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' (1), as a function of the number of global optimization iterations carried  out using the Nelder-Mead optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  iterations and the parameters resulting from each opti-  mization round were used as the starting parameters for  the following one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' In this study, 430 subsequent itera-  tions have been carried out to determine the parameters  of the force field, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The evolution of the force field parameters in the  course of the parameter optimization procedure is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The convergence is due to the decrease of σ  value, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The final values of the force field param-  eters and partial charges derived from the comparison  of NiCl2 geometrical parameters to the DFT results are  listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' These values have been employed in all  the simulations described below in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  RESULTS AND DISCUSSION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Benchmarking the accuracy of the constructed  force field  NiCl2 is a novel material for which very little infor-  mation has been obtained both experimentally and com-  putationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Hence, it is important to evaluate the ac-  curacy of the developed classical force field in order to  make computational predictions of structural and dy-  namical properties of NiCl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Table II lists the values  of Ni–Cl, Ni–Ni and Cl–Cl bond lengths and Cl–Ni–Cl  angles obtained via different optimization methods: clas-  sical force field optimization (column labeled ‘FF’) and  DFT optimization calculations using different exchange-  correlation functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' In addition to DFT-based cal-  culations described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' II A, several complementary  calculations have been carried out using the Gaussian  software package [44] to evaluate the optimal geometri-  cal parameters of NiCl2 with three different exchange-  correlation functionals (PBE0, HSE06 and LDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The results listed in Table II indicate that the Ni–  Cl bond length determined from classical structure opti-  mization calculations is close to the values obtained by  means of DFT with a relative discrepancy of 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The  equilibrium Ni–Ni bond length is longer than the values  obtained by DFT by about 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' At the same time, the  equilibrium Cl–Cl bond length is shorter than the values  obtained by DFT, with a relative deviation of about 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  A smaller Ni–Cl interplanar distance results in smaller  values of the Cl–Ni–Cl angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The discrepancy might be  attributed to limitations of the Nelder-Mead optimiza-  tion algorithm exploited in this study and a limited num-  ber of optimization iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Overall, the classical force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' (1), with the pa-  rameters listed in Table I describes with a reasonable ac-  curacy the geometrical parameters of a 2D NiCl2 sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The range of discrepancies between the results of classical  force field based and DFT-based geometry optimization  calculations is similar to that obtained in earlier studies  of transition-metal clusters and bulk materials [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Mechanical properties of NiCl2  In this study, mechanical properties of a NiCl2 sheet  have been studied by both DFT and classical force field  calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' DFT-based calculations have used as input  the supercell described above in Section II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The uni-  form stretching of the NiCl2 sheet has been simulated by  simultaneously increasing the lattice parameters of the  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='0-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='60-  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='9 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  11000  (eV)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='70-  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='8  (eV  Ni-CI  10500  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  C  α  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='75  A  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='80  10000  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='85  1550-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='78  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='8-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='80  1500-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='7  (eV)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='82  1  A  1450-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='84  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='6-  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='86-  1400-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='5-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='88 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  1350-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='4 -  400  0  100  200  300  400  0  100  200  300  0  100  200  300  400  Number of optimization iterations  Number of optimization iterations  Number of optimization iterations5  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Parameters of the force field, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' (1), derived in this study for the description of a 2D NiCl2 material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Aij (eV)  αij (Å−1)  C (eV Å4)  q, |e|  Ni–Cl  10804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='52  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='73  Ni  +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='718  Cl–Cl  1465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='99  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='55  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='805  Cl  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='859  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Comparison of geometrical parameters of a NiCl2 2D sheet after geometry optimization using the classical force  field, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' (1) (column labeled ‘FF’), and DFT with different exchange-correlation functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Optimization method  FF  DFT  PBE PBE0 HSE06 LDA  average  Ni–Cl, Å  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='38  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='39  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='40  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='34 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='03  Ni–Ni, Å  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='44  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='40  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='38  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='33 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='05  Cl–Cl, Å  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='28  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='41  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='38  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='33 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='06  Cl–Ni–Cl, deg  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='8  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='2  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='0  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='8  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='9  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='6  supercell along the x− and y−directions (parallel to the  NiCl2 plane) and further optimizing atomic positions in  the supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  To estimate elasticity of the sheet, the latter was bi-  axially stretched so that the lattice constant a increased  according to the relation  a = a0(1 + ε) ,  (3)  where a0 is the equilibrium lattice constant and ε is the  relative deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The deformation energy ∆E has  been calculated as follows:  ∆E = E − E0 ,  (4)  where E is the total energy of the deformed system and  E0 is the energy of the initial (unperturbed) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' As  the unit cell for DFT- and classical force field optimiza-  tion calculations contained different numbers of atoms,  the calculated deformation energy value was divided by  the number of atoms in the respective unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Figure 3 shows that the dependence of the NiCl2 sheet  deformation energy on ε is well described by a quadratic  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The threshold for the elastic deformation was  found to be 20 % and 18% for DFT-based and classical  force field based optimization calculations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Note that graphene possesses similar elastic properties  [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  For 2D materials,  the common definition of the  Young’s modulus is not applicable, since the thickness of  a 2D sheet cannot be determined unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' There-  fore, 2D materials should preferably be characterized by  the 2D Young’s modulus E2D, which does not depend  on the sheet thickness [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The value of E2D can be  evaluated using the formula  E2D =  1  2S0  (∆E)′′ ,  (5)  where S0 is the area of the relaxed unit cell and (∆E)′′  is the second derivative of the deformation energy with  respect to biaxial deformation ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The value of E2D =  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Deformation energy ∆E of a biaxially stretched NiCl2  sheet as a function of relative deformation ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='4 N/m has been obtained in this study through the  DFT calculations of a NiCl2 sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The classical force  field, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' (1), predicts a very close value of the 2D  Young’s modulus, E2D = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='9 N/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  To estimate the common 3D Young’s modulus, the ob-  tained value E2D should be divided by the effective sheet  thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' According to the results presented in Table II  the average NiCl2 sheet thickness (that is the average  Cl–Cl bond length obtained from the DFT calculations  with different exchange-correlation functionals), is equal  to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='35 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Using this value one obtains the Young’s mod-  ulus equal to 97 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' This value is close to the value of  107 GPa obtained for NiCl2 via DFT calculations in a re-  cent study [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Note that the Young’s modulus reported  for graphene is about 10 times higher [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 2D elastic  modulus of monolayers with similar structures, MoS2 and  WS2, are about 170 N/m [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Therefore, the results of  calculations carried out in this study predict that NiCl2  is a less rigid material compared to other well-studied 2D  sheets, and its Young’s modulus is close to that of GaP  monolayer [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Such a behavior is a consequence of the  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='2  口  FF optimization  DFT optimization  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='20  Relative deformation ε6  corrugated structure of 2D NiCl2 and high elasticity of  the Ni–Cl bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Thermal stability of NiCl2  The interatomic force field described in Section II C has  been utilized to study thermal properties of a 2D NiCl2  sheet and evaluate its melting temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  This has  been done by carrying out a series of constant tempera-  ture MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The system initially equilibrated  at 300 K was heated gradually by either 50 degrees (in  the temperature range from 300 to 2000 K) or 100 degrees  (in the range from 2000 to 5000 K) over 200 ps and then  equilibrated over 10 ns at each temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' A smaller  temperature increment of 20 K has been considered in  the temperature range from 1300 to 1500 K to evaluate  the melting temperature of NiCl2 more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The  first 4 ns of the simulated trajectories were excluded from  the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The remaining part of each trajectory was  used to evaluate the average value of the system’s to-  tal energy at a given temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The resulting caloric  curve, that is the dependence of the time-averaged total  energy of the system on temperature, ⟨Etot⟩(T ), is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 4(a) by symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Figure 4(b) shows the temper-  ature dependence of heat capacity at constant volume  Cv, defined as a derivative of the internal energy of the  system over temperature:  Cv =  �∂E  ∂T  �  v  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  (6)  The melting process in a macroscopic system occurs at a  specific temperature under fixed external pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' This  process reveals itself as a first-order phase transition via a  spike in the heat capacity of the system at the transition  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  At the initial phase of NiCl2 heating, the system’s to-  tal energy grows linearly with temperature, see the inset  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' At temperatures above ∼1200 K the NiCl2  sheet transforms into a pre-melted state when the crys-  talline structure is strongly deformed, but the 2D sheet  still maintains its integrity (see the simulation snapshot  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 7(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' A slight increase in the heat capacity as a  function of temperature also indicates the onset of sys-  tem’s pre-melting (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The calculated heat capacity curve has a sharp maxi-  mum at T ≈ 1315 K, indicating the melting temperature  of NiCl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' This value is considerably lower than the melt-  ing temperature for bulk nickel, Tm(Ni) = 1728 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' On  the other hand, the evaluated value of T ≈ 1315 K is  of the same order of magnitude as the melting temper-  ature of a well-studied 2D material tungsten disulfide,  Tm(WS2) ∼ 1520 K [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The evalualted melting temper-  ature of NiCl2 also lies within the melting temperature  range determined computationally for different 2D metal  monoxides and monochlorides such as BeO, MgO, LiCl,  and NaCl [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Caloric curve for NiCl2 (panel (a)) and the cor-  responding heat capacity Cv as a function of temperature  (panel (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The peak in the Cv(T ) dependence at T ≈ 1315 K  indicates the melting temperature of NiCl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  At temperatures above the melting point, the system’s  total energy ⟨Etot⟩ continues to grow linearly with T un-  til the onset of another phase transition takes place at  T ∼ 3200 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' In contrast to the melting phase transi-  tion at Tm ≈ 1315 K, this high-temperature transition  takes place over a broad temperature range from approx-  imately 3200 to 4400 K, as indicated by a broad peak  in the Cv(T ) dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' This transition corresponds  to the multifragmentation of NiCl2 into separate NiCl2  molecular fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The phase transitions corresponding to the peaks in  the heat capacity curve can be visualized by analyzing  the radial distribution function (RDF) of atoms in the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The RDF characterizes the number of atoms  located at certain radial distance r from a reference atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The RDF for Ni–Ni and Cl–Cl atomic pairs, evaluated  at different temperatures of the system, are plotted in  Figures 5(a) and 5(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' A rapid change of  the RDFs between 1310 and 1320 K indicates the melting  phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  It is of particular interest to explore, on the atomistic  level, the phase transition from the crystalline NiCl2 to  8600  7000  8800  Total energy (eV)  9000  500  1000  1500  8000  (a)  9000  (b)  4  Heat capacity (eV/K)  2  0  1000  2000  3000  4000  5000  Temperature (K)7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Radial distribution function for nickel (panel (a)) and clorine (panel (b)) atoms of NiCl2 at different temperatures of  the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' A rapid change of the RDFs between 1310 and 1320 K indicates the melting phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Variation of the system’s total energy, Etot, as a  function of simulation time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Colored curves show the Etot(t)  dependencies for different temperatures, namely T = 1310 K  (just below the phase transition temperature), T = 1320 K (in  the phase transition region), as well as T = 1360 K and 1380 K  (above the phase transition temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Labels (1) to (5)  correspond to different time instants for the curve at T =  1320 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' These instants correspond to the system’s snapshots  shown in panels (c)–(g) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  its molten state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Figure 6 shows the variation of Etot as  a function of simulation time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Colored curves show the  Etot(t) dependencies for different temperatures, namely  T = 1310 K (just below the phase transition tempera-  ture), T = 1320 K (in the phase transition region), as  well as T = 1360 K and 1380 K (above the phase transi-  tion temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  At T = 1310 K (black curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 6) the total energy  of NiCl2 fluctuates around the average value of about  −8800 eV over the 10 ns long simulation, which indi-  cates that the system remains in the crystalline state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  However, when the temperature increases to 1320 K the  system’s energy stays nearly constant within the first  ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='7 ns of the simulated trajectory, which follows by  a rapid rise of Etot(t) within ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='4 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  This jump in  the total energy is attributed to the formation of holes  in the NiCl2 sheet as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 7(c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' After that  the system’s total energy decreases as the system relaxes  into a highly porous 2D structure (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 7(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' This  structure is metastable, and it evolves quickly into an ar-  ray of quasi-1D structures, where denser regions are con-  nected by thin NiCl2 links (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 7(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' This structure  is also metastable, and it eventually relaxes into spheri-  cal droplets with a diameter of ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='5nm (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 7(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Interestingly, as the system’s temperature increases, ran-  dom evaporation of single NiCl2 units from the droplets  and the fast diffusion of Ni and Cl atoms occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' As a re-  sult of these processes, the droplets eventually merge into  a 1D NiCl2 nanowire which is thermally stable at tem-  peratures below ∼3000 K within the simulation times  considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' This behavior, however, may  be attributed to relative small sizes of the system and  the simulation box, which enable the interaction of Ni  and Cl atoms with their periodic images across the simu-  lation box boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' A detailed analysis of system’s  size effects on its thermal properties is an interesting  topic which however goes beyond the scope of the present  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' At higher temperatures, the NiCl2 nanowires dis-  integrate and the multifragmentation process takes place,  which is seen from the change of the ⟨Etot⟩(T ) slope  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  CONCLUSIONS  Two-dimensional (2D) materials possess properties  that are technologically relevant and differ from prop-  erties of the corresponding bulk counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Monolay-  ers of layered crystals are of particular interest because  they can be synthesized with a well-known top-down ap-  8550  1310K  2  1320 K  8600  1360K  1380 K  Total energy (eV  8650  (3)  8700  8750  8800  8850  0  2  6  8  Simulation time (ns)20  10  (b)  CI-CI  (a)  Ni-Ni  300 K  300 K  600 K  600 K  Radial distribution function  8  1000K  1000 K  15  1310 K  1310 K  1320 K  1320 K  6-  4500K  4500 K  10-  4-  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  0  2  8  2  4  6  10  6  8  10  Interatomic distance (A)  Interatomic distance (A)8  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Snapshots of the NiCl2 system at different temperatures T as indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Nickel and chlorine atoms are shown in blue  and green colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Panels (c) to (g) illustrate the system’s structure at different time instants corresponding to  labels (1)–(5) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Extensive experimental characterization of such  materials is expensive and time-consuming, so computa-  tional modeling may provide valuable support for exper-  imental research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  In this study we have reported a detailed computer  simulation of the recently proposed novel 2D material  NiCl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The combination of a rigorous DFT approach and  a classical approach based on an interatomic force field  provides insight into the mechanical and thermal prop-  erties of this material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' It has been found that the 2D  NiCl2 sheet is mechanically stable at the relative defor-  mation up to 18−20%, similar to graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' At the same  time, the obtained Young’s modulus for NiCl2 is 5 − 10  times lower than the Young’s modulus of other commonly  studied 2D materials such as graphene, MoS2 and WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  It has been found that the NiCl2 sheet is thermally sta-  ble up to the melting point temperature of 1315 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The  evaluated melting point for NiCl2 is of the same order  of magnitude as the melting temperature of WS2 and  lies within the melting temperature range determined  computationally for different 2D metal monoxides and  monochlorides, such as BeO, MgO, LiCl, and NaCl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' At  temperatures above the melting point structural degra-  dation of NiCl2 has been observed, which involves several  subsequent structural transformations, namely the for-  mation of a highly porous 2D sheet, 1D nanowires, and  nanodroplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  The classical force field developed in this study for sim-  ulating the 2D NiCl2 material describes reasonably accu-  rate the geometrical parameters of NiCl2, determined on  the basis of DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The force field parameters  for NiCl2 presented in this study might be useful for fur-  ther studies of more complex characteristics of this ma-  terial, such as thermal conductivity, defects and phonon  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' The general methodology presented through  an illustrative case study of NiCl2 can be utilized for the  computational characterization of other novel 2D materi-  als, including recently synthesized NiO2, NiS2 and NiSe2  materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The authors are grateful for the support of this work  by the Deutsche Forschungsgemeinschaft provided within  the collaborative project “Hydrogen adsorption on novel  two-dimensional materials: ab initio and molecular dy-  namics study” (project number 452691275).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Deng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Nie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Tian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Li, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' He,  Microchem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 160, 105744 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Karimi-Maleh,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Arıkan,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Savk,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Karimi, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Şen, Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 250, 123042  (f) T = 1320 K, t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='4 ns  = 1320 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='4 ns  (g) T = 1320 K, t = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='0 ns9  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Ahmad, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Bedük, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Majhi, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Salama,  Sensors and Actuators B: Chemical 286, 139 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Jagadeesh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Murugesan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 58, 5064 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Power Sources 402, 447  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Zhang, ACS Nano 12, 3875  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' You, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Zhou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Song, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Wang, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Gao, Small 15,  1805435 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [8] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Du, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Zheng,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Ma, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Interfaces 10, 2360 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Bykov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Ponomareva, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Prakapenka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Glazyrin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Smith,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Mahmood, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Abrikosov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Goncharov,  ACS Nano 15, 13539 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [11] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Phan and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Chung, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Hydrogen Energy  39, 20294 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Ren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Mao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Luo, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Liu, Carbon 149, 609  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 24, 6449  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Lin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Zhu,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Zhou, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' He, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Tsang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Ma, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Lu,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Yu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Teo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Liu, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Liu, Nature Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  8, 394 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [27] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Sharda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Grotta, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Ranalli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Sokolikova, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Pesci, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Palczynski, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Bemmer,  and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Mattevi, ACS Omega 3, 8655 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Shang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Wu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Cao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Peimyoo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Qiu,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Yu, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' 2, 131 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Ikeda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Irie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Naito, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Ya-  mamoto, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Express 9, 061101 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Luo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Lin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Tang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Feng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Gui, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Zhu,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Yang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Zhao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Zhou, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Song,  ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Interfaces 10, 34193 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Volkovets, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Solov’yov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Solov’yov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content='  Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Thonhauser, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Vast, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Wu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Baroni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Mat-  ter 29, 465901 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Fox, Gaussian 16 Revision C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' D 68, 246 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
+page_content=' Solov’yov,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQfWgAs/content/2301.02278v1.pdf'}
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+Zero-Shot Cross-Lingual Transfer Language Selection Using 
+Linguistic Similarity 
+Juuso Eronen**, Michal Ptaszynski* and Fumito Masui* 
+*Kitami Institute of Technology, 165, Koencho, Kitami, 090-0015, Hokkaido, Japan 
+  
+  
+ARTICLE INFO 
+ABSTRACT 
+Keywords: 
+We study the selection of transfer languages for different Natural Language Processing tasks, 
+Multilingual Natural Language Pro- 
+specifically sentiment analysis, named entity recognition and dependency parsing. In order to 
+cessing 
+select an optimal transfer language, we propose to utilize different linguistic similarity metrics 
+Zero-Shot Learning 
+to measure the distance between languages and make the choice of transfer language based on this 
+Transfer Learning 
+information instead of relying on intuition. We demonstrate that linguistic similarity correlates 
+Linguistics 
+with cross-lingual transfer performance for all of the proposed tasks. We also show that there 
+Language similarity 
+is a Statistically significant difference in choosing the optimal language as the transfer source 
+instead of English. This allows us to select a more suitable transfer language which can be used to 
+better leverage knowledge from high-resource languages in order to improve the performance of 
+language applications lacking data. For the study, we used datasets from eight different languages 
+from three language families. 
+1. Introduction 
+As with any other supervised learning problem, the tasks in Natural Language Processing (NLP) require sufficiently 
+large labeled datasets. What sets NLP apart from other fields is the presence of multiple languages the datasets can 
+appear in. This means that in order to successfully train models for all of the world’s 7,100 languages [1], one would 
+need to annotate a dataset for each language. This is a very difficult and costly task [2, 3] and has led to a small number 
+of high-resource languages to dominate the field [4, 5, 6]. This imbalance in the distribution of resources among 
+languages calls for the need to develop technologies that would make model development for low-resource languages 
+realistically feasible and efficient. 
+In order to address this problem, cross-lingual transfer been proposed as a solution. This means leveraging labeled 
+data from high-resource languages in order to improve the performance on lower-resource languages [7, 8, 9, 10]. 
+Particularly, the popularity of cross-lingual zero-shot learning, or training on one task/language and testing on a 
+different task/language completely unknown to the model, has increased greatly in the recent years. Zero-shot learning 
+has gained in popularity because it does not require any labeled data in the target language for training [11, 12]. 
+Moreover, zero-shot cross-lingual transfer utilizes large pre-trained multilingual transformer models like Multilingual 
+BERT [13] or XLM-RoBERTa [14]. These models are fine-tuned with training data in a language called the source 
+language (usually a high-resource language) and then used to predict entries from other languages than that used in 
+training, often with satisfying results [11, 15]. 
+The choice of transfer language is usually done by intuition [16] or simply defaults to English, as is the case 
+with popular multilingual benchmarks like XTREME [12] and XGLUE [17], even though there is no actual evidence 
+backing up these choices. Furthermore, in a survey of 157 cross-lingual learning papers by Pikuliak et al. [18] they 
+found out that English is used in 149 of those papers, followed by German with 82 papers in total. There has also 
+been some attempts in developing a more systematic transfer language selection method [19]. However, this method 
+requires training of a ranking model, limiting its use to the tasks and datasets used for training, making it unusable 
+off-the-shelf for other applications. 
+Choosing the optimal language for cross-lingual transfer remains widely an understudied problem. Usually, the 
+suitable source language candidate is decided experimentally or by pure intuition by the individual (researcher, or ML 
+practitioner) based on their own theoretical knowledge and experience in the field. One option to select the source 
+language would be by taking a look at languages that are from the same language group as the target language [20]. 
+  
+*Corresponding author 
+b4 eronen. juuso@gmail .com (J. Eronen) 
+ORCID(s): 0000-0001-9841-3652 (J. Eronen) 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 
+1 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using
+Linguistic Similarity
+Juuso Eronena,*, Michal Ptaszynskia and Fumito Masuia
+a Kitami Institute of Technology, 165, Koencho, Kitami, 090-0015, Hokkaido, Japan
+ARTICLE INFO
+ABSTRACT
+Keywords:
+We study the selection of transfer languages for different Natural Language Processing tasks,
+ Multilingual Natural Language Pro-
+specifically sentiment analysis, named entity recognition and dependency parsing. In order to
+cessing
+select an optimal transfer language, we propose to utilize different linguistic similarity metrics
+Zero-Shot Learning
+tomeasure the distance betweenlanguages andmake the choice oftransferlanguage basedon this
+Transfer Learning
+information instead of relying on intuition. We demonstrate that linguistic similarity correlates
+Linguistics
+with cross-lingual transfer performance for all of the proposed tasks. We also show that there
+Language similarity
+is a statistically significant difference in choosing the optimal language as the transfer source
+instead of English. This allows us to select a more suitable transfer language which can be used to
+better leverage knowledge from high-resource languages in order to improve the performance of
+from three language families.
+1. Introduction
+ As with any other supervised learning problem, the tasks in Natural Language Processing (NLP) require sufficiently
+large labeled datasets. What sets NLP apart from other fields is the presence of multiple languages the datasets can
+appear in. This means that in order to successfully train models for all of the world's 7,100 languages [1], one would
+need to annotate a dataset for each language. This is a very difficult and costly task [2, 3] and has led to a small number
+of high-resource languages to dominate the field [4, 5, 6]. This imbalance in the distribution of resources among
+languages calls for the need to develop technologies that would make model development for low-resource languages
+realistically feasible and efficient.
+In order to address this problem, cross-lingual transfer been proposed as a solution. This means leveraging labeled
+ s         e   
+Particularly, the popularity of cross-lingual zero-shot learning, or training on one task/language and testing on a
+different task/language completely unknown to the model, has increased greatly in the recent years. Zero-shot learning
+has gained in popularity because it does not require any labeled data in the target language for training [11, 12]
+BERT [13] or XLM-RoBERTa [14]. These models are fine-tuned with training data in a language called the source
+language (usually a high-resource language) and then used to predict entries from other languages than that used in
+training, often with satisfying results [11, 15].
+The choice of transfer language is usually done by intuition [16] or simply defaults to English, as is the case
+with popular multilingual benchmarks like XTREME [12] and XGLUE [17], even though there is no actual evidence
+backing up these choices. Furthermore, in a survey of 157 cross-lingual learning papers by Pikuliak et al. [18] they
+found out that English is used in 149 of those papers, followed by German with 82 papers in total. There has also
+been some attempts in developing a more systematic transfer language selection method [19]. However, this method
+requires training of a ranking model, limiting its use to the tasks and datasets used for training, making it unusable
+off-the-shelf for other applications.
+suitable source language candidate is decided experimentally or by pure intuition by the individual (researcher, or ML
+practitioner) based on their own theoretical knowledge and experience in the field. One option to select the source
+                    
+*Corresponding author 
+@ eronen. juuso@gmail.com (J. Eronen)
+ORCID(s): 0000-0001-9841-3652 (J. Eronen)
+Eronen et al.: Preprint submitted to Elsevier
+Page 1 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+However, this does not guarantee that that the linguistic features shared between the two languages would be similar 
+[21]. 
+In order to contribute to the further understanding and solving this problem, we propose a method for choosing the 
+source language for cross-lingual transfer. We show that there is a correlation between linguistic similarity and model 
+performance, allowing us to select the best transfer language by comparing the source and target languages using 
+different linguistic similarity measures. We also show that multilingual transformer models can be used to obtain good 
+performance on the target language in a zero-shot learning setting. To select the optimal source language for transfer, 
+we propose to quantify the features of languages to compute a metric that can be used in comparing the closeness of 
+languages using their linguistic properties. 
+There are some existing metrics that use linguistic features in order to measure the linguistic distance between 
+languages [22, 23, 24]. However, as these metrics simply take a handful or only a single linguistic feature into account, 
+we propose a new linguistic similarity metric, which contains almost two hundred different features, based on the World 
+Atlas of Language Structures (WALS) [25]. This allows us to not simply rely only on a single or a handful of features, 
+but to have a more robust metric by better quantifying all aspects of the languages. 
+In this research we concentrated on three different Natural Language Processing tasks, namely, sentiment analysis, 
+named entity recognition and dependency parsing. We used datasets from eight different languages, namely English, 
+German, Danish, Polish, Croatian, Russian, Japanese and Korean. The languages were chosen as they have relatively 
+high quality datasets available. Also, the languages represent different language families (English, German, Danish - 
+Germanic; Polish, Russian, Croatian - Slavic; Japanese, Korean - Koreano-Japonic language family). This also gives 
+us the opportunity to study the efficacy of cross-lingual transfer learning between and within language family groups. 
+In previous research [26] we showed that cross-lingual transfer performance correlates with the linguistic similarity 
+of the prediction target language and the source language used for fine-tuning the models. Our hypothesis is that this is 
+true for also other NLP tasks. In the experiments, we used multilingual transformer models, namely Multilingual BERT 
+and XLM-RoBERTa, which were fine-tuned by using each of the languages as source and target. We calculated the 
+linguistic similarity between all of our proposed languages using four different linguistic similarity metrics, EzGlot, 
+eLinguistics, a quantified model based on WALS and averaged lang2vec. To demonstrate the effectiveness of our 
+method, we then measured the correlation between the zero-shot cross-lingual transfer performance and linguistic 
+similarity. 
+The paper outline is as follows. In Section 2 we go through all areas of previous research that are addressed in 
+this paper. In Section 3 we describe all the tasks and datasets applied to this research and present their differences and 
+features. In Section 4 we describe the applied multilingual transformer models and the linguistic similarity metrics used 
+in this research. In Section 5 we describe our experiment workflow and go through all the results from the conducted 
+experiments. In Section 6 we discuss the results in general and bring out the most interesting findings in relation to the 
+research goals. 
+1.1. Contributions of This Study 
+The goal of this research is to develop a method for cross-lingual transfer language selection. Most often, the choice 
+of a transfer source is made purely by relying on the practitioner’s own judgement, using their accumulated experience 
+on the field and theoretical knowledge or simply choosing a language from the same language family as the target 
+[20]. The current methods have many problems as they are prone to bias from the practitioner and also completely 
+unoptimized. In fact, one could even say that there is no systematic method usable off-the-shelf that could be used to 
+determine, which languages should be considered as the cross-lingual transfer source. 
+We propose to investigate the possibility that different linguistic similarity metrics could be utilized when trying to 
+find possible source language candidates for cross-lingual transfer also for other tasks than abusive language detection. 
+We hypothesize that linguistic similarity correlates with cross-lingual transfer efficacy, meaning that by using more 
+similar languages, we would be able to achieve higher model performance. 
+This research is was conducted in order to confirm the findings of our previous research [26] also with other Natural 
+Language Processing tasks. We improved the calculation process of the linguistic similarity metric quantified from the 
+World Atlas of Language Structures. This was done by selecting all of the features that would have a defined value for 
+both languages in all possible language pairs instead of having to be shared between all of the languages, increasing 
+the robustness of the metric. Also, we applied a linguistic similarity metric based on lang2vec by Littell et al. [27]. 
+The main contributions of this work are as follows: 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 2 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+However, this does not guarantee that that the linguistic features shared between the two languages would be similar
+[21]. 
+In order to contribute to the further understanding and solving this problem, we propose a method for choosing the
+source language for cross-lingual transfer. We show that there is a correlation between linguistic similarity and model
+performance, allowing us to select the best transfer language by comparing the source and target languages using
+different linguistic similarity measures. We also show that multilingual transformer models can be used to obtain good
+performance on the target language in a zero-shot learning setting. To select the optimal source language for transfer,
+we propose to quantify the features of languages to compute a metric that can be used in comparing the closeness of
+languages using their linguistic properties.
+There are some existing metrics that use linguistic features in order to measure the linguistic distance between
+languages [22, 23, 24]. However, as these metrics simply take a handful or only a single linguistic feature into account,
+we propose a new linguistic similarity metric, which contains almost two hundred different features, based on the World
+Atlas of Language Structures (WALS) [25]. This allows us to not simply rely only on a single or a handful of features,
+but to have a more robust metric by better quantifying all aspects of the languages.
+In this research we concentrated on three different Natural Language Processing tasks, namely, sentiment analysis.
+named entity recognition and dependency parsing. We used datasets from eight different languages, namely English,
+German, Danish, Polish, Croatian, Russian, Japanese and Korean. The languages were chosen as they have relatively
+Germanic; Polish, Russian, Croatian - Slavic; Japanese, Korean - Koreano-Japonic language family). This also gives
+us the opportunity to study the efficacy of cross-lingual transfer learning between and within language family groups.
+In previous research [26] we showed that cross-lingual transfer performance correlates with the linguistic similarity
+of the prediction target language and the source language used for fine-tuning the models. Our hypothesis is that this is
+true for also other NLP tasks. In the experiments, we used multilingual transformer models, namely Multilingual BERT
+and XLM-RoBERTa, which were fine-tuned by using each of the languages as source and target. We calculated the
+linguistic similarity between all of our proposed languages using four different linguistic similarity metrics, EzGlot
+eLinguistics, a quantified model based on WALS and averaged lang2vec. To demonstrate the effectiveness of our
+method, we then measured the correlation between the zero-shot cross-lingual transfer performance and linguistic
+similarity.
+The paper outline is as follows. In Section 2 we go through all areas of previous research that are addressed in
+features. In Section 4 we describe the applied multilingual transformer models and the linguistic similarity metrics used
+in this research. In Section 5 we describe our experiment workflow and go through all the results from the conducted
+experiments. In Section 6 we discuss the results in general and bring out the most interesting findings in relation to the
+research goals.
+1.1. Contributions of This Study
+The goal of this research is to develop a method for cross-lingual transfer language selection. Most often, the choice
+of a transfer source is made purely by relying on the practitioner's own judgement, using their accumulated experience
+on the field and theoretical knowledge or simply choosing a language from the same language family as the target
+[20]. The current methods have many problems as they are prone to bias from the practitioner and also completely
+unoptimized. In fact, one could even say that there is no systematic method usable off-the-shelf that could be used to
+determine, which languages should be considered as the cross-lingual transfer source.
+We propose to investigate the possibility that different linguistic similarity metrics could be utilized when trying to
+find possible source language candidates for cross-lingual transfer also for other tasks than abusive language detection.
+We hypothesize that linguistic similarity correlates with cross-lingual transfer efficacy, meaning that by using more
+similar languages, we would be able to achieve higher model performance.
+This research is was conducted in order to confirm the findings of our previous research [26] also with other Natural
+Language Processing tasks. We improved the calculation process of the linguistic similarity metric quantified from the
+World Atlas of Language Structures. This was done by selecting all of the features that would have a defined value for
+both languages in all possible language pairs instead of having to be shared between all of the languages, increasing
+the robustness of the metric. Also, we applied a linguistic similarity metric based on lang2vec by Littell et al. [27].
+The main contributions of this work are as follows:
+Eronen et al.: Preprint submitted to Elsevier
+Page 2 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+e We confirm the transfer language selection method based on linguistic similarity with multiple NLP tasks. 
+e We demonstrate the efficacy of two multidomain linguistic similarity metrics: improved quantified WALS and 
+averaged lang2vec. 
+e We show that there is a significant difference in choosing an optimal transfer source language over English. 
+In practice, we propose to fine-tune cross-lingual pretrained transformer models, specifically mBERT and XLM-R, 
+on three different Natural Language Processing tasks (sentiment analysis, named entity recognition and dependency 
+parsing) using each of our proposed languages (English, German, Danish, Polish, Russian, Japanese, Korean) and then 
+perform zero-shot prediction on the rest of the languages of the proposed set. We calculated the linguistic similarity 
+between all of our proposed languages using four different linguistic similarity metrics, EzGlot, eLinguistics, quantified 
+World Atlas of Language Structures and an averaged lang2vec proximity vectors. We then calculated the correlation 
+between the zero-shot cross-lingual transfer performance and linguistic similarity to show the effectiveness of our 
+method. A block-diagram of the system is shown in Figure 1. 
+Source language 
+Target language 
+data 
+Linguistic 
+data 
+| 
+similarity 
+g — 
+ee 
+Output 
+Fine-tuning 
+= Fine-tuned model 
+Prediction 
+  
+  
+  
+  
+  
+Pre-trained 
+multilingual 
+transformer model 
+Figure 1: Block diagram of the proposed system 
+2. Previous Research 
+2.1. Measuring Linguistic Similarity 
+Already in 2006, the relation between the difficulty of language learning and the similarity of languages in general 
+was discussed in a book by Ringbom [28]. The Finnish language scene was presented as an example in order to 
+demonstrate the importance of cross-linguistic similarity in foreign language learning [29]. In short, he showed that 
+Finnish-speaking Finns have a harder time learning English than Swedish-speaking Finns. The reason behind this being 
+the closer relation between Swedish and English languages, giving an advantage to Swedish speakers when it comes 
+to transferring the existing linguistic knowledge. 
+Cottorell et al. [30] showed that not every language is equally difficult to model. It was also shown by them that there 
+is a correlation between the morphological richness of a language and the performance of the model. This means that 
+the more complex the language is, the more difficult it becomes to model. This is hinting that more simple languages 
+might not work so well when used as cross-lingual transfer sources for languages of higher complexity. This also implies 
+that the direct relatedness (for example, language family) of languages should not be the only criteria in deciding the 
+cross-lingual transfer source language as other features of the languages should also be thoroughly considered in order 
+to find the most optimal transfer language. 
+There has been some research in attempting to quantify a linguistic similarity metric from different linguistic 
+features. However, these metrics mostly commonly rely only on one or just a few different linguistic features. For 
+example, by comparing the consonants contained in a predefined set of words while taking into account the order 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 3 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+. We confirm the transfer language selection method based on linguistic similarity with multiple NLP tasks.
+. We demonstrate the efficacy of two multidomain linguistic similarity metrics: improved quantified WALS and
+averaged lang2vec.
+. We show that there is a significant difference in choosing an optimal transfer source language over English
+In practice, we propose to fine-tune cross-lingual pretrained transformer models, specifically mBERT and XLM-R,
+on three different Natural Language Processing tasks (sentiment analysis, named entity recognition and dependency
+parsing) using each of our proposed languages (English, German, Danish, Polish, Russian, Japanese, Korean) and then
+perform zero-shot prediction on the rest of the languages of the proposed set. We calculated the linguistic similarity
+between all of our proposed languages using four different linguistic similarity metrics, EzGlot, eLinguistics, quantified
+World Atlas of Language Structures and an averaged lang2vec proximity vectors. We then calculated the correlation
+between the zero-shot cross-lingual transfer performance and linguistic similarity to show the effectiveness of our
+method. A block-diagram of the system is shown in Figure 1.
+Source language
+Target language
+data
+data
+Linguistic
+ similarity
+indno
+Pre-trained
+Fine-tuning
+Fine-tuned model
+Prediction
+multilingual
+transformer model
+Figure 1: Block diagram of the proposed system
+2. Previous Research
+2.1. Measuring Linguistic Similarity
+Already in 2006, the relation between the difficulty of language learning and the similarity of languages in general
+was discussed in a book by Ringbom [28]. The Finnish language scene was presented as an example in order to
+demonstrate the importance of cross-linguistic similarity in foreign language learning [29]. In short, he showed that
+Finnish-speaking Finns have a harder time learning English than Swedish-speaking Finns. The reason behind this being
+the closer relation between Swedish and English languages, giving an advantage to Swedish speakers when it comes
+to transferring the existing linguistic knowledge.
+Cottorell et al. [3O] showed that not every language is equally difficult to model. It was also shown by them that there
+is a correlation between the morphological richness of a language and the performance of the model. This means that
+the more complex the language is, the more difficult it becomes to model. This is hinting that more simple languages
+that the direct relatedness (for example, language family) of languages should not be the only criteria in deciding the
+cross-lingual transfer source language as other features of the languages should also be thoroughly considered in order
+to find the most optimal transfer language.
+There has been some research in attempting to quantify a linguistic similarity metric from different linguistic
+features. However, these metrics mostly commonly rely only on one or just a few different linguistic features. For
+example, by comparing the consonants contained in a predefined set of words while taking into account the order
+Eronen et al.: Preprint submitted to Elsevier
+Page 3 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+in which these consonants appear in the words, one can calculate a genetic proximity score between two languages. 
+This is implemented as the eLinguistics [23] similarity metric. The metric makes it possible to get information about 
+the direct relatedness of the compared languages. However, once the used languages start to become more and more 
+distant, accidental similarities in consonants are introduced and there is a significant increase in the error rate. This is 
+also acknowledged by the authors. Even though the metric is easy to calculate, it completely ignores all other kinds of 
+linguistic features, for example, semantic, syntactic, or morphological. 
+Another method to calculate a similarity metric is to take a look at the vocabularies of two languages and concentrate 
+on their similarity. EzGlot [24] uses lexical similarity as its basis for computation. The metric uses lexical similarity 
+between the two compared languages while at the same time taking into account the amount of words the two languages 
+are sharing with other languages. This allows for the calculation of similarity between the two languages in relation to 
+the similarity with every other language. 
+Aggarwal et al. [22] proposed a linguistic similarity metric that utilizes multiple aspects of languages. Their metric, 
+called STL, is based on Semantic, Terminological (lexical) and Linguistic (syntactic) similarity of languages. The 
+method outperformed previous similarity metrics that concentrated only on one of the previously mentioned aspects 
+(31, 32]. They noticed that the terminological measures showed a much higher contribution when compared to the 
+other two features. However, in order to use the metric, the structure of the used vocabulary dataset needs to be in the 
+form of a complex ontology. Due to this fact and because of the dataset only consisting of German, French, Italian, 
+Dutch, Spanish and English, and due to the dataset used by the authors being no longer available, it was not feasible 
+to use the metric as a part of this research. 
+The lang2vec developed by Littell et al. [27] is a database that represents languages as typological, phylogenetic, 
+and geographical vectors, which are derived from a number of different linguistic resources, for example, WALS 
+[25], PHOIBLE [33], Ethnologue [1], and Glottolog [34]. Each of these utilize multiple different features, making 
+them more robust than the EzGlot or eLinguistics metrics. The lang2vec is a fully-fledged library that can be used to 
+query for different linguistic features and to get pre-computed distances between languages, based on some typological 
+information. 
+The World Atlas of Language Structures (WALS) project [25] consists of a database that catalogs phonological, 
+word semantic and grammatical knowledge for 2,662 languages with almost two hundred different linguistic features 
+from multiple domains. Using a linguistic similarity measure quantified from the WALS database into would allow 
+a more robust method to measure similarity and would aid capturing all aspects of the languages instead of relying 
+only on a single or a handful of linguistic features. Concentrating purely on using WALS to create a similarity metric 
+would also preserve homogeneity and allow a more explainable and controllable implementation. In previous research 
+[26] we proposed a novel linguistic similarity metric quantified from the WALS database. This metric proved to be 
+more robust compared to the other metrics, at least for the applied abusive language detection task, as it was based on 
+multiple kinds of linguistic features. 
+2.2. Transfer Language Selection 
+Selecting the optimal language for cross-lingual transfer remains mostly an unanswered question. Most of the time, 
+the decision of which language to use as the transfer source comes up to the practitioner’s consideration. This is usually 
+done experimentally or by intuition [20, 35, 16] or by simply relying on English [36, 37, 38]. For example, in order to 
+get a more successful transfer, Cottorell and Heigold [20] focused on using languages belonging to the same language 
+family as the cross-lingual transfer target. However, even though the languages are part of the same language family, 
+two languages could be very distant for example when looking at the complexity of grammar, which means that it does 
+not guarantee them sharing the same linguistic features [21]. 
+A common way for choosing the transfer language is to simply default to English. The reason being that it is the 
+de-facto highest resource language available for most NLP tasks [4]. This is also the case with popular multilingual 
+benchmarks like XTREME [12] and XGLUE [17]. Although, recently benchmarks like XTREME-R [39] have started 
+to include cross-lingual training sets. Furthermore, in a survey of 157 cross-lingual learning papers by Pikuliak et al. 
+[18] they found out that English was used in 149 papers, followed by German with 82 papers. Additionally, it has been 
+shown that other languages than English, for example, German and Russian tend to work better as transfer sources 
+[40]. 
+Duong et al. [41] found out that choosing the transfer language based on language family is not optimal for many 
+languages. For example, their experiments showed that the best source language for both Finnish and German is 
+Czech, even though being from a different language family than the targets. They concluded that apparently, the best 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 4 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+in which these consonants appear in the words, one can calculate a genetic proximity score between two languages.
+This is implemented as the eLinguistics [23] similarity metric. The metric makes it possible to get information about
+the direct relatedness of the compared languages. However, once the used languages start to become more and more
+distant, accidental similarities in consonants are introduced and there is a significant increase in the error rate. This is
+also acknowledged by the authors. Even though the metric is easy to calculate, it completely ignores all other kinds of
+linguistic features, for example, semantic, syntactic, or morphological.
+Another method to calculate a similarity metric is to take a look at the vocabularies of two languages and concentrate
+on their similarity. EzGlot [24] uses lexical similarity as its basis for computation. The metric uses lexical similarity
+between the two compared languages while at the same time taking into account the amount of words the two languages
+are sharing with other languages. This allows for the calculation of similarity between the two languages in relation to
+the similarity with every other language.
+Aggarwal et al. [22l proposed a linguistic similarity metric that utilizes multiple aspects of languages. Their metric,
+called STL, is based on Semantic, Terminological (lexical) and Linguistic (syntactic) similarity of languages. The
+method outperformed previous similarity metrics that concentrated only on one of the previously mentioned aspects
+[31, 32]. They noticed that the terminological measures showed a much higher contribution when compared to the
+other two features. However, in order to use the metric, the structure of the used vocabulary dataset needs to be in the
+form of a complex ontology. Due to this fact and because of the dataset only consisting of German, French, Italian,
+Dutch, Spanish and English, and due to the dataset used by the authors being no longer available, it was not feasible
+to use the metric as a part of this research
+The lang2vec developed by Littell et al. [27] is a database that represents languages as typological, phylogenetic,
+and geographical vectors, which are derived from a number of different linguistic resources, for example, WALS
+[25], PHOIBLE [33], Ethnologue [1], and Glottolog [34]. Each of these utilize multiple different features, making
+them more robust than the EzGlot or eLinguistics metrics. The lang2vec is a fully-fledged library that can be used to
+query for different linguistic features and to get pre-computed distances between languages, based on some typological
+information.
+The World Atlas of Language Structures (WALS) project [25] consists of a database that catalogs phonological,
+word semantic and grammatical knowledge for 2,662 languages with almost two hundred different linguistic features
+from multiple domains. Using a linguistic similarity measure quantified from the WALS database into would allow
+a more robust method to measure similarity and would aid capturing all aspects of the languages instead of relying
+only on a single or a handful of linguistic features. Concentrating purely on using WALS to create a similarity metric
+more robust compared to the other metrics, at least for the applied abusive language detection task, as it was based on
+multiple kinds of linguistic features.
+2.2. Transfer Language Selection
+ Selecting the optimal language for cross-lingual transfer remains mostly an unanswered question. Most of the time,
+the decision of which language to use as the transfer source comes up to the practitioner's consideration. This is usually
+done experimentally or by intuition [20, 35, 16] or by simply relying on English [36, 37, 38]. For example, in order to
+get a more successful transfer, Cottorell and Heigold [20] focused on using languages belonging to the same language
+family as the cross-lingual transfer target. However, even though the languages are part of the same language family,
+two languages could be very distant for example when looking at the complexity of grammar, which means that it does
+not guarantee them sharing the same linguistic features [21].
+A common way for choosing the transfer language is to simply default to English. The reason being that it is the
+de-facto highest resource language available for most NLP tasks [4]. This is also the case with popular multilingual
+benchmarks like XTREME [12] and XGLUE [17]. Although, recently benchmarks like XTREME-R [39] have started
+to include cross-lingual training sets. Furthermore, in a survey of 157 cross-lingual learning papers by Pikuliak et al.
+[18] they found out that English was used in 149 papers, followed by German with 82 papers. Additionally, it has been
+[40].
+Duong et al. [41] found out that choosing the transfer language based on language family is not optimal for many
+languages. For example, their experiments showed that the best source language for both Finnish and German is
+Czech, even though being from a different language family than the targets. They concluded that apparently, the best
+Eronen et al.: Preprint submitted to Elsevier
+Page 4 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+source language for cross-lingual transfer is not predictable from language family information. Instead, they proposed 
+two methods for transfer language selection. The first being based on the Jensen-Shannon divergence between the 
+distributions of parts-of-speech n-grams on a pair of languages. The second method was based on the word-order 
+information feature in WALS. Both of these methods showed improvements over choosing English or a language from 
+the same family as the target. They also experimented with using multiple source languages, which further improved 
+the performance. 
+It has been shown [42, 43, 44] that transferring from many high-resource languages at the same time can yield 
+higher results compared to selecting only a single language as the transfer source. However, these methods do not 
+consider the actual relation between the source and the target languages and the amount of contribution of each of the 
+languages to the total score. Also, Nooralahzadeh et al. [45] discovered that certain morphosyntactic features shared 
+between languages tend to give a boost to cross-lingual transfer performance. 
+Lin et al. [19] developed a ranking method for possible transfer language candidates using the lang2vec metrics 
+[27] together with dataset dependent features like word overlap and type-token ratio. they discovered that using both 
+the dataset independent linguistic features and database dependent features to train the ranking model yields the best 
+results. However, as their method requires training of the ranking model, it is dependent on the tasks and datasets used 
+for training and is not usable out of the box for other applications. 
+In another study [38] it was shown that the transfer performance with English as the source correlates with the 
+linguistic similarity metrics of lang2vec [27], meaning that target languages more similar to English yielded higher 
+scores. They found out that similarity of syntactic structures especially play an important role in selecting the source 
+language for tasks like parts-of-speech tagging (POS), named entity recognition (NER) and dependency parsing (DEP). 
+They also discovered that the fine-tuning corpus size of the target language also makes a difference considering the 
+cross-lingual transfer performance, especially for higher level tasks like question answering. However, their research 
+concentrated only on using English as the source language and the capabilities of other languages as the transfer source 
+were left completely unexplored. 
+Martinez et al. [46] found out that differences in language morphology in cross-lingual transfer generally lead to 
+a higher loss than when transferring between languages with the same morphological typology. Furthermore, they 
+showed that parts-of-speech tagging tends to be more sensitive towards changes in morphological typology compared 
+to sentiment analysis, which seems to be more sensitive to variables related to the fine-tuning data and the transfer 
+performance being generally harder to predict. 
+In their research, Gaikwad et al. [10] discovered that there could be a relation between cross-lingual transfer 
+performance and language similarity. They classified entries in the Marathi language using multiple languages, 
+specifically Bengali, Greek, English, Turkish and Hindi as cross-lingual transfer sources. Their results showed that 
+the closest language of these to Marathi, Hindi, also had the highest performance. This hints that a solution to the 
+problem of cross-lingual transfer language selection could be found with the aid of linguistic similarity. 
+In our previous research [26], we showed that there is a correlation between language similarity metrics and cross- 
+lingual transfer efficiency, at least for offensive language identification. This allows for choosing of an ideal transfer 
+language by using different metrics to compare the similarity languages without having to rely on one’s intuition. We 
+also showed that choosing a transfer language, for example, only by looking at the language family is not always the 
+best option. 
+3. Tasks 
+In this research, we concentrate on three different NLP tasks. Sentiment analysis as a document classification 
+task. Named entity recognition as a token classification task. And lastly, dependency parsing for understanding the 
+importance of syntax and grammar in cross-lingual transfer. 
+We hypothesize that the zero-shot cross-lingual transfer performance correlates with the linguistic similarity of the 
+source and target languages. In order to confirm our hypothesis, we used datasets from eight different languages, namely 
+English, German, Danish, Polish, Russian, Croatian, Japanese and Korean for all of the tasks. We chose these languages 
+as they had high quality datasets compared to other options and because the languages represent three different language 
+families (English, German, Danish - Germanic; Polish, Russian, Croatian - Slavic; Japanese, Korean - Koreano-Japonic 
+language family). This also gives us the opportunity to study the efficacy of cross-lingual transfer learning between 
+and within different language family groups. 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 5 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+source language for cross-lingual transfer is not predictable from language family information. Instead, they proposed
+two methods for transfer language selection. The first being based on the Jensen-Shannon divergence between the
+distributions of parts-of-speech n-grams on a pair of languages. The second method was based on the word-order
+information feature in WALS. Both of these methods showed improvements over choosing English or a language from
+the same family as the target. They also experimented with using multiple source languages, which further improved
+the performance.
+It has been shown [42, 43, 44] that transferring from many high-resource languages at the same time can yield
+higher results compared to selecting only a single language as the transfer source. However, these methods do not
+consider the actual relation between the source and the target languages and the amount of contribution of each of the
+languages to the total score. Also, Nooralahzadeh et al. [45] discovered that certain morphosyntactic features shared
+between languages tend to give a boost to cross-lingual transfer performance.
+Lin et al. [19] developed a ranking method for possible transfer language candidates using the lang2vec metrics
+[27] together with dataset dependent features like word overlap and type-token ratio. they discovered that using both
+the dataset independent linguistic features and database dependent features to train the ranking model yields the best
+results. However, as their method requires training of the ranking model, it is dependent on the tasks and datasets used
+for training and is not usable out of the box for other applications.
+In another study [38] it was shown that the transfer performance with English as the source correlates with the
+scores. They found out that similarity of syntactic structures especially play an important role in selecting the source
+language for tasks like parts-of-speech tagging (POS), named entity recognition (NER) and dependency parsing (DEP).
+They also discovered that the fine-tuning corpus size of the target language also makes a difference considering the
+cross-lingual transfer performance, especially for higher level tasks like question answering. However, their research
+concentrated only on using English as the source language and the capabilities of other languages as the transfer source
+were left completely unexplored.
+Martinez et al. [46] found out that differences in language morphology in cross-lingual transfer generally lead to
+a higher loss than when transferring between languages with the same morphological typology. Furthermore, they
+showed that parts-of-speech tagging tends to be more sensitive towards changes in morphological typology compared
+to sentiment analysis, which seems to be more sensitive to variables related to the fine-tuning data and the transfer
+performance being generally harder to predict.
+In their research, Gaikwad et al. [10] discovered that there could be a relation between cross-lingual transfer
+specifically Bengali, Greek, English, Turkish and Hindi as cross-lingual transfer sources. Their results showed that
+the closest language of these to Marathi, Hindi, also had the highest performance. This hints that a solution to the
+problem of cross-lingual transfer language selection could be found with the aid of linguistic similarity.
+In our previous research [26], we showed that there is a correlation between language similarity metrics and cross-
+lingual transfer efficiency, at least for offensive language identification. This allows for choosing of an ideal transfer
+language by using different metrics to compare the similarity languages without having to rely on one's intuition. We
+also showed that choosing a transfer language, for example, only by looking at the language family is not always the
+best option.
+3. Tasks
+In this research, we concentrate on three different NLP tasks. Sentiment analysis as a document classification
+task. Named entity recognition as a token classification task. And lastly, dependency parsing for understanding the
+importance of syntax and grammar in cross-lingual transfer.
+We hypothesize that the zero-shot cross-lingual transfer performance correlates with the linguistic similarity of the
+source and target languages. In order to confirm our hypothesis, we used datasets from eight different languages, namely
+English, German, Danish, Polish, Russian, Croatian, Japanese and Korean for all of the tasks. We chose these languages
+as they had high quality datasets compared to other options and because the languages represent three different language
+families (English, German, Danish - Germanic; Polish, Russian, Croatian - Slavic; Japanese, Korean - Koreano-Japonic
+language family). This also gives us the opportunity to study the efficacy of cross-lingual transfer learning between
+and within different language family groups.
+Eronen et al.: Preprint submitted to Elsevier
+Page 5 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+3.1. Sentiment Analysis 
+In the field of NLP, sentiment analysis is one of the most active research areas [47]. The recent research in sentiment 
+analysis, as with many other NLP tasks, has mainly focused on using deep neural networks and pretrained language 
+models [48, 49, 50, 51, 52]. The popularization of multilingual transformer models has made it possible to utilize 
+cross-lingual transfer in order to train models for low-resource languages. 
+Rasooli et al. [53] used a set of 16 languages from different language families, namely Indo-European, Turkic, 
+Afro-Asiatic, Uralic, and Sino-Tibetan, to learn a sentiment analysis model. Their experiments showed that for most 
+target languages the best result can be obtained by leveraging from multiple source languages at the same time. Also, 
+datasets of a similar genre and domain tended to yield higher results when compared to out-of-domain and dissimilar 
+genres. 
+Pelicon et al. [54] used zero-shot cross-lingual transfer to classify Croatian news articles with an mBERT model 
+fine-tuned using Slovene data with good results. In addition, Kumar et al. [55] used XLM-R and performed cross- 
+lingual transfer from English to Hindi. Their model compared favorably to the used benchmarks and gives an effective 
+solution to the analysis of sentiments in a resource-poor scenario. 
+The majority of the sentiment analysis datasets used in this research consists of product reviews, as we attempted 
+to keep the domain the same throughout the languages. However, for some languages, we were unable to find such 
+data, most notably Croatian, which consists of news articles. We also had to adjust the labels of some of the datasets so 
+that they would match among all of the languages. Training and evaluation splits were retained from original datasets 
+if possible, otherwise datasets were split to 80% training and 20% evaluation. 
+For this research we used the Multilingual Amazon Reviews Corpus [56], which covers English, Japanese and 
+German. The dataset contains over 200,000 reviews for each language collected between 2015 and 2019. The reviews 
+are labeled from one to five stars. However, as the other datasets used in this research used a two-point scale (positive, 
+negative), we adjusted the labels accordingly (positive: 5 and 4 stars, negative: 2 and 
+1 stars). 
+For Danish, we used a dataset containing almost 45,000 reviews crawled from Trustpilot by Alessandro Gianfelici! . 
+For Polish, we used the PolEmo 2.0 corpus [57]. This dataset contains over 8,000 reviews from the domains of medicine, 
+hotels, products and school. For both of these datasets, we also had to adjust the labels of this dataset to a two-point 
+scale similarly to the Amazon Reviews dataset. 
+The Russian dataset used in this research was a product review dataset by Smetanin et al. [58]. The dataset consists 
+of 90,000 automatically labeled reviews on the topic "Women’s Clothes and Accessories", split evenly among three 
+classes (positive, neutral, negative). The Croatian dataset is the same used by Pelicon et al. [54], containing around 
+2,000 news articles. The articles were collected from 24sata, one of the leading Croatian media companies. The 
+annotations were done by 6 people using a five-level Likert scale. The annotations were later adjusted to a three-point 
+scale by the authors. For the purpose of our experiments, in case of both datasets, we left out the neutral reviews in 
+order to binarize the labels. 
+The Korean dataset used in this research was Naver sentiment movie corpus v1.0*. The dataset consists of Naver 
+Movie reviews, with 100,000 positive and negative samples. The reviews were originally rated from one to ten, but the 
+creators binarized the dataset prior to publishing. 
+3.2. Named Entity Recognition 
+The research on Named Entity Recognition (NER) has also shifted towards using Deep Neural Networks and most 
+recently, pretrained transformer models [59, 60, 61]. Cross-lingual transfer has also been applied to NER in multiple 
+research. Fritzler et al. [62] used a metric-learning method to at the time outperform a state-of-the-art recurrent neural 
+network method and showed to be capable in both few-shot and zero-shot settings. Moon. et al. [63] used multilingual 
+BERT to fine-tune a NER model in multiple languages and showed it to be more effective than a model fine-tuned 
+only on a single language. This demonstrates that the model can leverage knowledge from other languages in order to 
+improve its performance on one. 
+Hvingelby et al. [64] presented a Danish NLP resource based on the Danish Universal Dependencies treebank and 
+showed that transferring from other Germanic languages, especially from English and Norwegian, to Danish can yield 
+good results when using mBERT. However, using other Germanic languages in addition to Danish did not give any 
+better results compared to fine-tuning only with Danish in their case. 
+  
+‘https ://github.com/AlessandroGianfelici/danish_reviews_dataset 
+*nttps: //github.com/e9t/nsmc 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 6 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+3.1. Sentiment Analysis
+In the field of NLP, sentiment analysis is one of the most active research areas [47]. The recent research in sentiment
+analysis, as with many other NLP tasks, has mainly focused on using deep neural networks and pretrained language
+models [48, 49, 50, 51, 52]. The popularization of multilingual transformer models has made it possible to utilize
+cross-lingual transfer in order to train models for low-resource languages
+Afro-Asiatic, Uralic, and Sino-Tibetan, to learn a sentiment analysis model. Their experiments showed that for most
+target languages the best result can be obtained by leveraging from multiple source languages at the same time. Also,
+datasets of a similar genre and domain tended to yield higher results when compared to out-of-domain and dissimilar
+genres.
+Pelicon et al. [54] used zero-shot cross-lingual transfer to classify Croatian news articles with an mBERT model
+fine-tuned using Slovene data with good results. In addition, Kumar et al. [55] used XLM-R and performed cross-
+lingual transfer from English to Hindi. Their model compared favorably to the used benchmarks and gives an effective
+solution to the analysis of sentiments in a resource-poor scenario.
+The majority of the sentiment analysis datasets used in this research consists of product reviews, as we attempted
+to keep the domain the same throughout the languages. However, for some languages, we were unable to find such
+data, most notably Croatian, which consists of news articles. We also had to adjust the labels of some of the datasets so
+that they would match among all of the languages. Training and evaluation splits were retained from original datasets
+if possible, otherwise datasets were split to 80% training and 20% evaluation.
+For this research we used the Multilingual Amazon Reviews Corpus [56], which covers English, Japanese and
+German. The dataset contains over 200,000 reviews for each language collected between 2015 and 2019. The reviews
+are labeled from one to five stars. However, as the other datasets used in this research used a two-point scale (positive,
+negative), we adjusted the labels accordingly (positive: 5 and 4 stars, negative: 2 and 1 stars).
+For Danish, we used a dataset containing almost 45,000 reviews crawled from Trustpilot by Alessandro Gianfelicil.
+For Polish, we used the PolEmo 2.0 corpus [57]. This dataset contains over 8,000 reviews from the domains of medicine,
+hotels, products and school. For both of these datasets, we also had to adjust the labels of this dataset to a two-point
+scale similarly to the Amazon Reviews dataset.
+The Russian dataset used in this research was a product review dataset by Smetanin et al. [58]. The dataset consists
+of 90,000 automatically labeled reviews on the topic "Women's Clothes and Accessories", split evenly among three
+classes (positive, neutral, negative). The Croatian dataset is the same used by Pelicon et al. [54], containing around
+2,000 news articles. The articles were collected from 24sata, one of the leading Croatian media companies. The
+annotations were done by 6 people using a five-level Likert scale. The annotations were later adjusted to a three-point
+scale by the authors. For the purpose of our experiments, in case of both datasets, we left out the neutral reviews in
+order to binarize the labels.
+The Korean dataset used in this research was Naver sentiment movie corpus v1.02. The dataset consists of Naver
+Movie reviews, with 100,000 positive and negative samples. The reviews were originally rated from one to ten, but the
+creators binarized the dataset prior to publishing.
+3.2. Named Entity Recognition
+The research on Named Entity Recognition (NER) has also shifted towards using Deep Neural Networks and most
+research. Fritzler et al. [62] used a metric-learning method to at the time outperform a state-of-the-art recurrent neural
+network method and showed to be capable in both few-shot and zero-shot settings. Moon. et al. [63] used multilingual
+BERT to fine-tune a NER model in multiple languages and showed it to be more effective than a model fine-tuned
+only on a single language. This demonstrates that the model can leverage knowledge from other languages in order to
+improve its performance on one.
+Hvingelby et al. [64] presented a Danish NLP resource based on the Danish Universal Dependencies treebank and
+showed that transferring from other Germanic languages, especially from English and Norwegian, to Danish can yield
+good results when using mBERT. However, using other Germanic languages in addition to Danish did not give any
+better results compared to fine-tuning only with Danish in their case.
+https://github.com/AlessandroGianfelici/danish_reviews_dataset
+2https://github.com/e9t/nsmc
+Eronen et al.: Preprint submitted to Elsevier
+Page 6 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+Entity projection [65, 66] has been used to generate pseudo-labeled datasets for low-resource NER datasets with 
+the help of parallel corpora. However, it has been shown by Weber and Steedman [67] that entity projection can 
+be outperformed by cross-lingual transfer and XLM-RoBERTa. The reason behind this could be explained by the 
+discovery by Lauscher et al. [38], who showed that transfer performance with English as the source correlates with the 
+similarity of the languages when dealing with a NER task. 
+In this study, we used the WikiANN [68] multilingual NER dataset also used by XTREME benchmark [12] for all 
+of the proposed languages. WikiANN consists of Wikipedia articles annotated with LOC (location), PER (person), and 
+ORG (organisation) NER tags. We used the version by Rahimi et al. [69], which has a balanced train, development, 
+and test splits and supports 176 of the 282 languages from the original WikiANN corpus. 
+3.3. Dependency Parsing 
+Cross-lingual transfer in dependency parsing (DEP) has been studied for some time before the advent of 
+multilingual transformer models [70, 71, 72, 73]. These studies mainly used deep neural network-based methods on 
+parallel corpora. The research by Duong et al. [41] discussed earlier in Section 2.2 was also conducted on a dependency 
+parsing task. Instead of using parallel corpora, their research was built around syntactic cross-lingual word embeddings 
+[74] trained over POS contexts to emphasize syntax. 
+Multilingual transformer models have also seen success in the dependency parsing task [15, 75, 76]. Most notably, 
+in their study, Lauscher et al. [38] discovered that structural and syntactic similarities between languages are the most 
+determining factor when it comes to the success of cross-lingual transfer for lower-level tasks like POS-tagging and 
+DEP. 
+The dataset used for all of the proposed languages in this study was the Universal Dependencies v2 [77], a widely 
+used resource in NLP as well as in linguistic research. The dataset was also used in the XTREME [12] benchmark 
+and in the research by Lauscher et al. [38] described earlier. Universal Dependencies is a framework for a consistent 
+annotation of grammar, including parts-of-speech, morphological features, and syntactic dependencies across a total 
+of more than 100 languages. 
+4. Methods 
+4.1. Models 
+For the experiments we used two pre-trained multilingual transformer models. The experiments were carried out 
+in a zero-shot cross-lingual setting [78], meaning that the fine-tuning is done using only data from another language 
+than the target language. 
+Multilingual BERT (mBERT) [13] is the multilingual version of BERT, which stands for Bidirectional Encoder 
+Representations from Transformers. It is based on an attention mechanism called the Transformer [79] that learns 
+contextual relations between words (or sub-words) in text. One of the features transformer models introduced is the 
+capability to read text input in both directions at once, instead of being able to only read it sequentially from left-to- 
+right or right-to-left. Taking advantage of this bidirectional capability, BERT is pre-trained on two NLP tasks, Masked 
+Language Modeling and Next Sentence Prediction. The objective of Masked Language Modeling is to mask a word in 
+a sentence and have the algorithm predict based on the word’s context what word has been hidden. In Next Sentence 
+Prediction, the algorithm takes two masked sentences and needs to predict if they have a sequential connection or not. 
+Although mBERT has not been trained using any cross-lingual data, it has showed cross-lingual capabilities and had 
+good results in many cross-lingual tasks [80]. This also includes various zero-shot transfer tasks. Multilingual BERT 
+has even been shown to outperform the usage of various cross-lingual embeddings [15]. This ability to generalize 
+could come from having word pieces used in all languages, for example, numbers, URLs, etc, mapped to a shared 
+space. This in turn forces the co-occurring pieces to also be mapped to a shared space, thus spreading the effect to 
+other word pieces, until different languages are close in a shared space [11]. 
+XLM-RoBERTa (XLM-R) [14] is a multi-lingual transformer model, also trained with the Masked Language 
+Model objective. XLM-R is trained on around a total of 2.5tb of CommonCrawl data in one hundred different languages. 
+The model is trained in the same way as the monolingual RoBERTa [81]. This means, that the only objective in its 
+pre-training is Masked Language Modeling. The model is not trained on the Next Sentence Prediction task like BERT 
+or using the parallel Translation Language Model objective of XLM. 
+XLM-R has been shown to outperform both mBERT and XLM on a many cross-lingual benchmarks, including 
+zero-shot cross-lingual transfer tasks [82]. It has also been shown to perform well on low-resource languages. A 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 7 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+Entity projection [65, 66] has been used to generate pseudo-labeled datasets for low-resource NER datasets with
+the help of parallel corpora. However, it has been shown by Weber and Steedman [67] that entity projection can
+be outperformed by cross-lingual transfer and XLM-RoBERTa. The reason behind this could be explained by the
+discovery by Lauscher et al. [38], who showed that transfer performance with English as the source correlates with the
+similarity of the languages when dealing with a NER task.
+In this study, we used the WikiANN [68] multilingual NER dataset also used by XTREME benchmark [12] for all
+of the proposed languages. WikiANN consists of Wikipedia articles annotated with LOC (location), PER (person), and
+ORG (organisation) NER tags. We used the version by Rahimi et al. [69], which has a balanced train, development.
+and test splits and supports 176 of the 282 languages from the original WikiANN corpus.
+3.3. Dependency Parsing
+Cross-lingual transfer in dependency parsing (DEP) has been studied for some time before the advent of
+multilingual transformer models [70, 71, 72, 73]. These studies mainly used deep neural network-based methods on
+parallel corpora. The research by Duong et al. [41] discussed earlier in Section 2.2 was also conducted on a dependency
+parsing task. Instead of using parallel corpora, their research was built around syntactic cross-lingual word embeddings
+[74] trained over POS contexts to emphasize syntax.
+Multilingual transformer models have also seen success in the dependency parsing task [15, 75, 76]. Most notably
+in their study, Lauscher et al. [38] discovered that structural and syntactic similarities between languages are the most
+determining factor when it comes to the success of cross-lingual transfer for lower-level tasks like POS-tagging and
+DEP.
+The dataset used for all of the proposed languages in this study was the Universal Dependencies v2 [77], a widely
+used resource in NLP as well as in linguistic research. The dataset was also used in the XTREME [12] benchmark
+and in the research by Lauscher et al. [38] described earlier. Universal Dependencies is a framework for a consistent
+annotation of grammar, including parts-of-speech, morphological features, and syntactic dependencies across a total
+of more than 100 languages.
+4. Methods
+4.1. Models 
+For the experiments we used two pre-trained multilingual transformer models. The experiments were carried out
+in a zero-shot cross-lingual setting [78], meaning that the fine-tuning is done using only data from another language
+than the target language.
+Multilingual BERT (mBERT) [13] is the multilingual version of BERT, which stands for Bidirectional Encoder
+contextual relations between words (or sub-words) in text. One of the features transformer models introduced is the
+capability to read text input in both directions at once, instead of being able to only read it sequentially from left-to-
+right or right-to-left. Taking advantage of this bidirectional capability, BERT is pre-trained on two NLP tasks, Masked
+Language Modeling and Next Sentence Prediction. The objective of Masked Language Modeling is to mask a word in
+a sentence and have the algorithm predict based on the word's context what word has been hidden. In Next Sentence
+Prediction, the algorithm takes two masked sentences and needs to predict if they have a sequential connection or not.
+Although mBERT has not been trained using any cross-lingual data, it has showed cross-lingual capabilities and had
+good results in many cross-lingual tasks [80]. This also includes various zero-shot transfer tasks. Multilingual BERT
+has even been shown to outperform the usage of various cross-lingual embeddings [15]. This ability to generalize
+could come from having word pieces used in all languages, for example, numbers, URLs, etc, mapped to a shared
+space. This in turn forces the co-occurring pieces to also be mapped to a shared space, thus spreading the effect to
+other word pieces, until different languages are close in a shared space [11].
+XLM-RoBERTa (XLM-R) [14] is a multi-lingual transformer model, also trained with the Masked Language
+Model objective. XLM-R is trained on around a total of 2.5tb of CommonCrawl data in one hundred different languages.
+The model is trained in the same way as the monolingual RoBERTa [81]. This means, that the only objective in its
+pre-training is Masked Language Modeling. The model is not trained on the Next Sentence Prediction task like BERT
+or using the parallel Translation Language Model objective of XLM.
+XLM-R has been shown to outperform both mBERT and XLM on a many cross-lingual benchmarks, including
+zero-shot cross-lingual transfer tasks [82]. It has also been shown to perform well on low-resource languages. A
+Eronen et al.: Preprint submitted to Elsevier
+Page 7 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+Table 1 
+eLinguistics metric between all applied languages 
+  
+Danish 
+English 
+German 
+Croatian 
+Polish 
+Russian 
+Japanese 
+Korean 
+  
+Danish 
+0.00 
+20.60 
+38.20 
+66.20 
+68.20 
+66.20 
+95.20 
+97.20 
+English 
+20.60 
+0.00 
+30.80 
+60.30 
+66.90 
+60.30 
+88.30 
+90.00 
+German 
+38.20 
+30.80 
+0.00 
+64.50 
+68.10 
+64.50 
+87.40 
+95.50 
+Croatian 
+66.20 
+60.30 
+64.50 
+0.00 
+10.70 
+5.60 
+90.70 
+87.20 
+Polish 
+68.20 
+66.90 
+68.10 
+10.70 
+0.00 
+5.10 
+93.30 
+89.50 
+Russian 
+66.20 
+60.30 
+64.50 
+5.60 
+5.10 
+0.00 
+93.30 
+89.50 
+Japanese 
+95.20 
+88.30 
+87.40 
+90.70 
+93.30 
+93.30 
+0.00 
+88.00 
+Korean 
+97.20 
+90.00 
+95.50 
+87.20 
+89.50 
+89.50 
+88.00 
+0.00 
+  
+notable feature of XLM-R is that it can also match the performance of to state-of-the-art monolingual models, 
+which demonstrates that it is possible to create multilingual models without sacrificing per-language performance 
+in a monolingual setting [14], most likely thanks to the sheer amount of data used in the pre-training. 
+4.2. Linguistic Similarity Metrics 
+To be able to calculate the correlation between cross-lingual zero-shot transfer performance and language similarity 
+for the proposed tasks, we needed 
+a way to quantify the aspects of all of the languages in our proposed set, 
+specifically, a language similarity metric. We utilized four language similarity measures, eLinguistics [23], EzGlot 
+[24], the multidomain metric we quantified from the linguistic features presented in WALS [25] and averaged genetic, 
+geographic, syntactic, inventory, phonological and featural metrics from lang2vec [27]. We propose that linguistic 
+similarity metrics could be utilized when trying to find optimal source language candidates for cross-lingual transfer. 
+We hypothesize that linguistic similarity correlates with cross-lingual transfer efficacy, meaning that by using more 
+similar languages, we would be able to achieve higher model performance. 
+eLinguistics [23] works by calculating a genetic proximity value for a pair of languages based on the use of phonetic 
+consonants. The score is calculated by taking a predefined word set and comparing the consonants contained in these 
+words. The method also takes into account the order of the consonants. This way, it is possible to get information 
+regarding the closeness of the phonetics of the pair of languages set for comparison. The assessment of the relationship 
+of the consonants is based on the research done by Brown et al [83]. 
+Even though completely disregarding semantic, morphological, and syntactic similarity and being very simple in 
+formulation, the similarity values produced by the method seemed to be in line with our expectations and the two 
+multidomain metrics (WALS, lang2vec) used in this research. However, as the distance between the two compared 
+languages increased, the method seemed to become increasingly more prone to errors. This is due to the surging 
+amount of accidental similarities in consonants. The similarity measure can be accessed from a web service*. The 
+similarity values between our proposed languages are shown in Table 1. 
+EzGlot [24] 
+is based on the similarity of vocabularies, or lexical similarity, of the two compared languages. 
+EzGlot’s similarity metric is computed by taking the lexical similarity between the two compared languages, while 
+in addition taking into account the number of words the pair of languages also have in common every other language. 
+This makes it possible to compute a similarity measure for a pair of languages in relation to their closeness with 
+every other language. Also, due to including the calculation of the number of words the languages share with all other 
+languages, the similarity measure becomes asymmetric between every pair of languages. This also supports studies 
+stating that mutual language intelligibility is being considered asymmetric as well [84, 21]. 
+A pre-computed language similarity matrix and the formula for its computation can be found on the EzGlot 
+similarity metric project’s web page*. However, the usability of the metric is hindered by the high amount of missing 
+values in the similarity matrix. For example taking a look at Japanese, which is one of the languages utilized in 
+our experiments, over half of the values are missing for our proposed languages. Also, the authors of the similarity 
+measure do not give away their data source. This means that we are unable to say anything regarding the quality of the 
+computations. This also makes it more difficult to fill in the missing values to the similarity matrix. We extracted the 
+  
+3nttp ://www.elinguistics.net/Compare_Languages. aspx 
+‘https: //www.ezglot.com/most-similar-languages .php 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 8 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+Table 1
+eLinguistics metric between all applied languages
+Danish
+Polish
+English
+German
+Croatian
+Russian
+Japanese
+Korean
+Danish
+0.00
+20.60
+38.20
+66.20
+68.20
+66.20
+95.20
+97.20
+English
+20.60
+0.00
+30.80
+60.30
+66.90
+60.30
+88.30
+90.00
+German
+38.20
+30.80
+0.00
+64.50
+68.10
+64.50
+87.40
+95.50
+Croatian
+60.30
+10.70
+5.60
+90.70
+66.20
+64.50
+0.00
+87.20
+Polish
+68.20
+66.90
+68.10
+10.70
+0.00
+5.10
+93.30
+89.50
+Russian
+66.20
+60.30
+64.50
+5.60
+5.10
+0.00
+93.30
+89.50
+Japanese
+95.20
+88.30
+87.40
+90.70
+93.30
+93.30
+0.00
+88.00
+Korean
+97.20
+90.00
+95.50
+87.20
+89.50
+89.50
+88.00
+0.00
+notable feature of XLM-R is that it can also match the performance of to state-of-the-art monolingual models
+which demonstrates that it is possible to create multilingual models without sacrificing per-language performance
+in a monolingual setting [14], most likely thanks to the sheer amount of data used in the pre-training.
+4.2. Linguistic Similarity Metrics
+ To be able to calculate the correlation between cross-lingual zero-shot transfer performance and language similarity
+for the proposed tasks, we needed a way to quantify the aspects of all of the languages in our proposed set
+specifically, a language similarity metric. We utilized four language similarity measures, eLinguistics [23], EzGlot
+[24], the multidomain metric we quantified from the linguistic features presented in WALS [25] and averaged genetic,
+geographic, syntactic, inventory, phonological and featural metrics from lang2vec [27]. We propose that linguistic
+similarity metrics could be utilized when trying to find optimal source language candidates for cross-lingual transfer.
+We hypothesize that linguistic similarity correlates with cross-lingual transfer efficacy, meaning that by using more
+similar languages, we would be able to achieve higher model performance.
+eLinguistics [23] works by calculating a genetic proximity value for a pair of languages based on the use of phonetic
+consonants. The score is calculated by taking a predefined word set and comparing the consonants contained in these
+words. The method also takes into account the order of the consonants. This way, it is possible to get information
+regarding the closeness of the phonetics of the pair of languages set for comparison. The assessment of the relationship
+of the consonants is based on the research done by Brown et al [83].
+Even though completely disregarding semantic, morphological, and syntactic similarity and being very simple in
+formulation, the similarity values produced by the method seemed to be in line with our expectations and the two
+multidomain metrics (WALS, lang2vec) used in this research. However, as the distance between the two compared
+languages increased, the method seemed to become increasingly more prone to errors. This is due to the surging
+amount of accidental similarities in consonants. The similarity measure can be accessed from a web service3. The
+similarity values between our proposed languages are shown in Table 1.
+EzGlot [24] is based on the similarity of vocabularies, or lexical similarity, of the two compared languages.
+EzGlot's similarity metric is computed by taking the lexical similarity between the two compared languages, while
+in addition taking into account the number of words the pair of languages also have in common every other language
+This makes it possible to compute a similarity measure for a pair of languages in relation to their closeness with
+every other language. Also, due to including the calculation of the number of words the languages share with all other
+languages, the similarity measure becomes asymmetric between every pair of languages. This also supports studies
+stating that mutual language intelligibility is being considered asymmetric as well [84, 21].
+A pre-computed language similarity matrix and the formula for its computation can be found on the EzGlot
+similarity metric project's web page4. However, the usability of the metric is hindered by the high amount of missing
+values in the similarity matrix. For example taking a look at Japanese, which is one of the languages utilized in
+our experiments, over half of the values are missing for our proposed languages. Also, the authors of the similarity
+measure do not give away their data source. This means that we are unable to say anything regarding the quality of the
+computations. This also makes it more difficult to fill in the missing values to the similarity matrix. We extracted the
+3http://www.elinguistics.net/Compare_Languages.aspx
+4https: //www.ezglot.com/most-similar-languages .php
+Eronen et al.: Preprint submitted to Elsevier
+Page 8 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+Table 2 
+EzGlot metric between all of the proposed languages 
+  
+Danish 
+English 
+German 
+Croatian 
+Polish 
+Russian 
+Japanese 
+Korean 
+  
+Danish 
+100 
+9 
+17 
+N/A 
+13 
+N/A 
+N/A 
+9 
+English 
+6 
+100 
+28 
+6 
+19 
+14 
+7 
+26 
+German 
+6 
+15 
+100 
+4 
+8 
+4 
+N/A 
+5 
+Croatian 
+N/A 
+4 
+5 
+100 
+14 
+9 
+N/A 
+5 
+Polish 
+6 
+12 
+9 
+14 
+100 
+15 
+N/A 
+5 
+Russian 
+N/A 
+11 
+7 
+11 
+19 
+100 
+N/A 
+11 
+Japanese 
+N/A 
+2 
+N/A 
+N/A 
+N/A 
+N/A 
+100 
+8 
+Korean 
+1 
+5 
+2 
+1 
+1 
+3 
+4 
+100 
+  
+Table 3 
+Averaged lang2vec metric between all of the proposed languages 
+  
+Danish 
+English 
+German 
+Croatian 
+Polish 
+Russian 
+Japanese 
+Korean 
+  
+Danish 
+0.000 
+0.511 
+0.487 
+0.550 
+0.565 
+0.597 
+0.694 
+0.691 
+English 
+0.511 
+0.000 
+0.352 
+0.578 
+0.486 
+0.488 
+0.635 
+0.578 
+German 
+0.487 
+0.352 
+0.000 
+0.550 
+0.470 
+0.471 
+0.594 
+0.579 
+Croatian 
+0.550 
+0.578 
+0.550 
+0.000 
+0.513 
+0.505 
+0.709 
+0.699 
+Polish 
+0.565 
+0.486 
+0.470 
+0.513 
+~=0.000 
+0.344 
+0.624 
+0.619 
+Russian 
+0.597 
+0.488 
+0.471 
+0.505 
+0.344 
+0.000 
+0.589 
+0.585 
+Japanese 
+0.694 
+0.635 
+0.594 
+0.709 
+0.624 
+0.589 
+0.000 
+0.518 
+Korean 
+0.691 
+0.578 
+0.579 
+0.699 
+0.619 
+0.585 
+0.518 
+0.000 
+  
+similarity values from the EzGlot’s similarity matrix for the proposed languages. These values are presented in Table 
+2. 
+Averaged lang2vec is calculated from genetic, geographic, syntactic, inventory, phonological and featural metrics 
+of lang2vec. Lang2vec [27] is a database that provides vector identifications of languages based on different linguistic 
+features based on various linguistic resources like WALS [25], PHOIBLE [33], Ethnologue [1], and Glottolog [34]. 
+The lang2vec is a fully-fledged library that can be used to query for different linguistic features and to get pre-computed 
+genetic, geographic, syntactic, inventory, phonological and featural distances between languages. In order to use 
+lang2vec as a multidomain linguistic similarity metric, we used an average value of these six categories. 
+The method is based on multiple types of linguistic features, making it naturally more robust than EzGlot or 
+eLinguistics similarity metrics, which only rely on a single kind of linguistic feature each. The method also uses a 
+larger amount of data compared to the previously described metric based on WALS. Additionally, lang2vec deals 
+with the missing values in linguistic resources by predicting them [85]. However, due to being based on multiple 
+sources, the heterogeneous nature of the method brings up many questions. For example, there might be incoherence 
+as we do not know how features are selected from different sources and how they are weighted. Also, the features are 
+one-hot encoded which causes a complete loss of ordinality between feature values. Additionally, using geographical 
+information as one of the vectors seems questionable as it was shown to be unreliable when predicting similarity of 
+languages [86]. The averaged distance matrix for lang2vec is shown in Table 3. 
+Quantified World Atlas of Language Structures is a similarity metric developed by us in previous research 
+[26]. It is based on The World Atlas of Language Structures (WALS) [25], which is a massive language database 
+that records phonological, word semantic and grammatical information for a total of 2,662 languages from over 200 
+different language families. There are 192 different linguistic features in the database currently (May 2022). However, 
+many of the linguistic features are missing for of the available languages. For example, one of the most extensively 
+documented language, English, has about 150 features documented in the database. This amount rapidly decreases for 
+languages studied less. Taking Danish as an example, it only 58 features documented>. Considering every language 
+and all of the features, this adds up to over 58,000 data points in total in the WALS database. This means the whole 
+  
+>Some even less studied languages have an even smaller number of features documented, e.g. Chuj language, spoken in Guatemala, has only 
+29, while the Indonesian Kutai language has only a single feature documented. 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 9 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+Table 2
+EzGlot metric between all of the proposed languages
+Danish
+English 
+German
+Croatian
+Polish
+Russian
+Japanese
+Korean
+Danish
+100
+9
+17
+N/A
+13 
+N/A
+N/A
+9
+English
+6
+100
+28
+19
+14
+26
+6
+German
+6
+15
+100
+4
+8
+4
+N/A
+5
+Croatian
+N/A
+4
+100
+14
+9
+N/A
+5
+5
+Polish
+6
+12
+9
+14
+100
+15
+N/A
+5
+Russian
+N/A
+11
+7
+11
+19
+100
+N/A
+11
+ Japanese
+N/A
+2
+N/A
+N/A
+N/A
+N/A
+100
+Korean
+1
+5
+2
+1
+1
+3
+4
+100
+Table 3
+Danish
+English
+German
+Croatian
+ Polish
+Russian
+Japanese
+Korean
+Danish
+0.000
+0.511
+0.487
+0.550
+0.565
+0.597
+0.694
+0.691
+English
+0.511
+0.000
+0.352
+0.578
+0.486
+0.488
+0.635
+0.578
+German
+0.487
+0.352
+0.000
+0.550
+0.470
+0.471
+0.594
+0.579
+Croatian
+0.550
+0.578
+0.550
+0.000
+0.513
+0.505
+0.709
+0.699
+Polish
+0.565
+0.486
+0.470
+0.513
+0.000
+0.344
+0.624
+0.619
+0.597
+Russian
+0.488
+0.471
+0.505
+0.344
+0.000
+0.589
+0.585
+ Japanese
+0.694
+0.635
+0.594
+0.709
+0.624
+0.589
+0.000
+0.518
+Korean
+0.691
+0.578
+0.579
+0.699
+0.619
+0.585
+0.518
+0.000
+similarity values from the EzGlot's similarity matrix for the proposed languages. These values are presented in Table
+Averaged lang2vec is calculated from genetic, geographic, syntactic, inventory, phonological and featural metrics
+of lang2vec. Lang2vec [27] is a database that provides vector identifications of languages based on different linguistic
+features based on various linguistic resources like WALS [25], PHOIBLE [33], Ethnologue [1], and Glottolog [34].
+The lang2vec is a fully-fledged library that can be used to query for different linguistic features and to get pre-computed
+genetic, geographic, syntactic, inventory, phonological and featural distances between languages. In order to use
+lang2vec as a multidomain linguistic similarity metric, we used an average value of these six categories.
+The method is based on multiple types of linguistic features, making it naturally more robust than EzGlot or
+eLinguistics similarity metrics, which only rely on a single kind of linguistic feature each. The method also uses a
+larger amount of data compared to the previously described metric based on WALS. Additionally, lang2vec deals
+with the missing values in linguistic resources by predicting them [85]. However, due to being based on multiple
+sources, the heterogeneous nature of the method brings up many questions. For example, there might be incoherence
+as we do not know how features are selected from different sources and how they are weighted. Also, the features are
+one-hot encoded which causes a complete loss of ordinality between feature values. Additionally, using geographical
+information as one of the vectors seems questionable as it was shown to be unreliable when predicting similarity of
+languages [86]. The averaged distance matrix for lang2vec is shown in Table 3.
+Quantified World Atlas of Language Structures is a similarity metric developed by us in previous research
+[26]. It is based on The World Atlas of Language Structures (WALS) [25], which is a massive language database
+that records phonological, word semantic and grammatical information for a total of 2,662 languages from over 200
+different language families. There are 192 different linguistic features in the database currently (May 2022). However,
+many of the linguistic features are missing for of the available languages. For example, one of the most extensively
+documented language, English, has about 150 features documented in the database. This amount rapidly decreases for
+languages studied less. Taking Danish as an example, it only 58 features documented5. Considering every language
+and all of the features, this adds up to over 58,000 data points in total in the WALS database. This means the whole
+ 5Some even less studied languages have an even smaller number of features documented, e.g. Chuj language, spoken in Guatemala, has only
+29, while the Indonesian Kutai language has only a single feature documented.
+Eronen et al.: Preprint submitted to Elsevier
+Page 9 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+Table 4 
+Quantified WALS metric between all of the proposed languages 
+  
+Danish 
+English 
+German 
+Croatian 
+Polish 
+Russian 
+Japanese 
+Korean 
+  
+Danish 
+0.000 
+0.109 
+0.140 
+0.167 
+~=0.197 
+0.155 
+0.236 
+0.202 
+English 
+0.109 
+0.000 
+0.136 
+0.179 
+0.164 
+0.141 
+0.252 
+0.209 
+German 
+0.140 
+0.136 
+0.000 
+0.221 
+0.196 
+0.140 
+0.248 
+0.225 
+Croatian 
+0.167 
+0.179 
+0.221 
+0.000 
+0.160 
+0.080 
+0.272 
+0.229 
+Polish 
+0.197 
+0.164 
+0.196 
+0.160 
+0.000 
+0.097 
+0.249 
+0.210 
+Russian 
+0.155 
+0.141 
+0.140 
+0.080 
+0.097 
+0.000 
+0.225 
+0.196 
+Japanese 
+0.236 
+0.252 
+0.248 
+0.272 
+0.249 
+0.225 
+0.000 
+0.108 
+Korean 
+0.202 
+0.209 
+0.225 
+0.229 
+0.210 
+0.196 
+0.108 
+0.000 
+  
+database is only approximately 12% populated, meaning a vast majority of the information is missing. Also many 
+major and widely studied languages are missing many features. For example, 25% of all of the features are missing for 
+English. These missing values and the sparsity of the data is the main point of concern when quantifying the WALS 
+database into a linguistic similarity metric as using lesser known and not so widely studied languages means having 
+less common features among them. 
+In previous research, we quantified a novel linguistic similarity metric from the WALS database based on the 
+features all of the proposed languages shared. One of the problems of the metric was that as the amount of languages 
+increased, the amount of features shared with them decreased due to missing values in the database. This time, we 
+improved the calculation process and increased the robustness of the metric. The improved version attempts to counter 
+the issue caused by the diminishing feature count. This was done by selecting all of the features that would have 
+a defined value for both languages in all possible language pairs instead of having to be shared between all of the 
+languages. The language pairs were formed from our proposed languages (English, German, Danish, Polish, Russian, 
+Croatian, Japanese and Korean). Otherwise, the process remained the same. In short, we converted the possible 
+feature values into numeric and calculated an average euclidean distance between all language pairs. This resulted 
+in a symmetric distance metric. The goal was to create a multidomain similarity metric that would also be coherent 
+and try to preserve the ordinality of the feature values. The finished distance matrix is shown in Table 4. 
+Lastly, our plan was to take a look at the STL similarity measure [22], which is based on multiple linguistic 
+features. The measure puts together three different aspects of language by using Semantic, Terminological (lexical) 
+and Linguistic (syntactic) similarity to form a single metric. According to the authors, the STL metric outperformed 
+many previous measures that were relying only on one of the previously mentioned feature types [31, 32]. However, in 
+order to be able to use the metric, the vocabulary dataset must be structured in the form of an ontology, which restricts 
+the metric’s use. Due to this fact and because of the lack of available languages for the used dataset, it was not possible 
+for us to utilize the metric in this research. 
+5. Experiments 
+5.1. Setup 
+We fine-tuned both of the models (mBERT, XLM-R) with all of the proposed languages (English, German, Danish, 
+Polish, Russian, Croatian, Japanese and Korean) for all of the tasks. Fine-tuning refers to the training of the parameters 
+of a pre-trained language model (like BERT) with task-specific labeled data. This produced 16 fine-tuned models for 
+each task, which sums up to a total of 48 fine-tuned models (two transformer models, eight languages, three tasks). 
+After fine-tuning, we evaluated the models with test datasets from all of the previously mentioned languages to compute 
+the cross-lingual zero-shot transfer scores. We did not use a train-dev-test, but only train-test scenario for evaluation, 
+because the test dataset has nothing to do with the training dataset in a zero-shot task. We also do not aim at optimizing 
+for each dataset, or creating a product, but rather study general properties. We evaluated the models with a macro 
+Fl-score for sentiment analysis and NER, and Label Attachment Score (LAS) for the dependency parsing task. 
+After finishing the evaluations for all of the fine-tuned models, we took a look at the correlation between the 
+zero-shot cross-lingual transfer scores and linguistic similarity. This was done by using the four previously introduced 
+linguistic similarity metrics (eLinguistics, EzGlot, WALS and lang2vec). We computed Pearson’s and Spearman’s 
+correlations between the models’ cross-lingual zero-shot transfer scores and the language similarity measures. The 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 10 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+Table 4 
+Quantified WALS metric between all of the proposed languages
+Danish
+English 
+German
+Croatian
+Polish
+Russian
+Korean
+Japanese
+Danish
+0.000
+0.109
+0.140
+0.167
+0.197
+0.155
+0.236
+0.202
+English
+0.109
+0.000
+0.136
+0.179
+0.164
+0.141
+0.252
+0.209
+German
+0.140
+0.136
+0.000
+0.221
+0.196
+0.140
+0.248
+0.225
+Croatian
+0.179
+0.221
+0.167
+0.000
+0.160
+0.080
+0.272
+0.229
+Polish
+0.197
+0.164
+0.196
+0.160
+0.000
+0.097
+0.249
+0.210
+Russian
+0.155
+0.141
+0.140
+0.080
+0.097
+0.000
+0.225
+0.196
+Japanese
+0.236
+0.252
+0.248
+0.272
+0.249
+0.225
+0.000
+0.108
+Korean
+0.202
+0.209
+0.225
+0.229
+0.210
+0.196
+0.108
+0.000
+database is only approximately 12% populated, meaning a vast majority of the information is missing. Also many
+major and widely studied languages are missing many features. For example, 25% of all of the features are missing for
+English. These missing values and the sparsity of the data is the main point of concern when quantifying the WALS
+database into a linguistic similarity metric as using lesser known and not so widely studied languages means having
+less common features among them.
+In previous research, we quantified a novel linguistic similarity metric from the WALS database based on the
+features all of the proposed languages shared. One of the problems of the metric was that as the amount of languages
+increased, the amount of features shared with them decreased due to missing values in the database. This time, we
+improved the calculation process and increased the robustness of the metric. The improved version attempts to counter
+the issue caused by the diminishing feature count. This was done by selecting all of the features that would have
+a defined value for both languages in all possible language pairs instead of having to be shared between all of the
+languages. The language pairs were formed from our proposed languages (English, German, Danish, Polish, Russian,
+Croatian, Japanese and Korean). Otherwise, the process remained the same. In short, we converted the possible
+feature values into numeric and calculated an average euclidean distance between all language pairs. This resulted
+in a symmetric distance metric. The goal was to create a multidomain similarity metric that would also be coherent
+and try to preserve the ordinality of the feature values. The finished distance matrix is shown in Table 4.
+Lastly, our plan was to take a look at the STL similarity measure [22], which is based on multiple linguistic
+features. The measure puts together three different aspects of language by using Semantic, Terminological (lexical)
+and Linguistic (syntactic) similarity to form a single metric. According to the authors, the STL metric outperformed
+many previous measures that were relying only on one of the previously mentioned feature types [31, 32]. However, in
+order to be able to use the metric, the vocabulary dataset must be structured in the form of an ontology, which restricts
+the metric's use. Due to this fact and because of the lack of available languages for the used dataset, it was not possible
+for us to utilize the metric in this research.
+5. Experiments
+5.1. Setup
+We fine-tuned both of the models (mBERT, XLM-R) with all of the proposed languages (English, German, Danish,
+Polish, Russian, Croatian, Japanese and Korean) for all of the tasks. Fine-tuning refers to the training of the parameters
+of a pre-trained language model (like BERT) with task-specific labeled data. This produced 16 fine-tuned models for
+each task, which sums up to a total of 48 fine-tuned models (two transformer models, eight languages, three tasks).
+After fine-tuning, we evaluated the models with test datasets from all of the previously mentioned languages to compute
+the cross-lingual zero-shot transfer scores. We did not use a train-dev-test, but only train-test scenario for evaluation,
+because the test dataset has nothing to do with the training dataset in a zero-shot task. We also do not aim at optimizing
+for each dataset, or creating a product, but rather study general properties. We evaluated the models with a macro
+F1-score for sentiment analysis and NER, and Label Attachment Score (LAS) for the dependency parsing task.
+After finishing the evaluations for all of the fine-tuned models, we took a look at the correlation between the
+zero-shot cross-lingual transfer scores and linguistic similarity. This was done by using the four previously introduced
+correlations between the models' cross-lingual zero-shot transfer scores and the language similarity measures. The
+Eronen et al.: Preprint submitted to Elsevier
+Page 10 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+  
+  
+Table 5 
+Tasks, models and linguistic similarity metrics used in the experiments 
+Tasks 
+Models 
+Linguistic Similarity Metrics 
+Sentiment Analysis 
+mBERT 
+EzGlot 
+Named Entity Recognition 
+XLM-RoBERTa _ eLinguistics 
+Dependency Parsing 
+Averaged lang2vec 
+Quantified WALS 
+  
+  
+  
+  
+  
+  
+Table 6 
+Sentiment analysis: F1-scores for mBERT 
+TARGET 
+Danish 
+English 
+German 
+Croatian 
+Polish 
+Russian 
+Japanese 
+(Korean 
+Danish 
+0.976 
+0.875 
+0.951 
+0.941 
+0.934 
+0.876 
+0.800 
+0.881 
+English 
+0.942 
+0.935 
+0.935 
+0.921 
+0.921 
+0.849 
+0.645 
+0.838 
+German 
+0.901 
+0.816 
+0.971 
+0.908 
+0.889 
+0.828 
+0.711 
+0.741 
+SOURCE 
+Croatian 
+0.952 
+0.883 
+0.948 
+0.973 
+0.940 
+0.863 
+0.802 
+0.881 
+Polish 
+0.952 
+0.876 
+0.948 
+0.948 
+0.967 
+0.861 
+0.771 
+0.878 
+Russian 
+0.949 
+0.862 
+0.939 
+0.938 
+0.933 
+0.957 
+0.774 
+0.867 
+Japanese 
+0.908 
+0.799 
+0.903 
+0.894 
+0.870 
+0.807 
+0.914 
+0.869 
+Korean 
+0.940 
+0.848 
+0.935 
+0.930 
+0.909 
+0.850 
+0.815 
+0.957 
+Table 7 
+Sentiment analysis: F1-scores for XLM-R 
+TARGET 
+Danish 
+English 
+German 
+Croatian 
+Polish 
+Russian 
+Japanese 
+Korean 
+Danish 
+0.972 
+0.857 
+0.932 
+0.934 
+0.926 
+0.869 
+0.749 
+0.846 
+English 
+0.939 
+0.925 
+0.920 
+0.922 
+0.916 
+0.844 
+0.705 
+0.816 
+German 
+0.882 
+0.791 
+0.966 
+0.890 
+0.871 
+0.816 
+0.671 
+0.711 
+SOURCE 
+Croatian 
+0.945 
+0.859 
+0.925 
+0.969 
+0.929 
+0.876 
+0.763 
+0.835 
+Polish 
+0.946 
+0.853 
+0.929 
+0.939 
+0.960 
+0.865 
+0.632 
+0.834 
+Russian 
+0.918 
+0.792 
+0.895 
+0.913 
+0.901 
+0.953 
+0.726 
+0.832 
+Japanese 
+0.888 
+0.793 
+0.880 
+0.871 
+0.851 
+0.773 
+0.905 
+0.832 
+Korean 
+0.921 
+0.804 
+0.903 
+0.911 
+0.880 
+0.824 
+0.690 
+0.953 
+  
+tasks, models and linguistic similarity metrics used in the experiments are listed in Table 5. The models were fine-tuned 
+by using PyTorch and the Huggingface Transformers library [87]. The hardware used was an Nvidia GTX 1080Ti GPU. 
+5.2. Results 
+Both of the multilingual transformer models (mnBERT, XLM-R) were fine-tuned with all of the proposed languages 
+for each task (sentiment analysis, NER, DEP) we introduced earlier. The models were fine-tuned using only the training 
+dataset from a single language before the evaluation step. The model evaluation scores are presented in Tables 6 and 
+7 for sentiment analysis, Tables 8 and 9 for NER and Tables 10 and 11 for DEP. 
+Looking at the results, we can clearly say that XLM-R outperformed mBERT in 
+all of the tasks. The only exception 
+to this was the sentiment analysis task, where mBERT slightly outperformed XLM-R. It can be noted from the results 
+that the highest transfer scores usually belong to the languages in the same language family as the source language 
+(English, German, Danish - Germanic; Croatian, Polish, Russian - Slavic; Japanese, Korean - Koreano-Japonic). Also, 
+most of the time there is a clear difference in the scores when evaluating with the same language as the source compared 
+to zero-shot cross-lingual transfer. The exceptions to this are the sentiment analysis task for both models and the NER 
+task for XLM-R. 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 11 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+Table 5
+Tasks, models and linguistic similarity metrics used in the experiments
+Tasks
+Models
+Linguistic Similarity Metrics
+mBERT
+EzGlot
+Sentiment Analysis
+Named Entity Recognition
+XLM-RoBERTa
+eLinguistics
+Dependency Parsing
+Averaged lang2vec
+Quantified WALS
+Table 6
+Sentiment analysis: F1-scores for mBERT
+TARGET
+Danish
+English
+German
+Croatian
+Polish
+Russian
+ Japanese
+Korean
+Danish
+0.976
+0.875
+0.951
+0.941
+0.934
+0.876
+0.800
+0.881
+English
+0.942
+0.935
+0.935
+0.921
+0.921
+0.849
+0.645
+0.838
+German
+0.901
+0.816
+0.971
+0.908
+0.889
+0.828
+0.711
+0.741
+SOURCE
+Croatian
+0.952
+0.883
+0.948
+0.973
+0.940
+0.863
+0.802
+0.881
+Polish
+0.952
+0.876
+0.948
+0.948
+0.967
+0.861
+0.771
+0.878
+Russian
+0.939
+0.949
+0.862
+0.938
+0.933
+0.957
+0.774
+0.867
+Japanese
+0.908
+0.799
+0.903
+0.894
+0.870
+0.807
+0.914
+0.869
+Korean
+0.940
+0.848
+0.935
+0.930
+0.909
+0.850
+0.815
+0.957
+Table 7
+Sentiment analysis: F1-scores for XLM-R
+TARGET
+Danish
+English
+Polish
+Russian
+Korean
+German
+Croatian
+Japanese
+Danish
+0.972
+0.857
+0.932
+0.934
+0.926
+0.869
+0.749
+0.846
+English
+0.939
+0.925
+0.920
+0.922
+0.916
+0.844
+0.705
+0.816
+German
+0.882
+0.791
+0.966
+0.890
+0.871
+0.816
+0.671
+0.711
+SOURCE
+Croatian
+0.945
+0.859
+0.925
+0.969
+0.929
+0.876
+0.763
+0.835
+Polish
+0.946
+0.853
+0.929
+0.939
+0.960
+0.865
+0.632
+0.834
+Russian
+0.918
+0.792
+0.895
+0.913
+0.901
+0.953
+0.726
+0.832
+Japanese
+0.888
+0.793
+0.880
+0.871
+0.851
+0.773
+0.905
+0.832
+Korean
+0.921
+0.804
+0.903
+0.911
+0.880
+0.824
+0.690
+0.953
+tasks, models and linguistic similarity metrics used in the experiments are listed in Table 5. The models were fine-tuned
+by using PyTorch and the Huggingface Transformers library [87]. The hardware used was an Nvidia GTX 1080Ti GPU.
+5.2. Results
+Both of the multilingual transformer models (mBERT, XLM-R) were fine-tuned with all of the proposed languages
+for each task (sentiment analysis, NER, DEP) we introduced earlier. The models were fine-tuned using only the training
+dataset from a single language before the evaluation step. The model evaluation scores are presented in Tables 6 and
+7 for sentiment analysis, Tables 8 and 9 for NER and Tables 10 and 11 for DEP.
+Looking at the results, we can clearly say that XLM-R outperformed mBERT in all of the tasks. The only exception
+to this was the sentiment analysis task, where mBERT slightly outperformed XLM-R. It can be noted from the results
+that the highest transfer scores usually belong to the languages in the same language family as the source language
+(English, German, Danish - Germanic; Croatian, Polish, Russian - Slavic; Japanese, Korean - Koreano-Japonic). Also,
+most of the time there is a clear difference in the scores when evaluating with the same language as the source compared
+to zero-shot cross-lingual transfer. The exceptions to this are the sentiment analysis task for both models and the NER
+task for XLM-R.
+Eronen et al.: Preprint submitted to Elsevier
+Page 11 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+  
+  
+  
+  
+  
+  
+  
+  
+Table 8 
+NER: Fi-scores for mBERT 
+TARGET 
+Danish 
+English 
+German 
+Croatian 
+Polish 
+Russian 
+Japanese 
+Korean 
+Danish 
+0.957 
+0.813 
+0.801 
+0.480 
+0.763 
+0.791 
+0.675 
+0.640 
+English 
+0.778 
+0.930 
+0.827 
+0.744 
+0.852 
+0.729 
+0.773 
+0.652 
+German 
+0.770 
+0.866 
+0.936 
+0.751 
+0.879 
+0.805 
+0.766 
+0.648 
+SOURCE 
+Croatian 
+0.667 
+0.710 
+0.748 
+0.876 
+0.727 
+0.770 
+0.707 
+0.622 
+Polish 
+0.691 
+0.676 
+0.702 
+0.695 
+0.956 
+0.648 
+0.754 
+0.625 
+Russian 
+0.759 
+0.825 
+0.764 
+0.761 
+0.867 
+0.946 
+0.759 
+0.564 
+Japanese 
+0.754 
+0.827 
+0.761 
+0.659 
+0.693 
+0.743 
+0.926 
+0.673 
+Korean 
+0.602 
+0.694 
+0.668 
+0.670 
+0.675 
+0.705 
+0.700 
+0.867 
+Table 9 
+NER: Fi-scores for XLM-R 
+TARGET 
+Danish 
+English 
+German 
+Croatian 
+Polish 
+Russian 
+Japanese 
+(Korean 
+Danish 
+0.975 
+0.868 
+0.876 
+0.723 
+0.958 
+0.910 
+0.873 
+0.755 
+English 
+0.955 
+0.941 
+0.941 
+0.820 
+0.961 
+0.934 
+0.920 
+0.781 
+German 
+0.960 
+0.926 
+0.948 
+0.846 
+0.975 
+0.930 
+0.918 
+0.765 
+SOURCE = _ Croatian 
+0.920 
+0.842 
+0.872 
+0.911 
+0.919 
+0.889 
+0.834 
+0.723 
+Polish 
+0.949 
+0.885 
+0.897 
+0.877 
+0.981 
+0.897 
+0.891 
+0.740 
+Russian 
+0.923 
+0.908 
+0.915 
+0.611 
+0.951 
+0.951 
+0.880 
+0.737 
+Japanese 
+0.955 
+0.909 
+0.920 
+0.847 
+0.961 
+0.926 
+0.936 
+0.789 
+Korean 
+0.827 
+0.752 
+0.799 
+0.491 
+0.751 
+0.820 
+0.840 
+0.900 
+Table 10 
+DEP: LAS-scores for mBERT 
+TARGET 
+Danish 
+English 
+German 
+Croatian 
+Polish 
+Russian 
+Japanese 
+Korean 
+Danish 
+0.860 
+0.545 
+0.631 
+0.619 
+0.556 
+0.647 
+0.092 
+0.026 
+English 
+0.652 
+0.891 
+0.670 
+0.624 
+0.570 
+0.653 
+0.165 
+0.021 
+German 
+0.635 
+0.603 
+0.842 
+0.672 
+0.613 
+0.733 
+0.130 
+0.062 
+SOURCE = Croatian 
+0.581 
+0.607 
+0.633 
+0.893 
+0.645 
+0.778 
+0.124 
+0.030 
+Polish 
+0.520 
+0.518 
+0.577 
+0.676 
+0.924 
+0.760 
+0.112 
+0.023 
+Russian 
+0.594 
+0.604 
+0.643 
+0.730 
+0.666 
+0.878 
+0.131 
+0.020 
+Japanese 
+0.132 
+0.148 
+0.163 
+0.114 
+0.117 
+0.126 
+0.926 
+0.033 
+Korean 
+0.058 
+0.065 
+0.060 
+0.035 
+0.045 
+0.054 
+0.035 
+0.293 
+  
+In dependency parsing, XLM-R slightly outperformed mBERT as expected. However, in the sentiment analysis 
+task mBERT scored slightly higher than XLM-R overall, with both models scoring high across all language pairs. 
+Some language pairs even achieving zero-shot cross-lingual transfer F-score of over 0.95. In this task, there seems not 
+to be a clear pattern what kind of language pairs tend to yield higher performance. For example, Slavic languages seem 
+to work better as sources for Danish compared to German languages in the case of both models. The scores are also 
+similarly high across the board for the NER task with XLM-R, with the model being able to achieve very high scores 
+with zero-shot transfer. The performance difference between mBERT and XLM-R is also more noticeable in the case 
+of NER. 
+As can be seen from Table 12, both Japanese and Korean worked decently well as cross-lingual transfer sources 
+for both sentiment analysis and NER tasks, even though being very different from the other languages used in the 
+experiments as they are the only non Indo-European languages. However, in the case of DEP their performance 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 12 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+Table 8
+NER: F1-scores for mBERT
+TARGET
+Danish
+English
+German
+Croatian
+Polish
+Russian
+ Japanese
+Korean
+Danish
+0.957
+0.813
+0.801
+0.480
+0.763
+0.791
+0.675
+0.640
+English
+0.778
+0.930
+0.827
+0.744
+0.852
+0.729
+0.773
+0.652
+German
+0.770
+0.866
+0.936
+0.751
+0.879
+0.805
+0.766
+0.648
+SOURCE
+Croatian
+0.667
+0.710
+0.748
+0.876
+0.727
+0.770
+0.707
+0.622
+Polish
+0.691
+0.676
+0.702
+0.695
+0.956
+0.648
+0.754
+0.625
+Russian
+0.759
+0.825
+0.764
+0.761
+0.867
+0.946
+0.759
+0.564
+0.754
+0.827
+0.761
+0.659
+0.693
+0.743
+0.926
+0.673
+Japanese
+Korean
+0.602
+0.694
+0.668
+0.670
+0.675
+0.705
+0.700
+0.867
+Table 9
+NER: F1-scores for XLM-R
+TARGET
+Danish
+English
+German
+Croatian
+Polish
+Russian
+ Japanese
+Korean
+Danish
+0.975
+0.868
+0.876
+0.723
+0.958
+0.910
+0.873
+0.755
+English
+0.955
+0.941
+0.941
+0.820
+0.961
+0.934
+0.920
+0.781
+German
+0.960
+0.926
+0.948
+0.846
+0.975
+0.930
+0.918
+0.765
+SOURCE
+Croatian
+0.920
+0.842
+0.872
+0.911
+0.919
+0.889
+0.834
+0.723
+Polish
+0.949
+0.885
+0.897
+0.877
+0.981
+0.897
+0.891
+0.740
+Russian
+0.923
+0.908
+0.915
+0.611
+0.951
+0.951
+0.880
+0.737
+Japanese
+0.955
+0.909
+0.920
+0.847
+0.961
+0.926
+0.936
+0.789
+Korean
+0.827
+0.752
+0.799
+0.491
+0.751
+0.820
+0.840
+0.900
+Table 10
+DEP: LAS-scores for mBERT
+TARGET
+Danish
+English
+German
+Croatian
+Polish
+Russian
+Korean
+Japanese
+Danish
+0.860
+0.545
+0.631
+0.619
+0.556
+0.647
+0.092
+0.026
+English
+0.652
+0.891
+0.670
+0.624
+0.570
+0.653
+0.165
+0.021
+German
+0.635
+0.603
+0.842
+0.672
+0.613
+0.733
+0.130
+0.062
+SOURCE
+Croatian
+0.581
+0.607
+0.633
+0.893
+0.645
+0.778
+0.124
+0.030
+Polish
+0.520
+0.577
+0.676
+0.924
+0.760
+0.112
+0.023
+0.518
+Russian
+0.594
+0.604
+0.643
+0.730
+0.666
+0.878
+0.131
+0.020
+ Japanese
+0.132
+0.148
+0.163
+0.114
+0.117
+0.126
+0.926
+0.033
+ Korean
+0.065
+0.060
+0.035
+0.035
+0.058
+0.045
+0.054
+0.293
+In dependency parsing, XLM-R slightly outperformed mBERT as expected. However, in the sentiment analysis
+task mBERT scored slightly higher than XLM-R overall, with both models scoring high across all language pairs
+Some language pairs even achieving zero-shot cross-lingual transfer F-score of over 0.95. In this task, there seems not
+to be a clear pattern what kind of language pairs tend to yield higher performance. For example, Slavic languages seem
+to work better as sources for Danish compared to German languages in the case of both models. The scores are also
+ similarly high across the board for the NER task with XLM-R, with the model being able to achieve very high scores
+with zero-shot transfer. The performance difference between mBERT and XLM-R is also more noticeable in the case
+of NER.
+As can be seen from Table 12, both Japanese and Korean worked decently well as cross-lingual transfer sources
+for both sentiment analysis and NER tasks, even though being very different from the other languages used in the
+experiments as they are the only non Indo-European languages. However, in the case of DEP their performance
+Eronen et al.: Preprint submitted to Elsevier
+Page 12 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+  
+  
+  
+  
+  
+  
+Table 11 
+DEP: LAS-scores for XLM-R 
+TARGET 
+Danish 
+English 
+German 
+Croatian 
+Polish 
+Russian 
+Japanese 
+Korean 
+Danish 
+0.888 
+0.679 
+0.725 
+0.706 
+0.672 
+0.715 
+0.095 
+0.366 
+English 
+0.733 
+0.911 
+0.728 
+0.720 
+0.700 
+0.716 
+0.112 
+0.364 
+German 
+0.712 
+0.681 
+0.854 
+0.751 
+0.732 
+0.784 
+0.066 
+0.405 
+SOURCE = Croatian 
+0.639 
+0.668 
+0.702 
+0.910 
+0.798 
+0.818 
+0.069 
+0.375 
+Polish 
+0.614 
+0.603 
+0.676 
+0.780 
+0.945 
+0.804 
+0.049 
+0.384 
+Russian 
+0.642 
+0.645 
+0.722 
+0.801 
+0.796 
+0.890 
+0.101 
+0.378 
+Japanese 
+0.118 
+0.132 
+0.172 
+0.098 
+0.122 
+0.106 
+0.937 
+0.317 
+Korean 
+0.326 
+0.292 
+0.381 
+0.298 
+0.325 
+0.304 
+0.183 
+0.877 
+Table 12 
+Average scores for each source language on each task 
+mBERT 
+XLM-R 
+Sentiment 
+NER’ 
+DEP __ 
+Sentiment 
+NER 
+DEP 
+Danish 
+0.904 
+0.740 
+0.497 
+0.886 
+0.867 
+0.606 
+English 
+0.873 
+0.786 
+0.531 
+0.874 
+0.907 
+0.623 
+German 
+0.845 
+0.803 
+0.536 
+0.825 
+0.909 
+0.623 
+Croatian 
+0.905 
+0.728 
+0.536 
+0.888 
+0.864 
+0.622 
+Polish 
+0.900 
+0.719 
+0.514 
+0.870 
+0.890 
+0.607 
+Russian 
+0.902 
+0.781 
+0.533 
+0.866 
+0.860 
+0.622 
+Japanese 
+0.871 
+0.754 
+0.220 
+0.849 
+0.905 
+0.250 
+Korean 
+0.898 
+0.698 
+0.081 
+0.861 
+0.773 
+0.373 
+  
+is extremely low. Except for this case with DEP, all of the proposed languages seem to be quite equal as cross- 
+lingual transfer sources in general. Interestingly, German, Croatian and Russian seem to perform slightly better overall 
+compared to the other languages, especially with mBERT. A similar phenomenon was also experienced by Turc et al. 
+[40] and in our previous research [26]. 
+5.3. Effect of Linguistic Similarity 
+We calculated the correlation between the zero-shot cross-lingual transfer results of the two models and each of the 
+four proposed linguistic similarity metrics (EzGlot, eLinguistics, WALS and lang2vec) in all proposed NLP tasks using 
+both Pearson’s and Spearman’s correlation coefficients (p-value). We were forced to ignore some of the language pairs 
+when calculating the correlations with the EzGlot metric as some of the similarity values were missing. The correlation 
+analysis results are shown in Table 13 for sentiment analysis, Table 14 for NER, Table 15 for DEP. 
+Looking at the results, one can say that there is mostly a strong correlation between lang2vec, WALS and 
+eLinguistics metrics and the cross-lingual zero-shot transfer score, and a strong-moderate correlation between the 
+EzGlot metric and the transfer scores for NER and DEP. In the case of sentiment analysis, the correlation is noticeably 
+lower with XLM-R, staying at a moderate level with all of the linguistic similarity metrics. The correlation is strongest 
+with the dependency parsing task with XLM-R, with the highest absolute Spearman’s correlation being 0.897 with 
+eLinguistics metric. Also, the results show p-value < 0.05 for all of the tasks, models and metrics, indicating statistical 
+significance. 
+For both of the models, the correlation for lang2vec, WALS and eLinguistics metrics are generally higher than 
+EzGlot, except in the case of sentiment analysis, where EzGlot’s correlation is slightly higher than WALS and 
+eLinguistics using both Pearson’s and Spearman’s correlation coefficients. Also, the correlations were generally slightly 
+stronger with mBERT in sentiment analysis, while XLM-R had higher correlations in both NER and DEP tasks. 
+However, the results changed drastically for all tasks except dependency parsing, when we removed the anchor 
+points of same source-target language pairs (monolingual scenarios), leaving only the zero-shot transfer results. This 
+was necessary to do in order remove the bias brought by the monolingual data points, as the scores are higher and the 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 13 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+Table 11
+DEP: LAS-scores for XLM-R
+TARGET
+Danish
+English
+Polish
+Russian
+Korean
+German
+Croatian
+Japanese
+Danish
+0.888
+0.679
+0.725
+0.706
+0.672
+0.715
+0.095
+0.366
+English
+0.733
+0.728
+0.720
+0.716
+0.112
+0.911
+0.700
+0.364
+German
+0.712
+0.681
+0.854
+0.751
+0.732
+0.784
+0.066
+0.405
+SOURCE
+Croatian
+0.639
+0.668
+0.702
+0.910
+0.798
+0.818
+0.069
+0.375
+Polish
+0.614
+0.603
+0.676
+0.780
+0.945
+0.804
+0.049
+0.384
+Russian
+0.642
+0.645
+0.722
+0.801
+0.796
+0.890
+0.101
+0.378
+Japanese
+0.118
+0.132
+0.172
+0.098
+0.122
+0.106
+0.937
+0.317
+Korean
+0.326
+0.292
+0.381
+0.298
+0.325
+0.304
+0.183
+0.877
+Table 12
+Average scores for each source language on each task
+mBERT
+XLM-R
+Sentiment
+NER
+DEP
+Sentiment
+NER
+DEP
+Danish
+0.904
+0.740
+0.497
+0.886
+0.867
+0.606
+English 
+0.873
+0.786
+0.531
+0.874
+0.907
+0.623
+German
+0.845
+0.803
+0.536
+0.825
+0.909
+0.623
+Croatian
+0.905
+0.728
+0.536
+0.888
+0.864
+0.622
+Polish
+0.900
+0.719
+0.514
+0.870
+0.890
+0.607
+Russian
+0.902
+0.781
+0.533
+0.866
+0.860
+0.622
+Japanese
+0.871
+0.754
+0.220
+0.849
+0.905
+0.250
+ Korean
+0.898
+0.698
+0.081
+0.861
+0.773
+0.373
+is extremely low. Except for this case with DEP, all of the proposed languages seem to be quite equal as cross-
+lingual transfer sources in general. Interestingly, German, Croatian and Russian seem to perform slightly better overall
+compared to the other languages, especially with mBERT. A similar phenomenon was also experienced by Turc et al.
+[40] and in our previous research [26].
+5.3. Effect of Linguistic Similarity
+We calculated the correlation between the zero-shot cross-lingual transfer results of the two models and each of the
+four proposed linguistic similarity metrics (EzGlot, eLinguistics, WALS and lang2vec) in all proposed NLP tasks using
+both Pearson's and Spearman's correlation coefficients (p-value). We were forced to ignore some of the language pairs
+when calculating the correlations with the EzGlot metric as some of the similarity values were missing. The correlation
+analysis results are shown in Table 13 for sentiment analysis, Table 14 for NER, Table 15 for DEP.
+Looking at the results, one can say that there is mostly a strong correlation between lang2vec, WALS and
+eLinguistics metrics and the cross-lingual zero-shot transfer score, and a strong-moderate correlation between the
+EzGlot metric and the transfer scores for NER and DEP. In the case of sentiment analysis, the correlation is noticeably
+lower with XLM-R, staying at a moderate level with all of the linguistic similarity metrics. The correlation is strongest
+with the dependency parsing task with XLM-R, with the highest absolute Spearman's correlation being O.897 with
+eLinguistics metric. Also, the results show p-value < 0.05 for all of the tasks, models and metrics, indicating statistical
+significance.
+For both of the models, the correlation for lang2vec, WALS and eLinguistics metrics are generally higher than
+EzGlot, except in the case of sentiment analysis, where EzGlot's correlation is slightly higher than WALS and
+eLinguistics using both Pearson's and Spearman's correlation coefficients. Also, the correlations were generally slightly
+stronger with mBERT in sentiment analysis, while XLM-R had higher correlations in both NER and DEP tasks.
+However, the results changed drastically for all tasks except dependency parsing, when we removed the anchor
+points of same source-target language pairs (monolingual scenarios), leaving only the zero-shot transfer results. This
+was necessary to do in order remove the bias brought by the monolingual data points, as the scores are higher and the
+Eronen et al.: Preprint submitted to Elsevier
+Page 13 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+Table 13 
+Sentiment analysis: Pearson’s and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics 
+Pearson 
+Spearman 
+XLM-R 
+mBERT 
+XLM-R 
+mBERT 
+p 
+p-value 
+p 
+p-value 
+p 
+p-value 
+p 
+p-value 
+WALS 
+-0.297 
+0.017 
+-0.645 
+0.000 = -0.331 
+0.008 
+-0.537 
+0.000 
+EzGlot 
+0.389 
+0.005 
+0.729 
+0.000 
+0.533 
+0.000 
+0.586 
+0.000 
+eLinguistics 
+= -0.355 
+0.004 
+-0.648 
+0.000 
+=-0.413 
+0.001 
+-0.652 
+0.000 
+lang2vec 
+-0.418 
+0.001 
+-0.746 
+0.000 
+-0.482 
+0.000 
+-0.623 
+0.000 
+Table 14 
+NER: Pearson's and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics 
+Pearson 
+Spearman 
+XLM-R 
+mBERT 
+XLM-R 
+mBERT 
+p 
+p-value 
+p 
+p-value 
+p 
+p-value 
+p 
+p-value 
+WALS 
+-0.514 
+0.000 
+-0.500 
+0.000 
+= -0.510 
+0.000 
+-0.486 
+0.000 
+EzGlot 
+0.494 
+0.000 
+0.427 
+0.002 
+0.464 
+0.001 
+0.401 
+0.004 
+eLinguistics 
+ -0.580 
+0.000 
+-0.517 
+0.000 
+-0.608 
+0.000 
+-0.553 
+0.000 
+lang2vec 
+-0.465 
+0.000 
+-0.432 
+0.000 
+= 
+-0.504 
+0.000 
+-0.461 
+0.000 
+Table 15 
+DEP: Pearson’s and Spearman's correlation coefficients for model LAS scores and linguistic similarity metrics 
+Pearson 
+Spearman 
+XLM-R 
+mBERT 
+XLM-R 
+mBERT 
+p 
+p-value 
+p 
+p-value 
+p 
+p-value 
+p 
+p-value 
+WALS 
+-0.781 
+0.000 
+-0.718 
+0.000 
+-0.844 
+0.000 
+-0.693 
+0.000 
+EzGlot 
+0.588 
+0.000 
+0.516 
+0.000 
+0.694 
+0.000 
+0.561 
+0.000 
+eLinguistics 
+-0.845 
+0.000 
+-0.840 
+0.000 
+=-0.897 
+0.000 
+-0.867 
+0.000 
+lang2vec 
+-0.702 
+0.000 
+-0.679 
+0.000 
+= -0.848 
+0.000 
+-0.754 
+0.000 
+  
+languages would also be the most similar (same). The results after removing the same source-target language pairs are 
+shown in Table 16 for sentiment analysis, Table 17 for NER, Table 18 for DEP. 
+First of all, for eLinguistics and WALS similarity metrics, the correlations generally dropped from strong to 
+moderate for the NER task while EzGlot’s correlation fell close to zero and also lost all statistical significance. The 
+correlation of lang2vec also plummeted and mostly lost statistical significance, but remained higher than EzGlot’s. 
+Interestingly, for sentiment analysis, the result looks completely opposite. The correlations of EzGlot and lang2vec 
+only fell slightly while WALS’ and eLinguistics’ correlation plummeted down and lost statistical significance for 
+XLM-R. The correlations in the dependency parsing task only dropped slightly for all of the linguistic similarity 
+metrics. Also, after removing the same source-target language pairs, the strongest correlations are still found in DEP, 
+followed by NER, while sentiment analysis still has the weakest correlations overall. However, as linguistic similarity 
+still correlates with cross-lingual transfer performance, we can get an improved model performance by using linguistic 
+similarity for transfer language selection. 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 14 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+Table 13
+Sentiment analysis: Pearson's and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics
+Pearson
+ Spearman
+XLM-R
+mBERT
+XLM-R
+mBERT
+ p-value
+p-value
+p
+p-value
+ p-value
+p
+p
+p
+WALS
+-0.297
+0.017
+-0.645
+0.000
+-0.331
+0.008
+-0.537
+0.000
+EzGlot
+0.389
+0.005
+0.729
+0.000
+0.533
+0.000
+0.586
+0.000
+eLinguistics
+-0.355
+0.004
+-0.648
+0.000
+-0.413
+0.001
+-0.652
+0.000
+lang2vec
+-0.418
+0.001
+-0.746
+0.000
+-0.482
+0.000
+-0.623
+0.000
+Table 14
+NER: Pearson's and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics
+Pearson
+ Spearman
+XLM-R
+mBERT
+XLM-R
+mBERT
+ p-value
+p-value
+p
+p-value
+ p-value
+p
+p
+p
+WALS
+-0.514
+0.000
+-0.500
+0.000
+-0.510
+0.000
+-0.486
+0.000
+EzGlot
+0.494
+0.000
+0.427
+0.002
+0.464
+0.001
+0.401
+0.004
+eLinguistics
+-0.580
+0.000
+-0.517
+0.000
+-0.608
+0.000
+-0.553
+0.000
+lang2vec
+-0.465
+0.000
+-0.432
+0.000
+-0.504
+0.000
+-0.461
+0.000
+Table 15
+DEP: Pearson's and Spearman's correlation coefficients for model LAS scores and linguistic similarity metrics
+Pearson
+ Spearman
+XLM-R
+mBERT
+XLM-R
+mBERT
+p
+p-value
+p
+p-value
+p
+p-value
+p
+p-value
+WALS
+-0.781
+0.000
+-0.718
+0.000
+-0.844
+0.000
+-0.693
+0.000
+EzGlot
+0.588
+0.000
+0.516
+0.000
+0.694
+0.000
+0.561
+0.000
+eLinguistics
+-0.845
+0.000
+-0.840
+0.000
+-0.897
+0.000
+-0.867
+0.000
+lang2vec
+-0.702
+0.000
+-0.679
+0.000
+-0.848
+0.000
+-0.754
+0.000
+languages would also be the most similar (same). The results after removing the same source-target language pairs are
+shown in Table 16 for sentiment analysis, Table 17 for NER, Table 18 for DEP.
+First of all, for eLinguistics and WALS similarity metrics, the correlations generally dropped from strong to
+moderate for the NER task while EzGlot's correlation fell close to zero and also lost all statistical significance. The
+correlation of lang2vec also plummeted and mostly lost statistical significance, but remained higher than EzGlot's
+only fell slightly while WALS' and eLinguistics' correlation plummeted down and lost statistical significance for
+XLM-R. The correlations in the dependency parsing task only dropped slightly for all of the linguistic similarity
+followed by NER, while sentiment analysis still has the weakest correlations overall. However, as linguistic similarity
+still correlates with cross-lingual transfer performance, we can get an improved model performance by using linguistic
+similarity for transfer language selection.
+Eronen et al.: Preprint submitted to Elsevier
+Page 14 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+Table 16 
+Sentiment analysis: Pearson’s and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics 
+for zero-shot only 
+  
+  
+  
+  
+Pearson 
+Spearman 
+XLM-R 
+mBERT 
+XLM-R 
+mBERT 
+p 
+p-value 
+p 
+p-value 
+p 
+p-value 
+p 
+p-value 
+WALS 
+-0.111 
+0.415 
+-0.262 
+0.051 
+= -0.168 
+0.216 
+-0.315 
+0.018 
+EzGlot 
+0.327 
+0.035 
+0.303 
+0.051 
+0.403 
+0.008 
+0.313 
+0.044 
+eLinguistics 
+ -0.229 
+0.090 
+-0.392 
+0.003 
+-0.284 
+0.034 
+-0.487 
+0.000 
+lang2vec 
+-0.380 
+0.004 
+-0.454 
+0.000 
+=-0.374 
+0.005 
+-0.440 
+0.001 
+  
+Table 17 
+NER: Pearson's and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics for zero-shot 
+only 
+  
+  
+  
+  
+Pearson 
+Spearman 
+XLM-R 
+mBERT 
+XLM-R 
+mBERT 
+p 
+p-value 
+p 
+p-value 
+p 
+p-value 
+p 
+p-value 
+WALS 
+-0.336 
+0.011 
+-0.347 
+0.009 
+-0.316 
+0.017 
+-0.293 
+0.028 
+EzGlot 
+0.173 
+0.274 
+0.120 
+0.448 
+0.170 
+0.282 
+0.095 
+0.549 
+eLinguistics 
+-0.453 
+0.000 
+-0.384 
+0.003 
+-0.458 
+0.000 
+-0.389 
+0.003 
+lang2vec 
+-0.234 
+0.082 
+-0.214 
+0.113 
+-0.307 
+0.021 
+-0.256 
+0.057 
+  
+Table 18 
+DEP: Pearson's and Spearman's correlation coefficients for model LAS scores and linguistic similarity metrics for zero-shot 
+only 
+  
+  
+  
+  
+Pearson 
+Spearman 
+XLM-R 
+mBERT 
+XLM-R 
+mBERT 
+p 
+p-value 
+p 
+p-value 
+p 
+p-value 
+p 
+p-value 
+WALS 
+-0.738 
+0.000 
+-0.661 
+0.000 
+-0.769 
+0.000 
+-0.588 
+0.000 
+EzGlot 
+0.421 
+0.005 
+0.373 
+0.015 
+0.488 
+0.001 
+0.349 
+0.024 
+eLinguistics 
+-0.795 
+0.000 
+-0.822 
+0.000 
+-0.848 
+0.000 
+-0.842 
+0.000 
+lang2vec 
+-0.714 
+0.000 
+-0.709 
+0.000 = -0.775 
+0.000 
+-0.676 
+0.000 
+  
+6. Discussion 
+6.1. Transfer Language Performance 
+XLM-R outperforming mBERT generally matches our expectations, as it also did so on a variety of benchmark 
+tasks [12, 17]. The reason behind this most likely is the fact that XLM-R uses a vastly larger amount of data for 
+pretraining compared to mBERT. The performance difference between the two models is the most clear in the NER 
+task. 
+According to the results, simply choosing English as the transfer source did not yield top results most of the time, 
+sometimes even as a source language to other languages in the Germanic language group. For example, it had a lower 
+than average performance in sentiment analysis and an average performance in the other two tasks probably due to 
+its simplicity when compared to both Danish and German. It was also slightly outperformed by Slavic languages in 
+some cases when used as a source for other Germanic languages. Another reason could be the influence of French 
+[88, 89, 90], which might further distance it from the other Germanic languages. Also, the differences in morphology 
+could be a factor here. Danish and German probably work better with each other due to a great amount of historic 
+mutual influence. 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 15 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+Table 16
+Sentiment analysis: Pearson's and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics
+for zero-shot only
+Pearson
+ Spearman
+XLM-R
+mBERT
+XLM-R
+mBERT
+p
+ p-value
+p
+p-value
+p
+ p-value
+ p-value
+p
+WALS
+-0.111
+0.415
+-0.262
+0.051
+-0.168
+0.216
+-0.315
+0.018
+EzGlot
+0.327
+0.035
+0.303
+0.051
+0.403
+0.008
+0.313
+0.044
+eLinguistics
+-0.229
+0.090
+-0.392
+0.003
+-0.284
+0.034
+-0.487
+0.000
+lang2vec
+-0.380
+0.004
+-0.454
+0.000
+-0.374
+0.005
+-0.440
+0.001
+Table 17
+NER: Pearson's and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics for zero-shot
+only
+Pearson
+ Spearman
+XLM-R
+mBERT
+XLM-R
+mBERT
+p
+ p-value
+p-value
+p-value
+ p-value
+p
+p
+p
+WALS
+-0.336
+0.011
+-0.347
+0.009
+-0.316
+0.017
+-0.293
+0.028
+EzGlot
+0.173
+0.274
+0.120
+0.448
+0.170
+0.282
+0.095
+0.549
+eLinguistics
+-0.453
+0.000
+-0.384
+0.003
+-0.458
+0.000
+-0.389
+0.003
+lang2vec
+-0.234
+0.082
+-0.214
+0.113
+-0.307
+0.021
+-0.256
+0.057
+Table 18
+DEP: Pearson's and Spearman's correlation coefficients for model LAS scores and linguistic similarity metrics for zero-shot
+only
+Pearson
+ Spearman
+XLM-R
+mBERT
+XLM-R
+mBERT
+p
+p-value
+ p-value
+p
+p-value
+ p-value
+p
+p
+WALS
+-0.738
+0.000
+-0.661
+0.000
+-0.769
+0.000
+-0.588
+0.000
+EzGlot
+0.421
+0.005
+0.373
+0.015
+0.488
+0.001
+0.349
+0.024
+eLinguistics
+-0.795
+0.000
+-0.822
+0.000
+-0.848
+0.000
+-0.842
+0.000
+lang2vec
+-0.714
+0.000
+-0.709
+0.000
+-0.775
+0.000
+-0.676
+0.000
+6. Discussion
+6.1. Transfer Language Performance
+XLM-R outperforming mBERT generally matches our expectations, as it also did so on a variety of benchmark
+tasks [12, 17]. The reason behind this most likely is the fact that XLM-R uses a vastly larger amount of data for
+pretraining compared to mBERT. The performance difference between the two models is the most clear in the NER
+task.
+According to the results, simply choosing English as the transfer source did not yield top results most of the time,
+than average performance in sentiment analysis and an average performance in the other two tasks probably due to
+its simplicity when compared to both Danish and German. It was also slightly outperformed by Slavic languages in
+some cases when used as a source for other Germanic languages. Another reason could be the influence of French
+[88, 89, 90], which might further distance it from the other Germanic languages. Also, the differences in morphology
+could be a factor here. Danish and German probably work better with each other due to a great amount of historic
+mutual influence.
+Eronen et al.: Preprint submitted to Elsevier
+Page 15 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+In sentiment analysis, the models achieved slightly better scores with English and it was on-par with other Germanic 
+languages. However, all of the Slavic languages still tended to work slightly better as transfer sources. These results 
+show that other languages should also be considered over English as the cross-lingual transfer source if available. For 
+the other two tasks, NER and DEP, English performed well and showed to be a good transfer source for the other two 
+Germanic languages. 
+In most cases, using languages from the same language family as the source language yielded the highest cross- 
+lingual transfer scores. This matches with the typical intuition-based selection process used to select source language for 
+cross-lingual transfer. However, relying only on intuition and looking purely at language families when when selecting 
+the transfer language will lead to diminished results in some cases. 
+One example would be taking Polish as the target language for NER task. One could expect that in this case, the 
+best transfer languages would be Croatian and Russian, but looking at the results (Tables 8 and 9) German had a 
+higher cross-lingual transfer score even though it is from the Germanic language family, not Slavic. This could be, for 
+example, due to mutual influence of these two languages. The grammar of both Danish and English is relatively simple 
+compared to German, which could aid them in generalizing better with one another. Looking at the scores, it can be 
+noted that German is a good source for both Germanic and Slavic languages, which could mean that the historical 
+mutual influence between the Germans and Slavs could be a factor here. Furthermore, German, in addition to having a 
+higher average performance on most tasks, tended to also work exceptionally well also as a source language for other 
+Slavic languages, most likely because of the reasons discussed above. 
+In addition, Japanese and Korean did not achieve comparably better scores with one another, contrary to our 
+expectations, and were even slightly outperformed by the other languages approximately half of the time, even though 
+being more similar with each other compared to any of the other proposed languages. Here the reason could be, for 
+example, the differences in the writing systems, as neither of these two languages use alphabets and their systems also 
+greatly differ from each other. 
+Also, both Russian and Croatian had a higher than average performance on most of the tasks. This was similar to 
+our previous research [26] where Russian performed exceptionally well as a transfer source for offensive language 
+identification. However, unlike in our previous research, Russian did not perform noticeably well as the transfer 
+language source for Korean and Japanese. Thus the phenomenon experienced previously is most likely related to 
+the topic of offensive language identification itself or to the properties of these specific datasets. We will investigate 
+this in later research. Also, Japanese and Korean had a satisfying performance as source languages for the Germanic 
+and Slavic languages in both sentiment analysis and NER tasks, even though Japanese and Korean are fundamentally 
+different from the languages of these two families, as they are the only non Indo-European languages in the proposed 
+set. This demonstrates that multilingual transformer models are also able to leverage knowledge even from very distant 
+languages. 
+6.2. Analysis of Linguistic Similarity Metrics 
+The correlation between cross-lingual transfer performance and the similarity metrics were strong or moderate 
+with all of the proposed metrics, which would suggest that using even a single feature such as lexical information or 
+by comparing phonetic consonants is still effective to some extent. 
+6.2.1. EzGlot 
+However, when considering only the zero-shot transfer results, EzGlot’s similarity metric’s correlation dropped 
+drastically and out of statistical significance in the NER task. This shows that it does not necessarily rely on lexical 
+features and that other linguistic features need to be considered when choosing the source language for NER. On 
+the other hand, the same happened with the lang2vec metric despite it being created using different features from 
+multiple domains. Also, the opposite happened in the sentiment analysis task, as both WALS and eLinguistics metrics’ 
+correlation dropped drastically and out of statistical significance. This hints the importance of lexical similarity when 
+choosing the source language for sentiment analysis tasks. 
+6.2.2. eLinguistics 
+Surprisingly, even though using only a predefined set of phonetic consonants for its calculation, the correlation of 
+eLinguistics’ similarity metric was stronger in all tasks compared to the the correlation of the WALS metric, which we 
+quantified from the WALS database using linguistic features from multiple domains. The correlation of eLinguistics 
+was also higher than the averaged lang2vec metric in both NER and DEP. The reason behind this could be that including 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 16 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+In sentiment analysis, the models achieved slightly better scores with English and it was on-par with other Germanic
+languages. However, all of the Slavic languages still tended to work slightly better as transfer sources. These results
+show that other languages should also be considered over English as the cross-lingual transfer source if available. For
+the other two tasks, NER and DEP, English performed well and showed to be a good transfer source for the other two
+Germanic languages.
+In most cases, using languages from the same language family as the source language yielded the highest cross-
+lingual transfer scores. This matches with the typicalintuition-based selection process used to select source language for
+cross-lingual transfer. However, relying only on intuition and looking purely at language families when when selecting
+the transfer language will lead to diminished results in some cases.
+One example would be taking Polish as the target language for NER task. One could expect that in this case, the
+best transfer languages would be Croatian and Russian, but looking at the results (Tables 8 and 9) German had a
+example, due to mutual influence of these two languages. The grammar of both Danish and English is relatively simple
+compared to German, which could aid them in generalizing better with one another. Looking at the scores, it can be
+noted that German is a good source for both Germanic and Slavic languages, which could mean that the historical
+mutual influence between the Germans and Slavs could be a factor here. Furthermore, German, in addition to having a
+higher average performance on most tasks, tended to also work exceptionally well also as a source language for other
+Slavic languages, most likely because of the reasons discussed above.
+In addition, Japanese and Korean did not achieve comparably better scores with one another, contrary to our
+expectations, and were even slightly outperformed by the other languages approximately half of the time, even though
+being more similar with each other compared to any of the other proposed languages. Here the reason could be, for
+example, the differences in the writing systems, as neither of these two languages use alphabets and their systems also
+greatly differ from each other.
+Also, both Russian and Croatian had a higher than average performance on most of the tasks. This was similar to
+our previous research [26] where Russian performed exceptionally well as a transfer source for offensive language
+identification. However, unlike in our previous research, Russian did not perform noticeably well as the transfer
+the topic of offensive language identification itself or to the properties of these specific datasets. We will investigate
+this in later research. Also, Japanese and Korean had a satisfying performance as source languages for the Germanic
+and Slavic languages in both sentiment analysis and NER tasks, even though Japanese and Korean are fundamentally
+different from the languages of these two families, as they are the only non Indo-European languages in the proposed
+set. This demonstrates that multilingual transformer models are also able to leverage knowledge even from very distant
+languages.
+6.2. Analysis of Linguistic Similarity Metrics
+The correlation between cross-lingual transfer performance and the similarity metrics were strong or moderate
+with all of the proposed metrics, which would suggest that using even a single feature such as lexical information or
+by comparing phonetic consonants is still effective to some extent.
+6.2.1. EzGlot
+However, when considering only the zero-shot transfer results, EzGlot's similarity metric's correlation dropped
+drastically and out of statistical significance in the NER task. This shows that it does not necessarily rely on lexical
+features and that other linguistic features need to be considered when choosing the source language for NER. On
+multiple domains. Also, the opposite happened in the sentiment analysis task, as both WALS and eLinguistics metrics'
+correlation dropped drastically and out of statistical significance. This hints the importance of lexical similarity when
+choosing the source language for sentiment analysis tasks.
+6.2.2. eLinguistics
+Surprisingly, even though using only a predefined set of phonetic consonants for its calculation, the correlation of
+eLinguistics' similarity metric was stronger in all tasks compared to the the correlation of the WALS metric, which we
+quantified from the WALS database using linguistic features from multiple domains. The correlation of eLinguistics
+was also higher than the averaged lang2vec metric in both NER and DEP. The reason behind this could be that including
+Eronen et al.: Preprint submitted to Elsevier
+Page 16 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+all of possible features between each language pair could have caused too many irrelevant features to be included. This 
+can cause a possible bias the metric calculation. 
+However, the eLinguistics metric also has its weak points as it is based on only a single aspect of language, even 
+though its correlation being the strongest. One can see from Table 
+1 that eLinguistics shows Japanese being very 
+distant from Korean, being at the same level as Polish and Russian, with Croatian being seemingly closer to Korean 
+than Japanese, which is not true due to the similarities in the vocabulary and grammar of Japanese and Korean. Taking 
+a look at Tables 4 and 3, it is clear that the WALS and lang2vec metrics are a lot more robust to this kind of errors. 
+The reason most likely is that instead of using only a single linguistic feature like the eLinguistics metric, the WALS 
+and lang2vec metrics are based on a large amount of features spanning over multiple domains. 
+6.2.3. Averaged lang2vec and Quantified WALS 
+Both lang2vec and WALS had a strong correlation with the DEP task. Although both metrics are based on a large 
+number of linguistic features spanning over multiple domains, their correlations varied greatly with the other two 
+tasks. Specifically, in NER, the correlation of lang2vec was noticeably lower. In sentiment analysis, the correlation of 
+WALS plummeted while lang2vec stayed at a moderate level. The reason is probably in the calculation of the metrics. 
+In the quantified WALS metric, features are treated as continuous whereas lang2vec uses one-hot encoding. Also, in 
+quantified WALS, every single feature has the same weight whereas in averaged lang2vec every category of features has 
+the same weight but might contain a different amount of features. Additionally, the features in lang2vec are collected 
+from multiple sources, which increases the amount of data while possibly introducing incoherence. Lastly, the number 
+of features used in the similarity calculations with the quantified WALS metric varies slightly between language pairs 
+due to missing values in the database. Lang2vec tries to counter this by using a model to predict the missing values, 
+although this might introduce more errors in some cases. 
+In the future, we will take another glance at the WALS database, aiming for a better quantification by looking at 
+the importance of each feature group (syntactic, lexical, phonetic, etc.) and weighing accordingly while filtering out 
+redundant features in order to develop an even more effective and comprehensive similarity metric. We will also take a 
+look at lang2vec, aiming to filter out redundant features and weigh the categories accordingly instead of simply taking 
+an average in order to make it better suited for transfer language selection. After all, both WALS and lang2vec metrics 
+are more robust thanks to being based on a large amount of features spanning over multiple domains instead of using 
+only a single linguistic feature like eLinguistics or EzGlot. 
+6.3. Task-Specific Analysis 
+6.3.1. Sentiment Analysis 
+Looking at the f-scores of the sentiment analysis task, it is clear that the results are very high across the board and 
+the score differences between language groups are also very small, with sometimes languages from other language 
+groups than the target emerging as the best performers. This is the case for example with Danish, as Croatian achieved 
+the highest zero-shot transfer scores for both mBERT and XLM-R instead of another Germanic language. 
+A trait only observed in this task was that lang2vec and EzGlot were the only metrics keeping a moderate correlation 
+in the zero-shot setting. The fact that Ezglot’s correlation stayed moderate hints the importance of lexical features. One 
+could argue that the reason behind the overall high scores might be due to the task being too easy, as it simply required 
+the classification of the entries into positive and negative. This has also been shown in other research [91]. However, this 
+also shows that it could be possible to achieve at least close to state-of-the-art results with multilingual transformer 
+models in a zero-shot cross-lingual setting. This raises questions about how to improve the cross-lingual models to 
+better utilize cross-lingual transfer. In the future, it would be useful to further investigate the models’ behaviour in 
+zero-shot setting. This could also be useful in the further development of measures to support low-resource languages. 
+6.3.2. Named Entity Recognition 
+For mBERT, the zero-shot results of the NER task look clearly lower than with same language pairs and quite 
+even across all of the proposed languages and the languages belonging to the same group having generally a slightly 
+higher score. However, the results of XLM-R closely resemble those of the sentiment analysis task as the results are 
+considerably high across all language pairs. This further shows the potential these models have in relieving the issues 
+with low-resource languages. Also, there is a moderate correlation between the zero-shot transfer performance and 
+linguistic similarity for both WALS and eLinguistics metrics, and a weak/moderate correlation with lang2vec. 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 17 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+all of possible features between each language pair could have caused too many irrelevant features to be included. This
+can cause a possible bias the metric calculation.
+However, the eLinguistics metric also has its weak points as it is based on only a single aspect of language, even
+though its correlation being the strongest. One can see from Table 1 that eLinguistics shows Japanese being very
+distant from Korean, being at the same level as Polish and Russian, with Croatian being seemingly closer to Korean
+than Japanese, which is not true due to the similarities in the vocabulary and grammar of Japanese and Korean. Taking
+a look at Tables 4 and 3, it is clear that the WALS and lang2vec metrics are a lot more robust to this kind of errors
+The reason most likely is that instead of using only a single linguistic feature like the eLinguistics metric, the WALS
+and lang2vec metrics are based on a large amount of features spanning over multiple domains.
+6.2.3. Averaged lang2vec and Quantified WALS
+Both lang2vec and WALS had a strong correlation with the DEP task. Although both metrics are based on a large
+number of linguistic features spanning over multiple domains, their correlations varied greatly with the other two
+tasks. Specifically, in NER, the correlation of lang2vec was noticeably lower. In sentiment analysis, the correlation of
+WALS plummeted while lang2vec stayed at a moderate level. The reason is probably in the calculation of the metrics.
+In the quantified WALS metric, features are treated as continuous whereas lang2vec uses one-hot encoding. Also, in
+quantified WALS, every single feature has the same weight whereas in averaged lang2vec every category of features has
+the same weight but might contain a different amount of features. Additionally, the features in lang2vec are collected
+from multiple sources, which increases the amount of data while possibly introducing incoherence. Lastly, the number
+of features used in the similarity calculations with the quantified WALS metric varies slightly between language pairs
+due to missing values in the database. Lang2vec tries to counter this by using a model to predict the missing values,
+although this might introduce more errors in some cases.
+In the future, we will take another glance at the WALS database, aiming for a better quantification by looking at
+the importance of each feature group (syntactic, lexical, phonetic, etc.) and weighing accordingly while filtering out
+redundant features in order to develop an even more effective and comprehensive similarity metric. We will also take a
+look at lang2vec, aiming to filter out redundant features and weigh the categories accordingly instead of simply taking
+an average in order to make it better suited for transfer language selection. After all, both WALS and lang2vec metrics
+are more robust thanks to being based on a large amount of features spanning over multiple domains instead of using
+only a single linguistic feature like eLinguistics or EzGlot.
+6.3. Task-Specific Analysis
+6.3.1. Sentiment Analysis
+Looking at the f-scores of the sentiment analysis task, it is clear that the results are very high across the board and
+the score differences between language groups are also very small, with sometimes languages from other language
+groups than the target emerging as the best performers. This is the case for example with Danish, as Croatian achieved
+the highest zero-shot transfer scores for both mBERT and XLM-R instead of another Germanic language.
+A trait only observed in this task was that lang2vec and EzGlot were the only metrics keeping a moderate correlation
+in the zero-shot setting. The fact that Ezglot's correlation stayed moderate hints the importance of lexical features. One
+could argue that the reason behind the overall high scores might be due to the task being too easy, as it simply required
+the classification of the entries into positive and negative. This has also been shown in other research [91]. However, this
+also shows that it could be possible to achieve at least close to state-of-the-art results with multilingual transformer
+models in a zero-shot cross-lingual setting. This raises questions about how to improve the cross-lingual models to
+better utilize cross-lingual transfer. In the future, it would be useful to further investigate the models' behaviour in
+zero-shot setting. This could also be useful in the further development of measures to support low-resource languages.
+6.3.2. Named Entity Recognition
+For mBERT, the zero-shot results of the NER task look clearly lower than with same language pairs and quite
+higher score. However, the results of XLM-R closely resemble those of the sentiment analysis task as the results are
+considerably high across all language pairs. This further shows the potential these models have in relieving the issues
+with low-resource languages. Also, there is a moderate correlation between the zero-shot transfer performance and
+linguistic similarity for both WALS and eLinguistics metrics, and a weak/moderate correlation with lang2vec.
+Eronen et al.: Preprint submitted to Elsevier
+Page 17 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+6.3.3. Dependency Parsing 
+The zero-shot results in DEP also seem clearly lower than with same language pairs and somewhat even across the 
+board. The languages belonging to the same language family also generally have a slightly higher score. As the task 
+requires the understanding of syntax and grammar and the scores are still reasonably high overall, the results could also 
+support studies claiming that cross-lingual transformer models are able to learn grammar without explicit information 
+[92]. 
+On the other hand, Japanese and Korean had very poor performance as source languages while also being very 
+difficult target languages in the DEP task, unlike in the sentiment analysis and NER tasks. This could mean that the 
+model is unable to generalize to the syntax and grammar of Indo-European languages with Japonic-Koreanic languages 
+and vice versa. The reason might be due to the differences in writing systems. In the DEP task, both XLM-R and 
+mBERT keep a strong correlation between the zero-shot transfer performance and linguistic similarity with WALS, 
+eLinguistics and lang2vec metrics and a moderate correlation with EzGlot in a zero-shot setting. 
+6.4. Impact 
+As there is a correlation with linguistic similarity and cross-lingual transfer performance for all of the tasks, 
+including the abusive language identification task used in our previous research [26], it is possible to use linguistic 
+similarity for transfer language selection, at least for these tasks. However, the correlation varied greatly from task 
+to task, which means there is a lot of room for improvement in developing an optimal similarity metric for transfer 
+language selection. 
+In order to confirm the efficacy of choosing another language over English as the cross-lingual transfer source, we 
+performed a z-test between the results of using English as transfer source and using the language with the highest score. 
+The test showed Z = —3.18 < —1.96 and p = 0.001 < 0.05 meaning that there is a significant difference between 
+using English and the optimal language as the cross-lingual transfer source. 
+Based on these results, as there is a significant difference between using English and the optimal language as the 
+cross-lingual transfer source, it is better to look for high-resource languages that have proper data available and are as 
+close as possible to the target language based on a similarity metric instead of making a decision based on intuition 
+or simply relying on English. This allows one to make a more informed and effective decision and makes model 
+development more efficient. 
+6.5. Future Research 
+In the future, we are planning to analyze, what kind of linguistic features are the most important from the 
+point of view of cross-lingual transfer. A solution could be grouping the features presented in WALS into syntactic, 
+lexical, phonetic, etc., and calculating, which feature group has the strongest correlation with the cross-lingual transfer 
+performance. We could then re-quantify the WALS database using this information in order to develop an even more 
+effective and comprehensive similarity metric. It would also be beneficial if the WALS project received more attention 
+and the feature matrix became more densely populated. Also, instead of taking an average of lang2vec’s categories, 
+they should be weighed by importance. 
+As shown by the DEP task, the models might be able to learn syntax and grammar without any explicit information. 
+This could mean that adding explicit syntactic and grammatical information to the pre-training process of the models 
+might also improve their performance. We will take a look at this in the future. Also, as the models achieved zero-shot 
+transfer scores rivaling those of the monolingual settings, especially in sentiment analysis, it would be useful to perform 
+an in-depth investigation about the models’ behaviour in a zero-shot transfer learning setting to possibly find insights 
+on how to improve their transfer learning capabilities. 
+7. Conclusions 
+In this research we studied cross-lingual transfer language selection for zero-shot learning using three different 
+NLP tasks, namely, sentiment analysis, NER, and dependency parsing. We showed the effectiveness of cross-lingual 
+zero-shot transfer learning with a total of eight languages from three language families. In this way, existing data from 
+higher-resource languages may be used to improve the performance of languages that lack sufficient data. 
+We found a strong correlation between the similarity of the used languages and cross-lingual transfer performance. 
+The transfer performance declines when the distance between the languages increases. This allows for the selection 
+of a more suitable transfer language by assessing linguistic similarities rather than simply depending on intuition 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 18 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+6.3.3. Dependency Parsing
+board. The languages belonging to the same language family also generally have a slightly higher score. As the task
+requires the understanding of syntax and grammar and the scores are still reasonably high overall, the results could also
+support studies claiming that cross-lingual transformer models are able to learn grammar without explicit information
+[92]
+On the other hand, Japanese and Korean had very poor performance as source languages while also being very
+difficult target languages in the DEP task, unlike in the sentiment analysis and NER tasks. This could mean that the
+model is unable to generalize to the syntax and grammar of Indo-European languages with Japonic-Koreanic languages
+and vice versa. The reason might be due to the differences in writing systems. In the DEP task, both XLM-R and
+mBERT keep a strong correlation between the zero-shot transfer performance and linguistic similarity with WALS,
+eLinguistics and lang2vec metrics and a moderate correlation with EzGlot in a zero-shot setting.
+6.4. Impact
+As there is a correlation with linguistic similarity and cross-lingual transfer performance for all of the tasks,
+including the abusive language identification task used in our previous research [26], it is possible to use linguistic
+similarity for transfer language selection, at least for these tasks. However, the correlation varied greatly from task
+to task, which means there is a lot of room for improvement in developing an optimal similarity metric for transfer
+language selection.
+In order to confirm the efficacy of choosing another language over English as the cross-lingual transfer source, we
+performed a z-test between the results of using English as transfer source and using the language with the highest score.
+The test showed Z = -3.18 < -1.96 and p = 0.001 < 0.05 meaning that there is a significant difference between
+using English and the optimal language as the cross-lingual transfer source.
+Based on these results, as there is a significant difference between using English and the optimal language as the
+cross-lingual transfer source, it is better to look for high-resource languages that have proper data available and are as
+close as possible to the target language based on a similarity metric instead of making a decision based on intuition
+or simply relying on English. This allows one to make a more informed and effective decision and makes model
+development more efficient.
+6.5. Future Research
+In the future, we are planning to analyze, what kind of linguistic features are the most important from the
+point of view of cross-lingual transfer. A solution could be grouping the features presented in WALS into syntactic,
+lexical, phonetic, etc., and calculating, which feature group has the strongest correlation with the cross-lingual transfer
+performance. We could then re-quantify the WALS database using this information in order to develop an even more
+effective and comprehensive similarity metric. It would also be beneficial if the WALS project received more attention
+and the feature matrix became more densely populated. Also, instead of taking an average of lang2vec's categories,
+they should be weighed by importance.
+ As shown by the DEP task, the models might be able to learn syntax and grammar without any explicit information.
+This could mean that adding explicit syntactic and grammatical information to the pre-training process of the models
+might also improve their performance. We will take a look at this in the future. Also, as the models achieved zero-shot
+transfer scores rivaling those of the monolingual settings, especially in sentiment analysis, it would be useful to perform
+an in-depth investigation about the models' behaviour in a zero-shot transfer learning setting to possibly find insights
+on how to improve their transfer learning capabilities.
+7. Conclusions
+In this research we studied cross-lingual transfer language selection for zero-shot learning using three different
+NLP tasks, namely, sentiment analysis, NER, and dependency parsing. We showed the effectiveness of cross-lingual
+zero-shot transfer learning with a total of eight languages from three language families. In this way, existing data from
+higher-resource languages may be used to improve the performance of languages that lack sufficient data.
+We found a strong correlation between the similarity of the used languages and cross-lingual transfer performance.
+The transfer performance declines when the distance between the languages increases. This allows for the selection
+of a more suitable transfer language by assessing linguistic similarities rather than simply depending on intuition
+Eronen et al.: Preprint submitted to Elsevier
+Page 18 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity 
+or defaulting to English. As our experiments have demonstrated, there is a significant difference in choosing the 
+optimal transfer language over defaulting to English. As there is a correlation between linguistic similarity and transfer 
+performance and a significant difference between using English and the optimal language as the cross-lingual transfer 
+source, one should instead choose the source language based on a linguistic similarity measure. Our experiments also 
+demonstrated that lexical information alone is insufficient to determine the optimal transfer languages at least for the 
+tasks of NER and DEP. 
+it is better to look for high-resource languages that have proper data available and are as close as possible to the target 
+language based on a similarity metric instead of making a decision based on intuition or simply relying on English. 
+This allows one to make a more informed and effective decision and makes model development more efficient. 
+The results showed that the proposed method for cross-lingual transfer language selection could also be useful as a 
+general method for other Natural Language Processing tasks, at least based on these tasks and our previous research. We 
+also showed that it is possible to achieve good performance on the target language in a zero-shot cross-lingual transfer 
+setting with multiple NLP tasks. This helps in developing better systems, especially when dealing with low-resource 
+languages. 
+We improved a novel linguistic similarity metric consisting of various linguistic features by using the WALS 
+database. Our proposed method did not show the strongest correlation with the transfer performance, but it still showed 
+potential as a metric that could be useful for the selection process, especially if given a more refined or inclusive feature 
+set. In the future, we will reassess the importances of the linguistic features used in the similarity metric calculation in 
+order to have a more refined feature set, aiming to create an even more effective and comprehensive linguistic similarity 
+metric. 
+Lastly, even though the overall high scores in the sentiment analysis task might be caused by the task being too 
+easy, it also shows that it could be possible to achieve results close to those of a monolingual fine-tuning in a zero-shot 
+cross-lingual transfer setting. This means it could be useful to thoroughly investigate the models’ behaviour in zero-shot 
+setting in order to find insights to improving their transfer capabilities. Also, as the DEP task demonstrated that the 
+models might have a capability to understand grammar, adding explicit syntactic and grammatical information to the 
+models’ pre-training could also increase performance. 
+CRediT authorship contribution statement 
+Juuso Eronen: Conceptualization, Methodology, Software, Writing - Original Draft, Writing - Review & Editing. 
+Michal Ptaszynski: Conceptualization, Methodology, Supervision, Data Curation. Fumito Masui: Conceptualization, 
+Methodology, Supervision, Data Curation. 
+References 
+[1] 
+David M. Eberhard, Gary F. Simons, and Charles D. Fennig, editors. Ethnologue: Languages of the World. 
+SIL International, Dallas, TX, 
+USA, twenty-fifth edition, 2022. 
+[2] 
+Sean P. Engelson and Ido Dagan. Minimizing manual annotation cost in supervised training from corpora. In Proceedings of the 34th Annual 
+Meeting on Association for Computational Linguistics, ACL ’96, page 319-326, USA, 1996. Association for Computational Linguistics. 
+[3] 
+Sandipan Dandapat, Priyanka Biswas, Monojit Choudhury, and Kalika Bali. Complex linguistic annotation—no easy way out! a case from 
+bangla and hindi pos labeling tasks. In Proceedings of the Third Linguistic Annotation Workshop (LAW III), pages 10-18, 2009. 
+[4] 
+Julia Hirschberg and Christopher D. Manning. Advances in natural language processing. Science, 349(6245):261-266, 2015. 
+[5] 
+Edoardo Maria Ponti, Helen O’Horan, Yevgeni Berzak, Ivan Vuli¢, Roi Reichart, Thierry Poibeau, Ekaterina Shutova, and Anna Korhonen. 
+Modeling language variation and universals: A survey on typological linguistics for natural language processing. Computational Linguistics, 
+45(3):559-601, September 2019. 
+[6] 
+Pratik Joshi, Sebastin Santy, Amar Budhiraja, Kalika Bali, and Monojit Choudhury. The state and fate of linguistic diversity and inclusion in 
+the NLP world. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 6282-6293, Online, July 
+2020. Association for Computational Linguistics. 
+[7] 
+Long Duong, Trevor Cohn, Steven Bird, and Paul Cook. 
+Low resource dependency parsing: Cross-lingual parameter sharing in a neural 
+network parser. 
+In Proceedings of the 53rd annual meeting of the Association for Computational Linguistics and the 7th international joint 
+conference on natural language processing (volume 2: short papers), pages 845-850, 2015. 
+[8] 
+Rouzbeh Ghasemi, Seyed Arad Ashrafi Asli, and Saeedeh Momtazi. Deep persian sentiment analysis: Cross-lingual training for low-resource 
+languages. Journal of Information Science, page 0165551520962781, 2020. 
+[9] 
+Raj Dabre, Chenhui Chu, and Anoop Kunchukuttan. 
+A survey of multilingual neural machine translation. 
+ACM Comput. Surv., 53(5), 
+September 2020. 
+[10] 
+Saurabh Gaikwad, Tharindu Ranasinghe, Marcos Zampieri, and Christopher M. Homan. Cross-lingual offensive language identification for 
+low resource languages: The case of marathi, 2021. 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 19 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+or defaulting to English. As our experiments have demonstrated, there is a significant difference in choosing the
+optimal transfer language over defaulting to English. As there is a correlation between linguistic similarity and transfer
+performance and a significant difference between using English and the optimal language as the cross-lingual transfer
+source. one should instead choose the source language based on a linguistic similarity measure. Our experiments also
+demonstrated that lexical information alone is insufficient to determine the optimal transfer languages at least for the
+tasks of NER and DEP.
+it is better to look for high-resource languages that have proper data available and are as close as possible to the target
+language based on a similarity metric instead of making a decision based on intuition or simply relying on English.
+This allows one to make a more informed and effective decision and makes model development more efficient.
+The results showed that the proposed method for cross-lingual transfer language selection could also be useful as a
+general method for other Natural Language Processing tasks, at least based on these tasks and our previous research. We
+also showed that it is possible to achieve good performance on the target language in a zero-shot cross-lingual transfer
+setting with multiple NLP tasks. This helps in developing better systems, especially when dealing with low-resource
+languages.
+We improved a novel linguistic similarity metric consisting of various linguistic features by using the WALS
+database. Our proposed method did not show the strongest correlation with the transfer performance, but it still showed
+potential as a metric that could be useful for the selection process, especially if given a more refined or inclusive feature
+set. In the future, we will reassess the importances of the linguistic features used in the similarity metric calculation in
+order to have a more refined feature set, aiming to create an even more effective and comprehensive linguistic similarity
+metric.
+Lastly, even though the overall high scores in the sentiment analysis task might be caused by the task being too
+easy, it also shows that it could be possible to achieve results close to those of a monolingual fine-tuning in a zero-shot
+cross-lingual transfer setting. This means it could be useful to thoroughly investigate the models' behaviour in zero-shot
+setting in order to find insights to improving their transfer capabilities. Also, as the DEP task demonstrated that the
+models might have a capability to understand grammar, adding explicit syntactic and grammatical information to the
+models' pre-training could also increase performance.
+CRediT authorship contribution statement
+Juuso Eronen: Conceptualization, Methodology, Software, Writing - Original Draft, Writing - Review & Editing.
+Michal Ptaszynski: Conceptualization, Methodology, Supervision, Data Curation. Fumito Masui: Conceptualization,
+Methodology, Supervision, Data Curation.
+References
+[1] David M. Eberhard, Gary F. Simons, and Charles D. Fennig, editors. Ethnologue: Languages of the World. SIL International, Dallas, TX,
+USA, twenty-fifth edition, 2022.
+[2] Sean P. Engelson and Ido Dagan. Minimizing manual annotation cost in supervised training from corpora. In Proceedings of the 34th Annual
+Meeting on Association for Computational Linguistics, ACL '96, page 319-326, USA, 1996. Association for Computational Linguistics
+bangla and hindi pos labeling tasks. In Proceedings of the Third Linguistic Annotation Workshop (LAW II), pages 10-18, 2009.
+[4] Julia Hirschberg and Christopher D. Manning. Advances in natural language processing. Science, 349(6245):261-266, 2015.
+[5] Edoardo Maria Ponti, Helen O'Horan, Yevgeni Berzak, Ivan Vulic, Roi Reichart, Thierry Poibeau, Ekaterina Shutova, and Anna Korhonen.
+Modeling language variation and universals: A survey on typological linguistics for natural language processing. Computational Linguistics,
+45(3):559-601, September 2019.
+[6] Pratik Joshi, Sebastin Santy, Amar Budhiraja, Kalika Bali, and Monojit Choudhury. The state and fate of linguistic diversity and inclusion in
+the NLP world. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 6282-6293, Online, July
+2020. Association for Computational Linguistics
+[7] Long Duong, Trevor Cohn, Steven Bird, and Paul Cook. Low resource dependency parsing: Cross-lingual parameter sharing in a neural
+network parser. In Proceedings of the 53rd annual meeting of the Association for Computational Linguistics and the 7th international joint
+conference on natural language processing (volume 2: short papers), pages 845-850, 2015.
+[8] Rouzbeh Ghasemi, Seyed Arad Ashrafi Asli, and Saeedeh Momtazi. Deep persian sentiment analysis: Cross-lingual training for low-resource
+languages. Journal of Information Science, page 0165551520962781, 2020.
+September 2020.
+[10] Saurabh Gaikwad, Tharindu Ranasinghe, Marcos Zampieri, and Christopher M. Homan. Cross-lingual offensive language identification for
+low resource languages: The case of marathi, 2021.
+Eronen et al.: Preprint submitted to Elsevier
+Page 19 of 23{11] 
+{12] 
+(13) 
+{14] 
+{15} 
+[16] 
+[17] 
+[18] 
+{19} 
+[20] 
+[21] 
+[22] 
+[23] 
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+Telmo Pires, Eva Schlinger, and Dan Garrette. How multilingual is multilingual bert? In ACL, 2019. 
+Junjie Hu, Sebastian Ruder, Aditya Siddhant, Graham Neubig, Orhan Firat, and Melvin Johnson. Xtreme: A massively multilingual multi-task 
+benchmark for evaluating cross-lingual generalisation. In International Conference on Machine Learning, pages 4411-4421. PMLR, 2020. 
+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 
+Bert: Pre-training of deep bidirectional transformers for language 
+understanding, 2018. 
+Alexis Conneau, Kartikay Khandelwal, Naman Goyal, Vishrav Chaudhary, Guillaume Wenzek, Francisco Guzman, Edouard Grave, Myle Ott, 
+Luke Zettlemoyer, and Veselin Stoyanov. 
+Unsupervised cross-lingual representation learning at scale. 
+In Proceedings of the 58th Annual 
+Meeting of the Association for Computational Linguistics, pages 8440-8451, Online, 2020. Association for Computational Linguistics. 
+Shijie Wu and Mark Dredze. Beto, bentz, becas: The surprising cross-lingual effectiveness of BERT. In Proceedings of the 2019 Conference 
+on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP- 
+IJCNLP), pages 833-844, Hong Kong, China, November 2019. Association for Computational Linguistics. 
+Edward P. Stabler and Edward L. Keenan. Structural similarity within and among languages. Theoretical Computer Science, 293(2):345-363, 
+2003. Algebraic Methods in Language Processing. 
+Yaobo Liang, Nan Duan, Yeyun Gong, Ning Wu, Fenfei Guo, Weizhen Qi, Ming Gong, Linjun Shou, Daxin Jiang, Guihong Cao, et al. Xglue: 
+A new benchmark datasetfor cross-lingual pre-training, understanding and generation. In Proceedings of the 2020 Conference on Empirical 
+Methods in Natural Language Processing (EMNLP), pages 6008-6018, 2020. 
+Matt Pikuliak, Marian Simko, and Maria Bielikova. Cross-lingual learning for text processing: A survey. Expert Systems with Applications, 
+165:113765, 2021. 
+Yu-Hsiang Lin, Chian-Yu Chen, Jean Lee, Zirui Li, Yuyan Zhang, Mengzhou Xia, Shruti Rijhwani, Junxian He, Zhisong Zhang, Xuezhe 
+Ma, Antonios Anastasopoulos, Patrick Littell, and Graham Neubig. Choosing transfer languages for cross-lingual learning. 
+In Proceedings 
+of the 57th Annual Meeting of the Association for Computational Linguistics, pages 3125-3135, Florence, Italy, July 2019. Association for 
+Computational Linguistics. 
+Ryan Cotterell and Georg Heigold. Cross-lingual character-level neural morphological tagging. 
+In Proceedings of the 2017 Conference on 
+Empirical Methods in Natural Language Processing, pages 748-759, Copenhagen, Denmark, September 2017. Association for Computational 
+Linguistics. 
+Charlotte Gooskens, Vincent J. van Heuven, Jelena Golubovi¢é, Anja Schiippert, Femke Swarte, and Stefanie Voigt. 
+Mutual intelligibility 
+between closely related languages in europe. International Journal of Multilingualism, 15(2):169-193, 2018. 
+Nitish Aggarwal, Tobias Wunner, Mihael Aréan, Paul Buitelaar, and Sean O’Riain. A similarity measure based on semantic, terminological 
+and linguistic information. In Proceedings of the 6th International Workshop on Ontology Matching, 01 2011. 
+Vincent Beaufils and Johannes Tomin. Stochastic approach to worldwide language classification: the signals and the noise towards long-range 
+exploration, Oct 2020. 
+Lazar Kovacevic, Vladimir Bradic, Gerard de Melo, Sinisa Zdravkovic, and Olga Ryzhova. Ezglot. https: //www.ezglot.com/, 2021. 
+Matthew S. Dryer and Martin Haspelmath, editors. WALS Online. Max Planck Institute for Evolutionary Anthropology, Leipzig, 2013. 
+Juuso Eronen, Michal Ptaszynski, Fumito Masui, Masaki Arata, Gniewosz Leliwa, and Michal Wroczynski. Transfer language selection for 
+zero-shot cross-lingual abusive language detection. Information Processing and Management, 59(4):102981, 2022. 
+Patrick Littell, David R. Mortensen, Ke Lin, Katherine Kairis, Carlisle Turner, and Lori Levin. URIEL and lang2vec: Representing languages 
+as typological, geographical, and phylogenetic vectors. In Proceedings of the 15th Conference of the European Chapter of the Association for 
+Computational Linguistics: Volume 2, Short Papers, pages 8-14, Valencia, Spain, April 2017. Association for Computational Linguistics. 
+Hakan Ringbom. Cross-linguistic Similarity in Foreign Language Learning. Multilingual Matters, 2006. 
+Robert Bley-Vroman. The evolving context of the fundamental difference hypothesis. Studies in Second Language Acquisition, 31(2):175—198, 
+2009. 
+Ryan Cotterell, Sabrina J. Mielke, Jason Eisner, and Brian Roark. Are all languages equally hard to language-model? In Proceedings of the 
+2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 
+2 (Short Papers), pages 536-541, New Orleans, Louisiana, 2018. Association for Computational Linguistics. 
+Palakorn Achananuparp, Xiaohua Hu, and Xiajiong Shen. The evaluation of sentence similarity measures. In Proceedings of the International 
+Conference on Data Warehousing and Knowledge Discovery, volume 5182, pages 305-316, 09 2008. 
+Aminul Islam and Diana Inkpen. Semantic text similarity using corpus-based word similarity and string similarity. ACM Trans. Knowl. Discov. 
+Data, 2(2), 2008. 
+Steven Moran, Daniel McCloy, and Richard Wright, editors. PHOIBLE Online. Max Planck Institute for Evolutionary Anthropology, Leipzig, 
+2014. 
+Harald Hammarstrém, Robert Forkel, Martin Haspelmath, and Sebastian Bank. glottolog/glottolog: Glottolog database 4.5, December 2021. 
+Melvin Johnson, Mike Schuster, Quoc V Le, Maxim Krikun, Yonghui Wu, Zhifeng Chen, Nikhil Thorat, Fernanda Viégas, Martin Wattenberg, 
+Greg Corrado, et al. Google’s multilingual neural machine translation system: Enabling zero-shot translation. Transactions of the Association 
+for Computational Linguistics, 5:339-351, 2017. 
+Yang Chen and Alan Ritter. Model selection for cross-lingual transfer. In Proceedings of the 2021 Conference on Empirical Methods in Natural 
+Language Processing, pages 5675-5687, Online and Punta Cana, Dominican Republic, November 2021. Association for Computational 
+Linguistics. 
+Jonathan H. Clark, Eunsol Choi, Michael Collins, Dan Garrette, Tom Kwiatkowski, Vitaly Nikolaev, and Jennimaria Palomaki. 
+TyDi 
+QA: A Benchmark for Information-Seeking Question Answering in Typologically Diverse Languages. Transactions of the Association for 
+Computational Linguistics, 8:454-470, 07 2020. 
+Anne Lauscher, Vinit Ravishankar, Ivan Vuli¢, and Goran Glava8. From zero to hero: On the limitations of zero-shot language transfer with 
+multilingual Transformers. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 
+4483-4499, Online, November 2020. Association for Computational Linguistics. 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 20 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+[11] Telmo Pires, Eva Schlinger, and Dan Garrette. How multilingual is multilingual bert? In ACL, 2019.
+[12] Junjie Hu, Sebastian Ruder, Aditya Siddhant, Graham Neubig, Orhan Firat, and Melvin Johnson. Xtreme: A massively multilingual multi-task
+benchmark for evaluating cross-lingual generalisation. In International Conference on Machine Learning, pages 4411-4421. PMLR, 2020.
+understanding, 2018.
+Luke Zettlemoyer
+and Veselin Stoyanov. Unsupervised cross-lingual repre
+esentation learning at scale. In Proceedings of the 58th Annual
+Meeting of the Association for Computational Linguistics, pages 8440-8451, Online, 2020. Association for Computational Linguistics
+[15] Shijie Wu and Mark Dredze. Beto, bentz, becas: The surprising cross-lingual effectiveness of BERT. In Proceedings of the 20i9 Conference
+on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-
+IJCNLP), pages 833-844, Hong Kong, China, November 2019. Association for Computational Linguistics
+[16] Edward P. Stabler and Edward L. Keenan. Structural similarity within and among languages. Theoretical Computer Science, 293(2):345-363,
+2003. Algebraic Methods in Language Processing.
+[17] Yaobo Liang, Nan Duan, Yeyun Gong, Ning Wu, Fenfei Guo, Weizhen Qi, Ming Gong, Linjun Shou, Daxin Jiang, Guihong Cao, et al. Xglue:
+A new benchmark datasetfor cross-lingual pre-training, understanding and generation. In Proceedings of the 2020 Conference on Empirical
+Methods in Natural Language Processing (EMNLP), pages 6008-6018, 2020.
+[18] Matus Pikuliak, Marian Simko, and Maria Bielikova. Cross-lingual learning for text processing: A survey. Expert Systems with Applications,
+165:113765, 2021
+[19] Yu-Hsiang Lin, Chian-Yu Chen, Jean Lee, Zirui Li, Yuyan Zhang, Mengzhou Xia, Shruti Rijhwani, Junxian He, Zhisong Zhang, Xuezhe
+of the 57th Annual Meeting of the Association for Computational Linguistics, pages 3125-3135, Florence, Italy, July 2019. Association for
+Computational Linguistics
+[20] Ryan Cotterell and Georg Heigold. Cross-lingual character-level neural morphological tagging. In Proceedings of the 2017 Conference on
+Empirical Methods in Natural Language Processing, pages 748-759, Copenhagen, Denmark, September 2017. Association for Computational
+Linguistics
+[21] Charlotte Gooskens, Vincent J. van Heuven, Jelena Golubovic, Anja Schuppert, Femke Swarte, and Stefanie Voigt. Mutual intelligibility
+between closely related languages in europe. International Journal of Multilingualism, 15(2):169-193, 2018
+[22] Nitish Aggarwal, Tobias Wunner, Mihael Arcan, Paul Buitelaar, and Sean O'Riain. A similarity measure based on semantic, terminological
+and linguistic information. In Proceedings of the 6th International Workshop on Ontology Matching, O1 2011
+[23] Vincent Beaufils and Johannes Tomin. Stochastic approach to worldwide language classification: the signals and the noise towards long-range
+exploration, Oct 2020.
+[24] Lazar Kovacevic, Vladimir Bradic, Gerard de Melo, Sinisa Zdravkovic, and Olga Ryzhova. Ezglot. https : / /www . ezglot . com/, 2021.
+[25] Matthew S. Dryer and Martin Haspelmath, editors. WALS Online. Max Planck Institute for Evolutionary Anthropology, Leipzig, 2013
+[27] Patrick Littell, David R. Mortensen, Ke Lin, Katherine Kairis, Carlisle Turner, and Lori Levin. URIEL and lang2vec: Representing languages
+as typological, geographical, and phylogenetic vectors. In Proceedings of the 15th Conference of the European Chapter of the Association for
+Computational Linguistics: Volume 2, Short Papers, pages 8-14, Valencia, Spain, April 2017. Association for Computational Linguistics.
+[28] Hakan Ringbom. Cross-linguistic Similarity in Foreign Language Learning. Multilingual Matters, 2006.
+[29] Robert Bley-Vroman. The evolving context of the fundamental difference hypothesis. Studies in Second Language Acquisition, 31(2):175-198,
+2009
+[3o] Ryan Cotterell, Sabrina J. Mielke, Jason Eisner, and Brian Roark. Are all languages equally hard to language-model? In Proceedings of the
+2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume
+[31] Palakorn Achananuparp, Xiaohua Hu, and Xiajiong Shen. The evaluation of sentence similarity measures. In Proceedings of the International
+Conference on Data Warehousing and Knowledge Discovery, volume 5182, pages 305-316, 09 2008.
+[32] Aminul Islam and Diana Inkpen. Semantic text similarity using corpus-based word similarity and string similarity. ACM Trans. Knowl. Discov.
+Data, 2(2), 2008.
+[33] Steven Moran, Daniel McCloy, and Richard Wright, editors. PHOIBLE Online. Max Planck Institute for Evolutionary Anthropology, Leipzig
+2014.
+341
+ Harald Hammarstrom, Robert Forkel, Martin Haspelmath, and Sebastian Bank. glottolog/glottolog: Glottolog database 4.5, December 2021
+[35] Melvin Johnson, Mike Schuster, Quoc V Le, Maxim Krikun, Yonghui Wu, Zhifeng Chen, Nikhil Thorat, Fernanda Viegas, Martin Wattenberg,
+Greg Corrado, et al. Google's multilingual neural machine translation system: Enabling zero-shot translation. Transactions of the Association
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+[36]  Yang Chen and Alan Ritter. Model selection for cross-lingual transfer. In Proceedings of the 2021 Conference on Empirical Methods in Natural
+Language Processing, pages 5675-5687, Online and Punta Cana, Dominican Republic, November 2021. Association for Computational
+Linguistics.
+[37] Jonathan H. Clark, Eunsol Choi, Michael Collins, Dan Garrette, Tom Kwiatkowski, Vitaly Nikolaev, and Jennimaria Palomaki. TyDi
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+Bing Li, Yujie He, and Wenjin Xu. Cross-lingual named entity recognition using parallel corpus: A new approach using xlm-roberta alignment. 
+arXiv preprint arXiv:2101.11112, 2021. 
+Sabine Weber and Mark Steedman. 
+Zero-shot cross-lingual transfer is a hard baseline to beat in German fine-grained entity typing. 
+In 
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+November 2021. Association for Computational Linguistics. 
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+1946-1958, Vancouver, Canada, July 2017. Association for Computational Linguistics. 
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+Massively multilingual transfer for NER. 
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+Min Xiao and Yuhong Guo. Distributed word representation learning for cross-lingual dependency parsing. In Proceedings of the Eighteenth 
+Conference on Computational Natural Language Learning, pages 119-129, Ann Arbor, Michigan, June 2014. Association for Computational 
+Linguistics. 
+Jérg Tiedemann. 
+Cross-lingual dependency parsing with universal dependencies and predicted pos labels. 
+In Proceedings of the Third 
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+Jiang Guo, Wanxiang Che, David Yarowsky, Haifeng Wang, and Ting Liu. 
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+Computational Linguistics. 
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+Human Language Technologies, pages 1058-1063, 2016. 
+Mohit Bansal. Dependency link embeddings: Continuous representations of syntactic substructures. 
+In Proceedings of the Ist Workshop 
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+Linguistics. 
+Dan Kondratyuk and Milan Straka. 75 languages, 1 model: Parsing Universal Dependencies universally. In Proceedings of the 2019 Conference 
+on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP- 
+IJCNLP), pages 2779-2795, Hong Kong, China, November 2019. Association for Computational Linguistics. 
+Matej Uléar and Marko Robnik-Sikonja. Finest bert and crosloengual bert. In International Conference on Text, Speech, and Dialogue, pages 
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+Antoine Bosselut, Emma Brunskill, Erik Brynjolfsson, Shyamal Buch, Dallas Card, Rodrigo Castellon, Niladri Chatterji, Annie Chen, Kathleen 
+Creel, Jared Quincy Davis, Dora Demszky, Chris Donahue, Moussa Doumbouya, Esin Durmus, Stefano Ermon, John Etchemendy, Kawin 
+Ethayarajh, Li Fei-Fei, Chelsea Finn, Trevor Gale, Lauren Gillespie, Karan Goel, Noah Goodman, Shelby Grossman, Neel Guha, Tatsunori 
+Hashimoto, Peter Henderson, John Hewitt, Daniel E. Ho, Jenny Hong, Kyle Hsu, Jing Huang, Thomas Icard, Saahil Jain, Dan Jurafsky, 
+Pratyusha Kalluri, Siddharth Karamcheti, Geoff Keeling, Fereshte Khani, Omar Khattab, Pang Wei Koh, Mark Krass, Ranjay Krishna, Rohith 
+Kuditipudi, Ananya Kumar, Faisal Ladhak, Mina Lee, Tony Lee, Jure Leskovec, Isabelle Levent, Xiang Lisa Li, Xuechen Li, Tengyu Ma, 
+Ali Malik, Christopher D. Manning, Suvir Mirchandani, Eric Mitchell, Zanele Munyikwa, Suraj Nair, Avanika Narayan, Deepak Narayanan, 
+Ben Newman, Allen Nie, Juan Carlos Niebles, Hamed Nilforoshan, Julian Nyarko, Giray Ogut, Laurel Orr, Isabel Papadimitriou, Joon Sung 
+Park, Chris Piech, Eva Portelance, Christopher Potts, Aditi Raghunathan, Rob Reich, Hongyu Ren, Frieda Rong, Yusuf Roohani, Camilo Ruiz, 
+Jack Ryan, Christopher Ré, Dorsa Sadigh, Shiori Sagawa, Keshav Santhanam, Andy Shih, Krishnan Srinivasan, Alex Tamkin, Rohan Taori, 
+Armin W. Thomas, Florian Tramér, Rose E. Wang, William Wang, Bohan Wu, Jiajun Wu, Yuhuai Wu, Sang Michael Xie, Michihiro Yasunaga, 
+Jiaxuan You, Matei Zaharia, Michael Zhang, Tianyi Zhang, Xikun Zhang, Yuhui Zhang, Lucia Zheng, Kaitlyn Zhou, and Percy Liang. On 
+the opportunities and risks of foundation models, 2021. 
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention 
+is all you need. Advances in neural information processing systems, 30, 2017. 
+Karthikeyan K, Zihan Wang, Stephen Mayhew, and Dan Roth. Cross-lingual ability of multilingual bert: An empirical study. In International 
+Conference on Learning Representations, 2020. 
+Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 
+Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692, 2019. 
+Alexis Conneau and Guillaume Lample. 
+Cross-lingual language model pretraining. Advances in Neural Information Processing Systems, 
+32:7059-7069, 2019. 
+Cecil Brown, Eric Holman, and Sgren Wichmann. Sound correspondences in the world’s languages. Language, 89:4—29, 03 2013. 
+Charlotte Gooskens. 
+The contribution of linguistic factors to the intelligibility of closely related languages. 
+Journal of Multilingual and 
+multicultural development, 28(6):445—467, 2007. 
+Chaitanya Malaviya, Graham Neubig, and Patrick Littell. 
+Learning language representations for typology prediction. 
+In Proceedings of 
+the 2017 Conference on Empirical Methods in Natural Language Processing, pages 2529-2535, Copenhagen, Denmark, September 2017. 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 22 of 23
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+[65] Alankar Jain, Bhargavi Paranjape, and Zachary C. Lipton. Entity projection via machine translation for cross-lingual NER. In Proceedings of
+the 2O19 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language
+[66] Bing Li, Yujie He, and Wenjin Xu. Cross-lingual named entity recognition using parallel corpus: A new approach using xlm-roberta alignment
+arXiv preprint arXiv:2101.11112, 2021.
+[67] Sabine Weber and Mark Steedman. Zero-shot cross-lingual transfer is a hard baseline to beat in German fine-grained entity typing.
+Proceedings of the Second Workshop on Insights from Negative Results in NLP, pages 42-48, Online and Punta Cana, Dominican Republic,
+November 2021. Association for Computational Linguistics.
+[68] Xiaoman Pan, Boliang Zhang, Jonathan May, Joel Nothman, Kevin Knight, and Heng Ji. Cross-lingual name tagging and linking for 282
+languages. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume I: Long Papers), pages
+[69] Afshin Rahimi, Yuan Li, and Trevor Cohn. Massively multilingual transfer for NER. In Proceedings of the 57th Annual Meeting of the
+[70] Min Xiao and Yuhong Guo. Distributed word representation learning for cross-lingual dependency parsing. In Proceedings of the Eighteenth
+Conference on Computational Natural Language Learning, pages 119-129, Ann Arbor, Michigan, June 2014. Association for Computational
+Linguistics
+[71] Jorg Tiedemann. Cross-lingual dependency parsing with universal dependencies and predicted pos labels. In Proceedings of the Third
+[72] Jiang Guo, Wanxiang Che, David Yarowsky, Haifeng Wang, and Ting Liu. Cross-lingual dependency parsing based on distributed
+epresentations. In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International
+Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 1234-1244, Beijing, China, July 2015. Association for
+Computational Linguistics.
+[73] Ophélie Lacroix, Lauriane Aufrant, Guillaume Wisniewski, and Frangois Yvon. Frustratingly easy cross-lingual transfer for transition-based
+dependency parsing. In Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics:
+Human Language Technologies, pages 1058-1063, 2016
+[74] Mohit Bansal. Dependency link embeddings: Continuous representations of syntactic substructures. In Proceedings of the Ist Workshop
+on Vector Space Modeling for Natural Language Processing, pages 102-108, Denver, Colorado, June 2015. Association for Computational
+Linguistics
+[75]  Dan Kondratyuk and Milan Straka. 75 languages, 1 model: Parsing Universal Dependencies universally. In Proceedings of the 2019 Conference
+on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-
+IJCNLP), pages 2779-2795, Hong Kong, China, November 2019. Association for Computational Linguistics
+[76] Matej Ulcar and Marko Robnik-Sikonja. Finest bert and crosloengual bert. In International Conference on Text, Speech, and Dialogue, pages
+104-111.Springer.2020
+[77] Joakim Nivre, Marie-Catherine de Marneffe, Filip Ginter, Jan Hajic, Christopher D. Manning, Sampo Pyysalo, Sebastian Schuster, Francis
+Resources and Evaluation Conference, pages 4034 4043, Marseille, France, May 2020. European Language Resources Association
+[78] Rishi Bommasani, Drew A. Hudson, Ehsan Adeli, Russ Altman, Simran Arora, Sydney von Arx, Michael S. Bernstein, Jeannette Bohg,
+Antoine Bosselut, Emma Brunskill, Erik Brynjolfsson, Shyamal Buch, Dallas Card, Rodrigo Castellon, Niladri Chatterji, Annie Chen, Kathleen
+Creel, Jared Quincy Davis, Dora Demszky, Chris Donahue, Moussa Doumbouya, Esin Durmus, Stefano Ermon, John Etchemendy, Kawin
+Ethayarajh, Li Fei-Fei, Chelsea Finn, Trevor Gale, Lauren Gillespie, Karan Goel, Noah Goodman, Shelby Grossman, Neel Guha, Tatsunori
+Hashimoto, Peter Henderson, John Hewitt, Daniel E. Ho, Jenny Hong, Kyle Hsu, Jing Huang, Thomas Icard, Saahil Jain, Dan Jurafsky,
+Pratyusha Kalluri. Siddharth Karamcheti. Geoff Keeling. Fereshte Khani, Omar Khattab. Pang Wei Koh. Mark Krass. Ranjay Krishna. Rohith
+Kuditipudi, Ananya Kumar, Faisal Ladhak, Mina Lee, Tony Lee, Jure Leskovec, Isabelle Levent, Xiang Lisa Li, Xuechen Li, Tengyu Ma,
+Ali Malik, Christopher D. Manning, Suvir Mirchandani, Eric Mitchell, Zanele Munyikwa, Suraj Nair, Avanika Narayan, Deepak Narayanan
+Ben Newman, Allen Nie, Juan Carlos Niebles, Hamed Nilforoshan, Julian Nyarko, Giray Ogut, Laurel Orr, Isabel Papadimitriou, Joon Sung
+Park, Chris Piech, Eva Portelance, Christopher Potts, Aditi Raghunathan, Rob Reich, Hongyu Ren, Frieda Rong, Yusuf Roohani, Camilo Ruiz.
+Jack Ryan, Christopher Ré, Dorsa Sadigh, Shiori Sagawa, Keshav Santhanam, Andy Shih, Krishnan Srinivasan, Alex Tamkin, Rohan Taori,
+Jiaxuan You, Matei Zaharia, Michael Zhang, Tianyi Zhang, Xikun Zhang, Yuhui Zhang, Lucia Zheng, Kaitlyn Zhou, and Percy Liang. On
+the opportunities and risks of foundation models, 2021.
+[79] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention
+is all you need. Advances in neural information processing systems, 30, 2017
+[80] Karthikeyan K, Zihan Wang, Stephen Mayhew, and Dan Roth. Cross-lingual ability of multilingual bert: An empirical study. In International
+Conference on Learning Representations, 2020.
+[81] 
+Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov.
+Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692, 2019.
+32:7059-7069, 2019
+[83] Cecil Brown, Eric Holman, and Soren Wichmann. 
+Sound c
+ribution of linguistic factors to the intelligibility of closely related languages. Journal of Multilingual and
+multicultural development, 28(6):445-467, 2007.
+[85] Chaitanya Malaviya, Graham Neubig, and Patrick Littell. Learning language representations for typology prediction. In Proceedings of
+the 2017 Conference on Empirical Methods in Natural Language Processing, pages 2529-2535, Copenhagen, Denmark, September 2017.
+Eronen et al.: Preprint submitted to Elsevier
+Page 22 of 23[86] 
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+Association for Computational Linguistics. 
+Benedikt Szmrecsanyi. Geography is overrated. Dialectological and folk dialectological concepts of space, pages 215-231, 2012. 
+Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf, 
+Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain 
+Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush. Transformers: State-of-the-art natural language processing. In Proceedings 
+of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations, pages 38-45, Online, October 2020. 
+Association for Computational Linguistics. 
+Leon Kellner. Historical outlines of English syntax. Macmillan, 1892. 
+Christiane Dalton-Puffer. The French influence on Middle English morphology: A corpus-based study on derivation, volume 20. Walter de 
+Gruyter, 2011. 
+Philip Durkin. Borrowed words: A history of loanwords in English. Oxford University Press, 2014. 
+Juuso Eronen, Michal Ptaszynski, Fumito Masui, Aleksander Smywiriski-Pohl, Gniewosz Leliwa, and Michal Wroczynski. Improving classifier 
+training efficiency for automatic cyberbullying detection with feature density. Information Processing & Management, 58(5):102616, 2021. 
+Ganesh Jawahar, Benoit Sagot, and Djamé Seddah. 
+What does BERT learn about the structure of language? 
+In Proceedings of the 57th 
+Annual Meeting of the Association for Computational Linguistics, pages 3651-3657, Florence, Italy, July 2019. Association for Computational 
+Linguistics. 
+  
+Eronen et al.: Preprint submitted to Elsevier 
+Page 23 of 23
+
+Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity
+Association for Computational Linguistics.
+[86] Benedikt Szmrecsanyi. Geography is overrated. Dialectological and folk dialectological concepts of space, pages 215-231, 2012.
+[87] Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf,
+Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain
+Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush. Transformers: State-of-the-art natural language processing. In Proceedings
+of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations, pages 38-45, Online, October 2020.
+Association for Computational Linguistics.
+[88]  Leon Kellner. Historical outlines of English syntax. Macmillan, 1892.
+[89] Christiane Dalton-Puffer. The French infuence on Middle English morphology: A corpus-based study on derivation, volume 20. Walter de
+Gruyter, 2011.
+[90]  
+Philip Durkin. Borrowed words: A history of loanwords in English. Oxford University Press, 2014.
+[91] Juuso Eronen, Michal Ptaszynski, Fumito Masui, Aleksander Smywinski-Pohl, Gniewosz Leliwa, and Michal Wroczynski. Improving classifier
+training efficiency for automatic cyberbullying detection with feature density. Information Processing & Management, 58(5):102616, 2021.
+[92]  C
+Annual Meeting of the Association for Computational Linguistics, pages 3651-3657, Florence, Italy, July 2019. Association for Computational
+Linguistics.
+Eronen et al.: Preprint submitted to Elsevier
+Page 23 of 23
\ No newline at end of file
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf,len=3668
+page_content='Zero-Shot Cross-Lingual Transfer Language Selection Using  Linguistic Similarity  Juuso Eronen**, Michal Ptaszynski* and Fumito Masui*  Kitami Institute of Technology, 165, Koencho, Kitami, 090-0015, Hokkaido, Japan  ARTICLE INFO  ABSTRACT  Keywords:  We study the selection of transfer languages for different Natural Language Processing tasks,  Multilingual Natural Language Pro-  specifically sentiment analysis, named entity recognition and dependency parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In order to  cessing  select an optimal transfer language, we propose to utilize different linguistic similarity metrics  Zero-Shot Learning  to measure the distance between languages and make the choice of transfer language based on this  Transfer Learning  information instead of relying on intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We demonstrate that linguistic similarity correlates  Linguistics  with cross-lingual transfer performance for all of the proposed tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We also show that there  Language similarity  is a Statistically significant difference in choosing the optimal language as the transfer source  instead of English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This allows us to select a more suitable transfer language which can be used to  better leverage knowledge from high-resource languages in order to improve the performance of  language applications lacking data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For the study, we used datasets from eight different languages  from three language families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Introduction  As with any other supervised learning problem, the tasks in Natural Language Processing (NLP) require sufficiently  large labeled datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' What sets NLP apart from other fields is the presence of multiple languages the datasets can  appear in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This means that in order to successfully train models for all of the world’s 7,100 languages [1], one would  need to annotate a dataset for each language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is a very difficult and costly task [2, 3] and has led to a small number  of high-resource languages to dominate the field [4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This imbalance in the distribution of resources among  languages calls for the need to develop technologies that would make model development for low-resource languages  realistically feasible and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In order to address this problem, cross-lingual transfer been proposed as a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This means leveraging labeled  data from high-resource languages in order to improve the performance on lower-resource languages [7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Particularly, the popularity of cross-lingual zero-shot learning, or training on one task/language and testing on a  different task/language completely unknown to the model, has increased greatly in the recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Zero-shot learning  has gained in popularity because it does not require any labeled data in the target language for training [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Moreover, zero-shot cross-lingual transfer utilizes large pre-trained multilingual transformer models like Multilingual  BERT [13] or XLM-RoBERTa [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' These models are fine-tuned with training data in a language called the source  language (usually a high-resource language) and then used to predict entries from other languages than that used in  training, often with satisfying results [11, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The choice of transfer language is usually done by intuition [16] or simply defaults to English, as is the case  with popular multilingual benchmarks like XTREME [12] and XGLUE [17], even though there is no actual evidence  backing up these choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Furthermore, in a survey of 157 cross-lingual learning papers by Pikuliak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [18] they  found out that English is used in 149 of those papers, followed by German with 82 papers in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' There has also  been some attempts in developing a more systematic transfer language selection method [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, this method  requires training of a ranking model, limiting its use to the tasks and datasets used for training, making it unusable  off-the-shelf for other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Choosing the optimal language for cross-lingual transfer remains widely an understudied problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Usually, the  suitable source language candidate is decided experimentally or by pure intuition by the individual (researcher, or ML  practitioner) based on their own theoretical knowledge and experience in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' One option to select the source  language would be by taking a look at languages that are from the same language group as the target language [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Corresponding author  b4 eronen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' juuso@gmail .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='com (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Eronen)  ORCID(s): 0000-0001-9841-3652 (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Eronen)  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page  1 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using  Linguistic Similarity  Juuso Eronena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Michal Ptaszynskia and Fumito Masuia  a Kitami Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' 165,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Koencho,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Kitami,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' 090-0015,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Hokkaido,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Japan  ARTICLE INFO  ABSTRACT  Keywords:  We study the selection of transfer languages for different Natural Language Processing tasks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Multilingual Natural Language Pro-  specifically sentiment analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' named entity recognition and dependency parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In order to  cessing  select an optimal transfer language, we propose to utilize different linguistic similarity metrics  Zero-Shot Learning  tomeasure the distance betweenlanguages andmake the choice oftransferlanguage basedon this  Transfer Learning  information instead of relying on intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We demonstrate that linguistic similarity correlates  Linguistics  with cross-lingual transfer performance for all of the proposed tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We also show that there  Language similarity  is a statistically significant difference in choosing the optimal language as the transfer source  instead of English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This allows us to select a more suitable transfer language which can be used to  better leverage knowledge from high-resource languages in order to improve the performance of  from three language families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Introduction  As with any other supervised learning problem, the tasks in Natural Language Processing (NLP) require sufficiently  large labeled datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' What sets NLP apart from other fields is the presence of multiple languages the datasets can  appear in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" This means that in order to successfully train models for all of the world's 7,100 languages [1], one would  need to annotate a dataset for each language." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is a very difficult and costly task [2, 3] and has led to a small number  of high-resource languages to dominate the field [4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This imbalance in the distribution of resources among  languages calls for the need to develop technologies that would make model development for low-resource languages  realistically feasible and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In order to address this problem, cross-lingual transfer been proposed as a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This means leveraging labeled  s         e  Particularly, the popularity of cross-lingual zero-shot learning, or training on one task/language and testing on a  different task/language completely unknown to the model, has increased greatly in the recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Zero-shot learning  has gained in popularity because it does not require any labeled data in the target language for training [11, 12]  BERT [13] or XLM-RoBERTa [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' These models are fine-tuned with training data in a language called the source  language (usually a high-resource language) and then used to predict entries from other languages than that used in  training, often with satisfying results [11, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The choice of transfer language is usually done by intuition [16] or simply defaults to English, as is the case  with popular multilingual benchmarks like XTREME [12] and XGLUE [17], even though there is no actual evidence  backing up these choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Furthermore, in a survey of 157 cross-lingual learning papers by Pikuliak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [18] they  found out that English is used in 149 of those papers, followed by German with 82 papers in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' There has also  been some attempts in developing a more systematic transfer language selection method [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, this method  requires training of a ranking model, limiting its use to the tasks and datasets used for training, making it unusable  off-the-shelf for other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  suitable source language candidate is decided experimentally or by pure intuition by the individual (researcher, or ML  practitioner) based on their own theoretical knowledge and experience in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' One option to select the source  Corresponding author  @ eronen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' juuso@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='com (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Eronen)  ORCID(s): 0000-0001-9841-3652 (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Eronen)  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 1 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  However, this does not guarantee that that the linguistic features shared between the two languages would be similar  [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In order to contribute to the further understanding and solving this problem, we propose a method for choosing the  source language for cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We show that there is a correlation between linguistic similarity and model  performance, allowing us to select the best transfer language by comparing the source and target languages using  different linguistic similarity measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We also show that multilingual transformer models can be used to obtain good  performance on the target language in a zero-shot learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' To select the optimal source language for transfer,  we propose to quantify the features of languages to compute a metric that can be used in comparing the closeness of  languages using their linguistic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  There are some existing metrics that use linguistic features in order to measure the linguistic distance between  languages [22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, as these metrics simply take a handful or only a single linguistic feature into account,  we propose a new linguistic similarity metric, which contains almost two hundred different features, based on the World  Atlas of Language Structures (WALS) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This allows us to not simply rely only on a single or a handful of features,  but to have a more robust metric by better quantifying all aspects of the languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In this research we concentrated on three different Natural Language Processing tasks, namely, sentiment analysis,  named entity recognition and dependency parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We used datasets from eight different languages, namely English,  German, Danish, Polish, Croatian, Russian, Japanese and Korean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The languages were chosen as they have relatively  high quality datasets available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, the languages represent different language families (English, German, Danish -  Germanic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Polish, Russian, Croatian - Slavic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Japanese, Korean - Koreano-Japonic language family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This also gives  us the opportunity to study the efficacy of cross-lingual transfer learning between and within language family groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In previous research [26] we showed that cross-lingual transfer performance correlates with the linguistic similarity  of the prediction target language and the source language used for fine-tuning the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Our hypothesis is that this is  true for also other NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In the experiments, we used multilingual transformer models, namely Multilingual BERT  and XLM-RoBERTa, which were fine-tuned by using each of the languages as source and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We calculated the  linguistic similarity between all of our proposed languages using four different linguistic similarity metrics, EzGlot,  eLinguistics, a quantified model based on WALS and averaged lang2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' To demonstrate the effectiveness of our  method, we then measured the correlation between the zero-shot cross-lingual transfer performance and linguistic  similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The paper outline is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Section 2 we go through all areas of previous research that are addressed in  this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Section 3 we describe all the tasks and datasets applied to this research and present their differences and  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Section 4 we describe the applied multilingual transformer models and the linguistic similarity metrics used  in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Section 5 we describe our experiment workflow and go through all the results from the conducted  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Section 6 we discuss the results in general and bring out the most interesting findings in relation to the  research goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Contributions of This Study  The goal of this research is to develop a method for cross-lingual transfer language selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Most often, the choice  of a transfer source is made purely by relying on the practitioner’s own judgement, using their accumulated experience  on the field and theoretical knowledge or simply choosing a language from the same language family as the target  [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The current methods have many problems as they are prone to bias from the practitioner and also completely  unoptimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In fact, one could even say that there is no systematic method usable off-the-shelf that could be used to  determine, which languages should be considered as the cross-lingual transfer source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  We propose to investigate the possibility that different linguistic similarity metrics could be utilized when trying to  find possible source language candidates for cross-lingual transfer also for other tasks than abusive language detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  We hypothesize that linguistic similarity correlates with cross-lingual transfer efficacy, meaning that by using more  similar languages, we would be able to achieve higher model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  This research is was conducted in order to confirm the findings of our previous research [26] also with other Natural  Language Processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We improved the calculation process of the linguistic similarity metric quantified from the  World Atlas of Language Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This was done by selecting all of the features that would have a defined value for  both languages in all possible language pairs instead of having to be shared between all of the languages, increasing  the robustness of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, we applied a linguistic similarity metric based on lang2vec by Littell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The main contributions of this work are as follows:  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 2 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  However, this does not guarantee that that the linguistic features shared between the two languages would be similar  [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In order to contribute to the further understanding and solving this problem, we propose a method for choosing the  source language for cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We show that there is a correlation between linguistic similarity and model  performance, allowing us to select the best transfer language by comparing the source and target languages using  different linguistic similarity measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We also show that multilingual transformer models can be used to obtain good  performance on the target language in a zero-shot learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' To select the optimal source language for transfer,  we propose to quantify the features of languages to compute a metric that can be used in comparing the closeness of  languages using their linguistic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  There are some existing metrics that use linguistic features in order to measure the linguistic distance between  languages [22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, as these metrics simply take a handful or only a single linguistic feature into account,  we propose a new linguistic similarity metric, which contains almost two hundred different features, based on the World  Atlas of Language Structures (WALS) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This allows us to not simply rely only on a single or a handful of features,  but to have a more robust metric by better quantifying all aspects of the languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In this research we concentrated on three different Natural Language Processing tasks, namely, sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  named entity recognition and dependency parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We used datasets from eight different languages, namely English,  German, Danish, Polish, Croatian, Russian, Japanese and Korean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The languages were chosen as they have relatively  Germanic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Polish, Russian, Croatian - Slavic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Japanese, Korean - Koreano-Japonic language family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This also gives  us the opportunity to study the efficacy of cross-lingual transfer learning between and within language family groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In previous research [26] we showed that cross-lingual transfer performance correlates with the linguistic similarity  of the prediction target language and the source language used for fine-tuning the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Our hypothesis is that this is  true for also other NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In the experiments, we used multilingual transformer models, namely Multilingual BERT  and XLM-RoBERTa, which were fine-tuned by using each of the languages as source and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We calculated the  linguistic similarity between all of our proposed languages using four different linguistic similarity metrics, EzGlot  eLinguistics, a quantified model based on WALS and averaged lang2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' To demonstrate the effectiveness of our  method, we then measured the correlation between the zero-shot cross-lingual transfer performance and linguistic  similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The paper outline is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Section 2 we go through all areas of previous research that are addressed in  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Section 4 we describe the applied multilingual transformer models and the linguistic similarity metrics used  in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Section 5 we describe our experiment workflow and go through all the results from the conducted  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Section 6 we discuss the results in general and bring out the most interesting findings in relation to the  research goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Contributions of This Study  The goal of this research is to develop a method for cross-lingual transfer language selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" Most often, the choice  of a transfer source is made purely by relying on the practitioner's own judgement, using their accumulated experience  on the field and theoretical knowledge or simply choosing a language from the same language family as the target  [20]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The current methods have many problems as they are prone to bias from the practitioner and also completely  unoptimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In fact, one could even say that there is no systematic method usable off-the-shelf that could be used to  determine, which languages should be considered as the cross-lingual transfer source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  We propose to investigate the possibility that different linguistic similarity metrics could be utilized when trying to  find possible source language candidates for cross-lingual transfer also for other tasks than abusive language detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  We hypothesize that linguistic similarity correlates with cross-lingual transfer efficacy, meaning that by using more  similar languages, we would be able to achieve higher model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  This research is was conducted in order to confirm the findings of our previous research [26] also with other Natural  Language Processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We improved the calculation process of the linguistic similarity metric quantified from the  World Atlas of Language Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This was done by selecting all of the features that would have a defined value for  both languages in all possible language pairs instead of having to be shared between all of the languages, increasing  the robustness of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, we applied a linguistic similarity metric based on lang2vec by Littell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The main contributions of this work are as follows:  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 2 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  e We confirm the transfer language selection method based on linguistic similarity with multiple NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  e We demonstrate the efficacy of two multidomain linguistic similarity metrics: improved quantified WALS and  averaged lang2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  e We show that there is a significant difference in choosing an optimal transfer source language over English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In practice, we propose to fine-tune cross-lingual pretrained transformer models, specifically mBERT and XLM-R,  on three different Natural Language Processing tasks (sentiment analysis, named entity recognition and dependency  parsing) using each of our proposed languages (English, German, Danish, Polish, Russian, Japanese, Korean) and then  perform zero-shot prediction on the rest of the languages of the proposed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We calculated the linguistic similarity  between all of our proposed languages using four different linguistic similarity metrics, EzGlot, eLinguistics, quantified  World Atlas of Language Structures and an averaged lang2vec proximity vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We then calculated the correlation  between the zero-shot cross-lingual transfer performance and linguistic similarity to show the effectiveness of our  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' A block-diagram of the system is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Source language  Target language  data  Linguistic  data  |  similarity  g —  ee  Output  Fine-tuning  = Fine-tuned model  Prediction  Pre-trained  multilingual  transformer model  Figure 1: Block diagram of the proposed system  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Previous Research  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Measuring Linguistic Similarity  Already in 2006, the relation between the difficulty of language learning and the similarity of languages in general  was discussed in a book by Ringbom [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The Finnish language scene was presented as an example in order to  demonstrate the importance of cross-linguistic similarity in foreign language learning [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In short, he showed that  Finnish-speaking Finns have a harder time learning English than Swedish-speaking Finns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason behind this being  the closer relation between Swedish and English languages, giving an advantage to Swedish speakers when it comes  to transferring the existing linguistic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Cottorell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [30] showed that not every language is equally difficult to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' It was also shown by them that there  is a correlation between the morphological richness of a language and the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This means that  the more complex the language is, the more difficult it becomes to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is hinting that more simple languages  might not work so well when used as cross-lingual transfer sources for languages of higher complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This also implies  that the direct relatedness (for example, language family) of languages should not be the only criteria in deciding the  cross-lingual transfer source language as other features of the languages should also be thoroughly considered in order  to find the most optimal transfer language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  There has been some research in attempting to quantify a linguistic similarity metric from different linguistic  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, these metrics mostly commonly rely only on one or just a few different linguistic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For  example, by comparing the consonants contained in a predefined set of words while taking into account the order  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 3 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We confirm the transfer language selection method based on linguistic similarity with multiple NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We demonstrate the efficacy of two multidomain linguistic similarity metrics: improved quantified WALS and  averaged lang2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We show that there is a significant difference in choosing an optimal transfer source language over English  In practice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' we propose to fine-tune cross-lingual pretrained transformer models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' specifically mBERT and XLM-R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  on three different Natural Language Processing tasks (sentiment analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' named entity recognition and dependency  parsing) using each of our proposed languages (English,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' German,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Danish,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Polish,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Russian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Japanese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Korean) and then  perform zero-shot prediction on the rest of the languages of the proposed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We calculated the linguistic similarity  between all of our proposed languages using four different linguistic similarity metrics, EzGlot, eLinguistics, quantified  World Atlas of Language Structures and an averaged lang2vec proximity vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We then calculated the correlation  between the zero-shot cross-lingual transfer performance and linguistic similarity to show the effectiveness of our  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' A block-diagram of the system is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Source language  Target language  data  data  Linguistic  similarity  indno  Pre-trained  Fine-tuning  Fine-tuned model  Prediction  multilingual  transformer model  Figure 1: Block diagram of the proposed system  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Previous Research  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Measuring Linguistic Similarity  Already in 2006, the relation between the difficulty of language learning and the similarity of languages in general  was discussed in a book by Ringbom [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The Finnish language scene was presented as an example in order to  demonstrate the importance of cross-linguistic similarity in foreign language learning [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In short, he showed that  Finnish-speaking Finns have a harder time learning English than Swedish-speaking Finns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason behind this being  the closer relation between Swedish and English languages, giving an advantage to Swedish speakers when it comes  to transferring the existing linguistic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Cottorell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [3O] showed that not every language is equally difficult to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' It was also shown by them that there  is a correlation between the morphological richness of a language and the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This means that  the more complex the language is, the more difficult it becomes to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is hinting that more simple languages  that the direct relatedness (for example, language family) of languages should not be the only criteria in deciding the  cross-lingual transfer source language as other features of the languages should also be thoroughly considered in order  to find the most optimal transfer language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  There has been some research in attempting to quantify a linguistic similarity metric from different linguistic  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, these metrics mostly commonly rely only on one or just a few different linguistic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For  example, by comparing the consonants contained in a predefined set of words while taking into account the order  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 3 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  in which these consonants appear in the words, one can calculate a genetic proximity score between two languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  This is implemented as the eLinguistics [23] similarity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The metric makes it possible to get information about  the direct relatedness of the compared languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, once the used languages start to become more and more  distant, accidental similarities in consonants are introduced and there is a significant increase in the error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is  also acknowledged by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Even though the metric is easy to calculate, it completely ignores all other kinds of  linguistic features, for example, semantic, syntactic, or morphological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Another method to calculate a similarity metric is to take a look at the vocabularies of two languages and concentrate  on their similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' EzGlot [24] uses lexical similarity as its basis for computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The metric uses lexical similarity  between the two compared languages while at the same time taking into account the amount of words the two languages  are sharing with other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This allows for the calculation of similarity between the two languages in relation to  the similarity with every other language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [22] proposed a linguistic similarity metric that utilizes multiple aspects of languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Their metric,  called STL, is based on Semantic, Terminological (lexical) and Linguistic (syntactic) similarity of languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The  method outperformed previous similarity metrics that concentrated only on one of the previously mentioned aspects  (31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' They noticed that the terminological measures showed a much higher contribution when compared to the  other two features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, in order to use the metric, the structure of the used vocabulary dataset needs to be in the  form of a complex ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Due to this fact and because of the dataset only consisting of German, French, Italian,  Dutch, Spanish and English, and due to the dataset used by the authors being no longer available, it was not feasible  to use the metric as a part of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The lang2vec developed by Littell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [27] is a database that represents languages as typological, phylogenetic,  and geographical vectors, which are derived from a number of different linguistic resources, for example, WALS  [25], PHOIBLE [33], Ethnologue [1], and Glottolog [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Each of these utilize multiple different features, making  them more robust than the EzGlot or eLinguistics metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The lang2vec is a fully-fledged library that can be used to  query for different linguistic features and to get pre-computed distances between languages, based on some typological  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The World Atlas of Language Structures (WALS) project [25] consists of a database that catalogs phonological,  word semantic and grammatical knowledge for 2,662 languages with almost two hundred different linguistic features  from multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Using a linguistic similarity measure quantified from the WALS database into would allow  a more robust method to measure similarity and would aid capturing all aspects of the languages instead of relying  only on a single or a handful of linguistic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Concentrating purely on using WALS to create a similarity metric  would also preserve homogeneity and allow a more explainable and controllable implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In previous research  [26] we proposed a novel linguistic similarity metric quantified from the WALS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This metric proved to be  more robust compared to the other metrics, at least for the applied abusive language detection task, as it was based on  multiple kinds of linguistic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Transfer Language Selection  Selecting the optimal language for cross-lingual transfer remains mostly an unanswered question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Most of the time,  the decision of which language to use as the transfer source comes up to the practitioner’s consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is usually  done experimentally or by intuition [20, 35, 16] or by simply relying on English [36, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example, in order to  get a more successful transfer, Cottorell and Heigold [20] focused on using languages belonging to the same language  family as the cross-lingual transfer target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, even though the languages are part of the same language family,  two languages could be very distant for example when looking at the complexity of grammar, which means that it does  not guarantee them sharing the same linguistic features [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  A common way for choosing the transfer language is to simply default to English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason being that it is the  de-facto highest resource language available for most NLP tasks [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is also the case with popular multilingual  benchmarks like XTREME [12] and XGLUE [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Although, recently benchmarks like XTREME-R [39] have started  to include cross-lingual training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Furthermore, in a survey of 157 cross-lingual learning papers by Pikuliak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  [18] they found out that English was used in 149 papers, followed by German with 82 papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Additionally, it has been  shown that other languages than English, for example, German and Russian tend to work better as transfer sources  [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Duong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [41] found out that choosing the transfer language based on language family is not optimal for many  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example, their experiments showed that the best source language for both Finnish and German is  Czech, even though being from a different language family than the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' They concluded that apparently, the best  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 4 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  in which these consonants appear in the words, one can calculate a genetic proximity score between two languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  This is implemented as the eLinguistics [23] similarity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The metric makes it possible to get information about  the direct relatedness of the compared languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, once the used languages start to become more and more  distant, accidental similarities in consonants are introduced and there is a significant increase in the error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is  also acknowledged by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Even though the metric is easy to calculate, it completely ignores all other kinds of  linguistic features, for example, semantic, syntactic, or morphological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Another method to calculate a similarity metric is to take a look at the vocabularies of two languages and concentrate  on their similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' EzGlot [24] uses lexical similarity as its basis for computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The metric uses lexical similarity  between the two compared languages while at the same time taking into account the amount of words the two languages  are sharing with other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This allows for the calculation of similarity between the two languages in relation to  the similarity with every other language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [22l proposed a linguistic similarity metric that utilizes multiple aspects of languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Their metric,  called STL, is based on Semantic, Terminological (lexical) and Linguistic (syntactic) similarity of languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The  method outperformed previous similarity metrics that concentrated only on one of the previously mentioned aspects  [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' They noticed that the terminological measures showed a much higher contribution when compared to the  other two features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, in order to use the metric, the structure of the used vocabulary dataset needs to be in the  form of a complex ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Due to this fact and because of the dataset only consisting of German, French, Italian,  Dutch, Spanish and English, and due to the dataset used by the authors being no longer available, it was not feasible  to use the metric as a part of this research  The lang2vec developed by Littell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [27] is a database that represents languages as typological, phylogenetic,  and geographical vectors, which are derived from a number of different linguistic resources, for example, WALS  [25], PHOIBLE [33], Ethnologue [1], and Glottolog [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Each of these utilize multiple different features, making  them more robust than the EzGlot or eLinguistics metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The lang2vec is a fully-fledged library that can be used to  query for different linguistic features and to get pre-computed distances between languages, based on some typological  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The World Atlas of Language Structures (WALS) project [25] consists of a database that catalogs phonological,  word semantic and grammatical knowledge for 2,662 languages with almost two hundred different linguistic features  from multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Using a linguistic similarity measure quantified from the WALS database into would allow  a more robust method to measure similarity and would aid capturing all aspects of the languages instead of relying  only on a single or a handful of linguistic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Concentrating purely on using WALS to create a similarity metric  more robust compared to the other metrics, at least for the applied abusive language detection task, as it was based on  multiple kinds of linguistic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Transfer Language Selection  Selecting the optimal language for cross-lingual transfer remains mostly an unanswered question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" Most of the time,  the decision of which language to use as the transfer source comes up to the practitioner's consideration." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is usually  done experimentally or by intuition [20, 35, 16] or by simply relying on English [36, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example, in order to  get a more successful transfer, Cottorell and Heigold [20] focused on using languages belonging to the same language  family as the cross-lingual transfer target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, even though the languages are part of the same language family,  two languages could be very distant for example when looking at the complexity of grammar, which means that it does  not guarantee them sharing the same linguistic features [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  A common way for choosing the transfer language is to simply default to English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason being that it is the  de-facto highest resource language available for most NLP tasks [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is also the case with popular multilingual  benchmarks like XTREME [12] and XGLUE [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Although, recently benchmarks like XTREME-R [39] have started  to include cross-lingual training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Furthermore, in a survey of 157 cross-lingual learning papers by Pikuliak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  [18] they found out that English was used in 149 papers, followed by German with 82 papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Additionally, it has been  [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Duong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [41] found out that choosing the transfer language based on language family is not optimal for many  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example, their experiments showed that the best source language for both Finnish and German is  Czech, even though being from a different language family than the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' They concluded that apparently, the best  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 4 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  source language for cross-lingual transfer is not predictable from language family information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Instead, they proposed  two methods for transfer language selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The first being based on the Jensen-Shannon divergence between the  distributions of parts-of-speech n-grams on a pair of languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The second method was based on the word-order  information feature in WALS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Both of these methods showed improvements over choosing English or a language from  the same family as the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' They also experimented with using multiple source languages, which further improved  the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  It has been shown [42, 43, 44] that transferring from many high-resource languages at the same time can yield  higher results compared to selecting only a single language as the transfer source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, these methods do not  consider the actual relation between the source and the target languages and the amount of contribution of each of the  languages to the total score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, Nooralahzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [45] discovered that certain morphosyntactic features shared  between languages tend to give a boost to cross-lingual transfer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [19] developed a ranking method for possible transfer language candidates using the lang2vec metrics  [27] together with dataset dependent features like word overlap and type-token ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' they discovered that using both  the dataset independent linguistic features and database dependent features to train the ranking model yields the best  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, as their method requires training of the ranking model, it is dependent on the tasks and datasets used  for training and is not usable out of the box for other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In another study [38] it was shown that the transfer performance with English as the source correlates with the  linguistic similarity metrics of lang2vec [27], meaning that target languages more similar to English yielded higher  scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' They found out that similarity of syntactic structures especially play an important role in selecting the source  language for tasks like parts-of-speech tagging (POS), named entity recognition (NER) and dependency parsing (DEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  They also discovered that the fine-tuning corpus size of the target language also makes a difference considering the  cross-lingual transfer performance, especially for higher level tasks like question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, their research  concentrated only on using English as the source language and the capabilities of other languages as the transfer source  were left completely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [46] found out that differences in language morphology in cross-lingual transfer generally lead to  a higher loss than when transferring between languages with the same morphological typology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Furthermore, they  showed that parts-of-speech tagging tends to be more sensitive towards changes in morphological typology compared  to sentiment analysis, which seems to be more sensitive to variables related to the fine-tuning data and the transfer  performance being generally harder to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In their research, Gaikwad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [10] discovered that there could be a relation between cross-lingual transfer  performance and language similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' They classified entries in the Marathi language using multiple languages,  specifically Bengali, Greek, English, Turkish and Hindi as cross-lingual transfer sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Their results showed that  the closest language of these to Marathi, Hindi, also had the highest performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This hints that a solution to the  problem of cross-lingual transfer language selection could be found with the aid of linguistic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In our previous research [26], we showed that there is a correlation between language similarity metrics and cross-  lingual transfer efficiency, at least for offensive language identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This allows for choosing of an ideal transfer  language by using different metrics to compare the similarity languages without having to rely on one’s intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We  also showed that choosing a transfer language, for example, only by looking at the language family is not always the  best option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Tasks  In this research, we concentrate on three different NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Sentiment analysis as a document classification  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Named entity recognition as a token classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' And lastly, dependency parsing for understanding the  importance of syntax and grammar in cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  We hypothesize that the zero-shot cross-lingual transfer performance correlates with the linguistic similarity of the  source and target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In order to confirm our hypothesis, we used datasets from eight different languages, namely  English, German, Danish, Polish, Russian, Croatian, Japanese and Korean for all of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We chose these languages  as they had high quality datasets compared to other options and because the languages represent three different language  families (English, German, Danish - Germanic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Polish, Russian, Croatian - Slavic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Japanese, Korean - Koreano-Japonic  language family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This also gives us the opportunity to study the efficacy of cross-lingual transfer learning between  and within different language family groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 5 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  source language for cross-lingual transfer is not predictable from language family information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Instead, they proposed  two methods for transfer language selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The first being based on the Jensen-Shannon divergence between the  distributions of parts-of-speech n-grams on a pair of languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The second method was based on the word-order  information feature in WALS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Both of these methods showed improvements over choosing English or a language from  the same family as the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' They also experimented with using multiple source languages, which further improved  the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  It has been shown [42, 43, 44] that transferring from many high-resource languages at the same time can yield  higher results compared to selecting only a single language as the transfer source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, these methods do not  consider the actual relation between the source and the target languages and the amount of contribution of each of the  languages to the total score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, Nooralahzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [45] discovered that certain morphosyntactic features shared  between languages tend to give a boost to cross-lingual transfer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [19] developed a ranking method for possible transfer language candidates using the lang2vec metrics  [27] together with dataset dependent features like word overlap and type-token ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' they discovered that using both  the dataset independent linguistic features and database dependent features to train the ranking model yields the best  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, as their method requires training of the ranking model, it is dependent on the tasks and datasets used  for training and is not usable out of the box for other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In another study [38] it was shown that the transfer performance with English as the source correlates with the  scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' They found out that similarity of syntactic structures especially play an important role in selecting the source  language for tasks like parts-of-speech tagging (POS), named entity recognition (NER) and dependency parsing (DEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  They also discovered that the fine-tuning corpus size of the target language also makes a difference considering the  cross-lingual transfer performance, especially for higher level tasks like question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, their research  concentrated only on using English as the source language and the capabilities of other languages as the transfer source  were left completely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [46] found out that differences in language morphology in cross-lingual transfer generally lead to  a higher loss than when transferring between languages with the same morphological typology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Furthermore, they  showed that parts-of-speech tagging tends to be more sensitive towards changes in morphological typology compared  to sentiment analysis, which seems to be more sensitive to variables related to the fine-tuning data and the transfer  performance being generally harder to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In their research, Gaikwad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [10] discovered that there could be a relation between cross-lingual transfer  specifically Bengali, Greek, English, Turkish and Hindi as cross-lingual transfer sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Their results showed that  the closest language of these to Marathi, Hindi, also had the highest performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This hints that a solution to the  problem of cross-lingual transfer language selection could be found with the aid of linguistic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In our previous research [26], we showed that there is a correlation between language similarity metrics and cross-  lingual transfer efficiency, at least for offensive language identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" This allows for choosing of an ideal transfer  language by using different metrics to compare the similarity languages without having to rely on one's intuition." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We  also showed that choosing a transfer language, for example, only by looking at the language family is not always the  best option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Tasks  In this research, we concentrate on three different NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Sentiment analysis as a document classification  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Named entity recognition as a token classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' And lastly, dependency parsing for understanding the  importance of syntax and grammar in cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  We hypothesize that the zero-shot cross-lingual transfer performance correlates with the linguistic similarity of the  source and target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In order to confirm our hypothesis, we used datasets from eight different languages, namely  English, German, Danish, Polish, Russian, Croatian, Japanese and Korean for all of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We chose these languages  as they had high quality datasets compared to other options and because the languages represent three different language  families (English, German, Danish - Germanic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Polish, Russian, Croatian - Slavic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Japanese, Korean - Koreano-Japonic  language family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This also gives us the opportunity to study the efficacy of cross-lingual transfer learning between  and within different language family groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 5 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Sentiment Analysis  In the field of NLP, sentiment analysis is one of the most active research areas [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The recent research in sentiment  analysis, as with many other NLP tasks, has mainly focused on using deep neural networks and pretrained language  models [48, 49, 50, 51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The popularization of multilingual transformer models has made it possible to utilize  cross-lingual transfer in order to train models for low-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Rasooli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [53] used a set of 16 languages from different language families, namely Indo-European, Turkic,  Afro-Asiatic, Uralic, and Sino-Tibetan, to learn a sentiment analysis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Their experiments showed that for most  target languages the best result can be obtained by leveraging from multiple source languages at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also,  datasets of a similar genre and domain tended to yield higher results when compared to out-of-domain and dissimilar  genres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Pelicon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [54] used zero-shot cross-lingual transfer to classify Croatian news articles with an mBERT model  fine-tuned using Slovene data with good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In addition, Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [55] used XLM-R and performed cross-  lingual transfer from English to Hindi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Their model compared favorably to the used benchmarks and gives an effective  solution to the analysis of sentiments in a resource-poor scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The majority of the sentiment analysis datasets used in this research consists of product reviews, as we attempted  to keep the domain the same throughout the languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, for some languages, we were unable to find such  data, most notably Croatian, which consists of news articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We also had to adjust the labels of some of the datasets so  that they would match among all of the languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Training and evaluation splits were retained from original datasets  if possible, otherwise datasets were split to 80% training and 20% evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  For this research we used the Multilingual Amazon Reviews Corpus [56], which covers English, Japanese and  German.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The dataset contains over 200,000 reviews for each language collected between 2015 and 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reviews  are labeled from one to five stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, as the other datasets used in this research used a two-point scale (positive,  negative), we adjusted the labels accordingly (positive: 5 and 4 stars, negative: 2 and  1 stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  For Danish, we used a dataset containing almost 45,000 reviews crawled from Trustpilot by Alessandro Gianfelici!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  For Polish, we used the PolEmo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='0 corpus [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This dataset contains over 8,000 reviews from the domains of medicine,  hotels, products and school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For both of these datasets, we also had to adjust the labels of this dataset to a two-point  scale similarly to the Amazon Reviews dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The Russian dataset used in this research was a product review dataset by Smetanin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The dataset consists  of 90,000 automatically labeled reviews on the topic "Women’s Clothes and Accessories", split evenly among three  classes (positive, neutral, negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The Croatian dataset is the same used by Pelicon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [54], containing around  2,000 news articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The articles were collected from 24sata, one of the leading Croatian media companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The  annotations were done by 6 people using a five-level Likert scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The annotations were later adjusted to a three-point  scale by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For the purpose of our experiments, in case of both datasets, we left out the neutral reviews in  order to binarize the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The Korean dataset used in this research was Naver sentiment movie corpus v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='0*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The dataset consists of Naver  Movie reviews, with 100,000 positive and negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reviews were originally rated from one to ten, but the  creators binarized the dataset prior to publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Named Entity Recognition  The research on Named Entity Recognition (NER) has also shifted towards using Deep Neural Networks and most  recently, pretrained transformer models [59, 60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Cross-lingual transfer has also been applied to NER in multiple  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Fritzler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [62] used a metric-learning method to at the time outperform a state-of-the-art recurrent neural  network method and showed to be capable in both few-shot and zero-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [63] used multilingual  BERT to fine-tune a NER model in multiple languages and showed it to be more effective than a model fine-tuned  only on a single language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This demonstrates that the model can leverage knowledge from other languages in order to  improve its performance on one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Hvingelby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [64] presented a Danish NLP resource based on the Danish Universal Dependencies treebank and  showed that transferring from other Germanic languages, especially from English and Norwegian, to Danish can yield  good results when using mBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, using other Germanic languages in addition to Danish did not give any  better results compared to fine-tuning only with Danish in their case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  ‘https ://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='com/AlessandroGianfelici/danish_reviews_dataset  nttps: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='com/e9t/nsmc  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 6 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Sentiment Analysis  In the field of NLP, sentiment analysis is one of the most active research areas [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The recent research in sentiment  analysis, as with many other NLP tasks, has mainly focused on using deep neural networks and pretrained language  models [48, 49, 50, 51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The popularization of multilingual transformer models has made it possible to utilize  cross-lingual transfer in order to train models for low-resource languages  Afro-Asiatic, Uralic, and Sino-Tibetan, to learn a sentiment analysis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Their experiments showed that for most  target languages the best result can be obtained by leveraging from multiple source languages at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also,  datasets of a similar genre and domain tended to yield higher results when compared to out-of-domain and dissimilar  genres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Pelicon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [54] used zero-shot cross-lingual transfer to classify Croatian news articles with an mBERT model  fine-tuned using Slovene data with good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In addition, Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [55] used XLM-R and performed cross-  lingual transfer from English to Hindi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Their model compared favorably to the used benchmarks and gives an effective  solution to the analysis of sentiments in a resource-poor scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The majority of the sentiment analysis datasets used in this research consists of product reviews, as we attempted  to keep the domain the same throughout the languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, for some languages, we were unable to find such  data, most notably Croatian, which consists of news articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We also had to adjust the labels of some of the datasets so  that they would match among all of the languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Training and evaluation splits were retained from original datasets  if possible, otherwise datasets were split to 80% training and 20% evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  For this research we used the Multilingual Amazon Reviews Corpus [56], which covers English, Japanese and  German.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The dataset contains over 200,000 reviews for each language collected between 2015 and 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reviews  are labeled from one to five stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, as the other datasets used in this research used a two-point scale (positive,  negative), we adjusted the labels accordingly (positive: 5 and 4 stars, negative: 2 and 1 stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  For Danish, we used a dataset containing almost 45,000 reviews crawled from Trustpilot by Alessandro Gianfelicil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  For Polish, we used the PolEmo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='0 corpus [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This dataset contains over 8,000 reviews from the domains of medicine,  hotels, products and school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For both of these datasets, we also had to adjust the labels of this dataset to a two-point  scale similarly to the Amazon Reviews dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The Russian dataset used in this research was a product review dataset by Smetanin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The dataset consists  of 90,000 automatically labeled reviews on the topic "Women\'s Clothes and Accessories", split evenly among three  classes (positive, neutral, negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The Croatian dataset is the same used by Pelicon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [54], containing around  2,000 news articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The articles were collected from 24sata, one of the leading Croatian media companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The  annotations were done by 6 people using a five-level Likert scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The annotations were later adjusted to a three-point  scale by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For the purpose of our experiments, in case of both datasets, we left out the neutral reviews in  order to binarize the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The Korean dataset used in this research was Naver sentiment movie corpus v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The dataset consists of Naver  Movie reviews, with 100,000 positive and negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reviews were originally rated from one to ten, but the  creators binarized the dataset prior to publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Named Entity Recognition  The research on Named Entity Recognition (NER) has also shifted towards using Deep Neural Networks and most  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Fritzler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [62] used a metric-learning method to at the time outperform a state-of-the-art recurrent neural  network method and showed to be capable in both few-shot and zero-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [63] used multilingual  BERT to fine-tune a NER model in multiple languages and showed it to be more effective than a model fine-tuned  only on a single language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This demonstrates that the model can leverage knowledge from other languages in order to  improve its performance on one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Hvingelby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [64] presented a Danish NLP resource based on the Danish Universal Dependencies treebank and  showed that transferring from other Germanic languages, especially from English and Norwegian, to Danish can yield  good results when using mBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, using other Germanic languages in addition to Danish did not give any  better results compared to fine-tuning only with Danish in their case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='com/AlessandroGianfelici/danish_reviews_dataset  2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='com/e9t/nsmc  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 6 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Entity projection [65, 66] has been used to generate pseudo-labeled datasets for low-resource NER datasets with  the help of parallel corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, it has been shown by Weber and Steedman [67] that entity projection can  be outperformed by cross-lingual transfer and XLM-RoBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason behind this could be explained by the  discovery by Lauscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [38], who showed that transfer performance with English as the source correlates with the  similarity of the languages when dealing with a NER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In this study, we used the WikiANN [68] multilingual NER dataset also used by XTREME benchmark [12] for all  of the proposed languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' WikiANN consists of Wikipedia articles annotated with LOC (location), PER (person), and  ORG (organisation) NER tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We used the version by Rahimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [69], which has a balanced train, development,  and test splits and supports 176 of the 282 languages from the original WikiANN corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Dependency Parsing  Cross-lingual transfer in dependency parsing (DEP) has been studied for some time before the advent of  multilingual transformer models [70, 71, 72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' These studies mainly used deep neural network-based methods on  parallel corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The research by Duong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [41] discussed earlier in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2 was also conducted on a dependency  parsing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Instead of using parallel corpora, their research was built around syntactic cross-lingual word embeddings  [74] trained over POS contexts to emphasize syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Multilingual transformer models have also seen success in the dependency parsing task [15, 75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Most notably,  in their study, Lauscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [38] discovered that structural and syntactic similarities between languages are the most  determining factor when it comes to the success of cross-lingual transfer for lower-level tasks like POS-tagging and  DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The dataset used for all of the proposed languages in this study was the Universal Dependencies v2 [77], a widely  used resource in NLP as well as in linguistic research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The dataset was also used in the XTREME [12] benchmark  and in the research by Lauscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [38] described earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Universal Dependencies is a framework for a consistent  annotation of grammar, including parts-of-speech, morphological features, and syntactic dependencies across a total  of more than 100 languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Methods  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Models  For the experiments we used two pre-trained multilingual transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The experiments were carried out  in a zero-shot cross-lingual setting [78], meaning that the fine-tuning is done using only data from another language  than the target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Multilingual BERT (mBERT) [13] is the multilingual version of BERT, which stands for Bidirectional Encoder  Representations from Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' It is based on an attention mechanism called the Transformer [79] that learns  contextual relations between words (or sub-words) in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' One of the features transformer models introduced is the  capability to read text input in both directions at once, instead of being able to only read it sequentially from left-to-  right or right-to-left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Taking advantage of this bidirectional capability, BERT is pre-trained on two NLP tasks, Masked  Language Modeling and Next Sentence Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The objective of Masked Language Modeling is to mask a word in  a sentence and have the algorithm predict based on the word’s context what word has been hidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Next Sentence  Prediction, the algorithm takes two masked sentences and needs to predict if they have a sequential connection or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Although mBERT has not been trained using any cross-lingual data, it has showed cross-lingual capabilities and had  good results in many cross-lingual tasks [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This also includes various zero-shot transfer tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Multilingual BERT  has even been shown to outperform the usage of various cross-lingual embeddings [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This ability to generalize  could come from having word pieces used in all languages, for example, numbers, URLs, etc, mapped to a shared  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This in turn forces the co-occurring pieces to also be mapped to a shared space, thus spreading the effect to  other word pieces, until different languages are close in a shared space [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  XLM-RoBERTa (XLM-R) [14] is a multi-lingual transformer model, also trained with the Masked Language  Model objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' XLM-R is trained on around a total of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='5tb of CommonCrawl data in one hundred different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The model is trained in the same way as the monolingual RoBERTa [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This means, that the only objective in its  pre-training is Masked Language Modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The model is not trained on the Next Sentence Prediction task like BERT  or using the parallel Translation Language Model objective of XLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  XLM-R has been shown to outperform both mBERT and XLM on a many cross-lingual benchmarks, including  zero-shot cross-lingual transfer tasks [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' It has also been shown to perform well on low-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' A  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 7 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Entity projection [65, 66] has been used to generate pseudo-labeled datasets for low-resource NER datasets with  the help of parallel corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, it has been shown by Weber and Steedman [67] that entity projection can  be outperformed by cross-lingual transfer and XLM-RoBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason behind this could be explained by the  discovery by Lauscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [38], who showed that transfer performance with English as the source correlates with the  similarity of the languages when dealing with a NER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In this study, we used the WikiANN [68] multilingual NER dataset also used by XTREME benchmark [12] for all  of the proposed languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' WikiANN consists of Wikipedia articles annotated with LOC (location), PER (person), and  ORG (organisation) NER tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We used the version by Rahimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [69], which has a balanced train, development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  and test splits and supports 176 of the 282 languages from the original WikiANN corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Dependency Parsing  Cross-lingual transfer in dependency parsing (DEP) has been studied for some time before the advent of  multilingual transformer models [70, 71, 72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' These studies mainly used deep neural network-based methods on  parallel corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The research by Duong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [41] discussed earlier in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2 was also conducted on a dependency  parsing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Instead of using parallel corpora, their research was built around syntactic cross-lingual word embeddings  [74] trained over POS contexts to emphasize syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Multilingual transformer models have also seen success in the dependency parsing task [15, 75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Most notably  in their study, Lauscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [38] discovered that structural and syntactic similarities between languages are the most  determining factor when it comes to the success of cross-lingual transfer for lower-level tasks like POS-tagging and  DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The dataset used for all of the proposed languages in this study was the Universal Dependencies v2 [77], a widely  used resource in NLP as well as in linguistic research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The dataset was also used in the XTREME [12] benchmark  and in the research by Lauscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' [38] described earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Universal Dependencies is a framework for a consistent  annotation of grammar, including parts-of-speech, morphological features, and syntactic dependencies across a total  of more than 100 languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Methods  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Models  For the experiments we used two pre-trained multilingual transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The experiments were carried out  in a zero-shot cross-lingual setting [78], meaning that the fine-tuning is done using only data from another language  than the target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Multilingual BERT (mBERT) [13] is the multilingual version of BERT, which stands for Bidirectional Encoder  contextual relations between words (or sub-words) in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' One of the features transformer models introduced is the  capability to read text input in both directions at once, instead of being able to only read it sequentially from left-to-  right or right-to-left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Taking advantage of this bidirectional capability, BERT is pre-trained on two NLP tasks, Masked  Language Modeling and Next Sentence Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" The objective of Masked Language Modeling is to mask a word in  a sentence and have the algorithm predict based on the word's context what word has been hidden." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Next Sentence  Prediction, the algorithm takes two masked sentences and needs to predict if they have a sequential connection or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Although mBERT has not been trained using any cross-lingual data, it has showed cross-lingual capabilities and had  good results in many cross-lingual tasks [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This also includes various zero-shot transfer tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Multilingual BERT  has even been shown to outperform the usage of various cross-lingual embeddings [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This ability to generalize  could come from having word pieces used in all languages, for example, numbers, URLs, etc, mapped to a shared  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This in turn forces the co-occurring pieces to also be mapped to a shared space, thus spreading the effect to  other word pieces, until different languages are close in a shared space [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  XLM-RoBERTa (XLM-R) [14] is a multi-lingual transformer model, also trained with the Masked Language  Model objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' XLM-R is trained on around a total of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='5tb of CommonCrawl data in one hundred different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The model is trained in the same way as the monolingual RoBERTa [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This means, that the only objective in its  pre-training is Masked Language Modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The model is not trained on the Next Sentence Prediction task like BERT  or using the parallel Translation Language Model objective of XLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  XLM-R has been shown to outperform both mBERT and XLM on a many cross-lingual benchmarks, including  zero-shot cross-lingual transfer tasks [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' It has also been shown to perform well on low-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' A  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 7 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Table 1  eLinguistics metric between all applied languages  Danish  English  German  Croatian  Polish  Russian  Japanese  Korean  Danish  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='00  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='60  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='20  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='00  notable feature of XLM-R is that it can also match the performance of to state-of-the-art monolingual models,  which demonstrates that it is possible to create multilingual models without sacrificing per-language performance  in a monolingual setting [14], most likely thanks to the sheer amount of data used in the pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Linguistic Similarity Metrics  To be able to calculate the correlation between cross-lingual zero-shot transfer performance and language similarity  for the proposed tasks, we needed  a way to quantify the aspects of all of the languages in our proposed set,  specifically, a language similarity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We utilized four language similarity measures, eLinguistics [23], EzGlot  [24], the multidomain metric we quantified from the linguistic features presented in WALS [25] and averaged genetic,  geographic, syntactic, inventory, phonological and featural metrics from lang2vec [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We propose that linguistic  similarity metrics could be utilized when trying to find optimal source language candidates for cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  We hypothesize that linguistic similarity correlates with cross-lingual transfer efficacy, meaning that by using more  similar languages, we would be able to achieve higher model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  eLinguistics [23] works by calculating a genetic proximity value for a pair of languages based on the use of phonetic  consonants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The score is calculated by taking a predefined word set and comparing the consonants contained in these  words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The method also takes into account the order of the consonants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This way, it is possible to get information  regarding the closeness of the phonetics of the pair of languages set for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The assessment of the relationship  of the consonants is based on the research done by Brown et al [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Even though completely disregarding semantic, morphological, and syntactic similarity and being very simple in  formulation, the similarity values produced by the method seemed to be in line with our expectations and the two  multidomain metrics (WALS, lang2vec) used in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, as the distance between the two compared  languages increased, the method seemed to become increasingly more prone to errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is due to the surging  amount of accidental similarities in consonants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The similarity measure can be accessed from a web service*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The  similarity values between our proposed languages are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  EzGlot [24]  is based on the similarity of vocabularies, or lexical similarity, of the two compared languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  EzGlot’s similarity metric is computed by taking the lexical similarity between the two compared languages, while  in addition taking into account the number of words the pair of languages also have in common every other language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  This makes it possible to compute a similarity measure for a pair of languages in relation to their closeness with  every other language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, due to including the calculation of the number of words the languages share with all other  languages, the similarity measure becomes asymmetric between every pair of languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This also supports studies  stating that mutual language intelligibility is being considered asymmetric as well [84, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  A pre-computed language similarity matrix and the formula for its computation can be found on the EzGlot  similarity metric project’s web page*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, the usability of the metric is hindered by the high amount of missing  values in the similarity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example taking a look at Japanese, which is one of the languages utilized in  our experiments, over half of the values are missing for our proposed languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, the authors of the similarity  measure do not give away their data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This means that we are unable to say anything regarding the quality of the  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This also makes it more difficult to fill in the missing values to the similarity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We extracted the  3nttp ://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='elinguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' : Preprint submitted to Elsevier  Page 8 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Table 1  eLinguistics metric between all applied languages  Danish  Polish  English  German  Croatian  Russian  Japanese  Korean  Danish  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='00  notable feature of XLM-R is that it can also match the performance of to state-of-the-art monolingual models  which demonstrates that it is possible to create multilingual models without sacrificing per-language performance  in a monolingual setting [14], most likely thanks to the sheer amount of data used in the pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Linguistic Similarity Metrics  To be able to calculate the correlation between cross-lingual zero-shot transfer performance and language similarity  for the proposed tasks, we needed a way to quantify the aspects of all of the languages in our proposed set  specifically, a language similarity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We utilized four language similarity measures, eLinguistics [23], EzGlot  [24], the multidomain metric we quantified from the linguistic features presented in WALS [25] and averaged genetic,  geographic, syntactic, inventory, phonological and featural metrics from lang2vec [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We propose that linguistic  similarity metrics could be utilized when trying to find optimal source language candidates for cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  We hypothesize that linguistic similarity correlates with cross-lingual transfer efficacy, meaning that by using more  similar languages, we would be able to achieve higher model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  eLinguistics [23] works by calculating a genetic proximity value for a pair of languages based on the use of phonetic  consonants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The score is calculated by taking a predefined word set and comparing the consonants contained in these  words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The method also takes into account the order of the consonants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This way, it is possible to get information  regarding the closeness of the phonetics of the pair of languages set for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The assessment of the relationship  of the consonants is based on the research done by Brown et al [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Even though completely disregarding semantic, morphological, and syntactic similarity and being very simple in  formulation, the similarity values produced by the method seemed to be in line with our expectations and the two  multidomain metrics (WALS, lang2vec) used in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, as the distance between the two compared  languages increased, the method seemed to become increasingly more prone to errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is due to the surging  amount of accidental similarities in consonants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The similarity measure can be accessed from a web service3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The  similarity values between our proposed languages are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  EzGlot [24] is based on the similarity of vocabularies, or lexical similarity, of the two compared languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content="  EzGlot's similarity metric is computed by taking the lexical similarity between the two compared languages, while  in addition taking into account the number of words the pair of languages also have in common every other language  This makes it possible to compute a similarity measure for a pair of languages in relation to their closeness with  every other language." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, due to including the calculation of the number of words the languages share with all other  languages, the similarity measure becomes asymmetric between every pair of languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This also supports studies  stating that mutual language intelligibility is being considered asymmetric as well [84, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content="  A pre-computed language similarity matrix and the formula for its computation can be found on the EzGlot  similarity metric project's web page4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, the usability of the metric is hindered by the high amount of missing  values in the similarity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example taking a look at Japanese, which is one of the languages utilized in  our experiments, over half of the values are missing for our proposed languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, the authors of the similarity  measure do not give away their data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This means that we are unable to say anything regarding the quality of the  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This also makes it more difficult to fill in the missing values to the similarity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We extracted the  3http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='elinguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='net/Compare_Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='aspx  4https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='ezglot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='com/most-similar-languages .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='php  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Page 8 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Table 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='EzGlot metric between all of the proposed languages  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Danish  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Polish  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Russian  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Japanese  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Korean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Table 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Averaged lang2vec metric between all of the proposed languages  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Danish  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='English  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='German  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Croatian  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Polish  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Russian  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Japanese  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Korean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Danish  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='511  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='487  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='550  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='565  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='597  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='694  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='691  English  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='511  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='352  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='578  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='486  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='488  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='635  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='578  German  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='487  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='352  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='550  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='470  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='471  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='594  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='579  Croatian  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='550  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='578  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='550  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='513  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='505  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='709  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='699  Polish  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='565  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='486  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='470  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='513  ~=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='344  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='624  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='619  Russian  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='597  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='488  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='471  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='505  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='344  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='589  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='585  Japanese  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='694  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='635  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='594  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='709  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='624  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='589  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='518  Korean  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='691  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='578  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='579  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='699  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='619  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='585  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='518  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='000  similarity values from the EzGlot’s similarity matrix for the proposed languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' These values are presented in Table  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Averaged lang2vec is calculated from genetic, geographic, syntactic, inventory, phonological and featural metrics  of lang2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Lang2vec [27] is a database that provides vector identifications of languages based on different linguistic  features based on various linguistic resources like WALS [25], PHOIBLE [33], Ethnologue [1], and Glottolog [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The lang2vec is a fully-fledged library that can be used to query for different linguistic features and to get pre-computed  genetic, geographic, syntactic, inventory, phonological and featural distances between languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In order to use  lang2vec as a multidomain linguistic similarity metric, we used an average value of these six categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The method is based on multiple types of linguistic features, making it naturally more robust than EzGlot or  eLinguistics similarity metrics, which only rely on a single kind of linguistic feature each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The method also uses a  larger amount of data compared to the previously described metric based on WALS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Additionally, lang2vec deals  with the missing values in linguistic resources by predicting them [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, due to being based on multiple  sources, the heterogeneous nature of the method brings up many questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example, there might be incoherence  as we do not know how features are selected from different sources and how they are weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, the features are  one-hot encoded which causes a complete loss of ordinality between feature values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Additionally, using geographical  information as one of the vectors seems questionable as it was shown to be unreliable when predicting similarity of  languages [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The averaged distance matrix for lang2vec is shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Quantified World Atlas of Language Structures is a similarity metric developed by us in previous research  [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' It is based on The World Atlas of Language Structures (WALS) [25], which is a massive language database  that records phonological, word semantic and grammatical information for a total of 2,662 languages from over 200  different language families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' There are 192 different linguistic features in the database currently (May 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However,  many of the linguistic features are missing for of the available languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example, one of the most extensively  documented language, English, has about 150 features documented in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This amount rapidly decreases for  languages studied less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Taking Danish as an example, it only 58 features documented>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Considering every language  and all of the features, this adds up to over 58,000 data points in total in the WALS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This means the whole  >Some even less studied languages have an even smaller number of features documented, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Chuj language, spoken in Guatemala, has only  29, while the Indonesian Kutai language has only a single feature documented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Page 9 of 23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Table 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='EzGlot metric between all of the proposed languages  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Danish  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='English  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='German  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Croatian  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Polish  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Russian  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Japanese  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Korean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Danish  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='English  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='German  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Croatian  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Polish  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Russian  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Japanese  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='N/A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Korean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Table 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Danish  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='English  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='German  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Croatian  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Polish  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Russian  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Japanese  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Korean  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='Danish  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content="000  similarity values from the EzGlot's similarity matrix for the proposed languages." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' These values are presented in Table  Averaged lang2vec is calculated from genetic, geographic, syntactic, inventory, phonological and featural metrics  of lang2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Lang2vec [27] is a database that provides vector identifications of languages based on different linguistic  features based on various linguistic resources like WALS [25], PHOIBLE [33], Ethnologue [1], and Glottolog [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The lang2vec is a fully-fledged library that can be used to query for different linguistic features and to get pre-computed  genetic, geographic, syntactic, inventory, phonological and featural distances between languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In order to use  lang2vec as a multidomain linguistic similarity metric, we used an average value of these six categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The method is based on multiple types of linguistic features, making it naturally more robust than EzGlot or  eLinguistics similarity metrics, which only rely on a single kind of linguistic feature each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The method also uses a  larger amount of data compared to the previously described metric based on WALS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Additionally, lang2vec deals  with the missing values in linguistic resources by predicting them [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, due to being based on multiple  sources, the heterogeneous nature of the method brings up many questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example, there might be incoherence  as we do not know how features are selected from different sources and how they are weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, the features are  one-hot encoded which causes a complete loss of ordinality between feature values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Additionally, using geographical  information as one of the vectors seems questionable as it was shown to be unreliable when predicting similarity of  languages [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The averaged distance matrix for lang2vec is shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Quantified World Atlas of Language Structures is a similarity metric developed by us in previous research  [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' It is based on The World Atlas of Language Structures (WALS) [25], which is a massive language database  that records phonological, word semantic and grammatical information for a total of 2,662 languages from over 200  different language families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' There are 192 different linguistic features in the database currently (May 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However,  many of the linguistic features are missing for of the available languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example, one of the most extensively  documented language, English, has about 150 features documented in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This amount rapidly decreases for  languages studied less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Taking Danish as an example, it only 58 features documented5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Considering every language  and all of the features, this adds up to over 58,000 data points in total in the WALS database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This means the whole  5Some even less studied languages have an even smaller number of features documented, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Chuj language, spoken in Guatemala, has only  29, while the Indonesian Kutai language has only a single feature documented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' : Preprint submitted to Elsevier  Page 9 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Table 4  Quantified WALS metric between all of the proposed languages  Danish  English  German  Croatian  Polish  Russian  Japanese  Korean  Danish  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='000  database is only approximately 12% populated, meaning a vast majority of the information is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also many  major and widely studied languages are missing many features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example, 25% of all of the features are missing for  English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' These missing values and the sparsity of the data is the main point of concern when quantifying the WALS  database into a linguistic similarity metric as using lesser known and not so widely studied languages means having  less common features among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In previous research, we quantified a novel linguistic similarity metric from the WALS database based on the  features all of the proposed languages shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' One of the problems of the metric was that as the amount of languages  increased, the amount of features shared with them decreased due to missing values in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This time, we  improved the calculation process and increased the robustness of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The improved version attempts to counter  the issue caused by the diminishing feature count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This was done by selecting all of the features that would have  a defined value for both languages in all possible language pairs instead of having to be shared between all of the  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The language pairs were formed from our proposed languages (English, German, Danish, Polish, Russian,  Croatian, Japanese and Korean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Otherwise, the process remained the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In short, we converted the possible  feature values into numeric and calculated an average euclidean distance between all language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This resulted  in a symmetric distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The goal was to create a multidomain similarity metric that would also be coherent  and try to preserve the ordinality of the feature values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The finished distance matrix is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Lastly, our plan was to take a look at the STL similarity measure [22], which is based on multiple linguistic  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The measure puts together three different aspects of language by using Semantic, Terminological (lexical)  and Linguistic (syntactic) similarity to form a single metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' According to the authors, the STL metric outperformed  many previous measures that were relying only on one of the previously mentioned feature types [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, in  order to be able to use the metric, the vocabulary dataset must be structured in the form of an ontology, which restricts  the metric’s use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Due to this fact and because of the lack of available languages for the used dataset, it was not possible  for us to utilize the metric in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Experiments  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Setup  We fine-tuned both of the models (mBERT, XLM-R) with all of the proposed languages (English, German, Danish,  Polish, Russian, Croatian, Japanese and Korean) for all of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Fine-tuning refers to the training of the parameters  of a pre-trained language model (like BERT) with task-specific labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This produced 16 fine-tuned models for  each task, which sums up to a total of 48 fine-tuned models (two transformer models, eight languages, three tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  After fine-tuning, we evaluated the models with test datasets from all of the previously mentioned languages to compute  the cross-lingual zero-shot transfer scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We did not use a train-dev-test, but only train-test scenario for evaluation,  because the test dataset has nothing to do with the training dataset in a zero-shot task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We also do not aim at optimizing  for each dataset, or creating a product, but rather study general properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We evaluated the models with a macro  Fl-score for sentiment analysis and NER, and Label Attachment Score (LAS) for the dependency parsing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  After finishing the evaluations for all of the fine-tuned models, we took a look at the correlation between the  zero-shot cross-lingual transfer scores and linguistic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This was done by using the four previously introduced  linguistic similarity metrics (eLinguistics, EzGlot, WALS and lang2vec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We computed Pearson’s and Spearman’s  correlations between the models’ cross-lingual zero-shot transfer scores and the language similarity measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 10 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Table 4  Quantified WALS metric between all of the proposed languages  Danish  English  German  Croatian  Polish  Russian  Korean  Japanese  Danish  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='000  database is only approximately 12% populated, meaning a vast majority of the information is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also many  major and widely studied languages are missing many features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example, 25% of all of the features are missing for  English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' These missing values and the sparsity of the data is the main point of concern when quantifying the WALS  database into a linguistic similarity metric as using lesser known and not so widely studied languages means having  less common features among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In previous research, we quantified a novel linguistic similarity metric from the WALS database based on the  features all of the proposed languages shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' One of the problems of the metric was that as the amount of languages  increased, the amount of features shared with them decreased due to missing values in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This time, we  improved the calculation process and increased the robustness of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The improved version attempts to counter  the issue caused by the diminishing feature count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This was done by selecting all of the features that would have  a defined value for both languages in all possible language pairs instead of having to be shared between all of the  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The language pairs were formed from our proposed languages (English, German, Danish, Polish, Russian,  Croatian, Japanese and Korean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Otherwise, the process remained the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In short, we converted the possible  feature values into numeric and calculated an average euclidean distance between all language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This resulted  in a symmetric distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The goal was to create a multidomain similarity metric that would also be coherent  and try to preserve the ordinality of the feature values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The finished distance matrix is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Lastly, our plan was to take a look at the STL similarity measure [22], which is based on multiple linguistic  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The measure puts together three different aspects of language by using Semantic, Terminological (lexical)  and Linguistic (syntactic) similarity to form a single metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' According to the authors, the STL metric outperformed  many previous measures that were relying only on one of the previously mentioned feature types [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" However, in  order to be able to use the metric, the vocabulary dataset must be structured in the form of an ontology, which restricts  the metric's use." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Due to this fact and because of the lack of available languages for the used dataset, it was not possible  for us to utilize the metric in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Experiments  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Setup  We fine-tuned both of the models (mBERT, XLM-R) with all of the proposed languages (English, German, Danish,  Polish, Russian, Croatian, Japanese and Korean) for all of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Fine-tuning refers to the training of the parameters  of a pre-trained language model (like BERT) with task-specific labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This produced 16 fine-tuned models for  each task, which sums up to a total of 48 fine-tuned models (two transformer models, eight languages, three tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  After fine-tuning, we evaluated the models with test datasets from all of the previously mentioned languages to compute  the cross-lingual zero-shot transfer scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We did not use a train-dev-test, but only train-test scenario for evaluation,  because the test dataset has nothing to do with the training dataset in a zero-shot task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We also do not aim at optimizing  for each dataset, or creating a product, but rather study general properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We evaluated the models with a macro  F1-score for sentiment analysis and NER, and Label Attachment Score (LAS) for the dependency parsing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  After finishing the evaluations for all of the fine-tuned models, we took a look at the correlation between the  zero-shot cross-lingual transfer scores and linguistic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" This was done by using the four previously introduced  correlations between the models' cross-lingual zero-shot transfer scores and the language similarity measures." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 10 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Table 5  Tasks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' models and linguistic similarity metrics used in the experiments  Tasks  Models  Linguistic Similarity Metrics  Sentiment Analysis  mBERT  EzGlot  Named Entity Recognition  XLM-RoBERTa _ eLinguistics  Dependency Parsing  Averaged lang2vec  Quantified WALS  Table 6  Sentiment analysis: F1-scores for mBERT  TARGET  Danish  English  German  Croatian  Polish  Russian  Japanese  (Korean  Danish  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='953  tasks, models and linguistic similarity metrics used in the experiments are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The models were fine-tuned  by using PyTorch and the Huggingface Transformers library [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Results  Both of the multilingual transformer models (mnBERT, XLM-R) were fine-tuned with all of the proposed languages  for each task (sentiment analysis, NER, DEP) we introduced earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' The model evaluation scores are presented in Tables 6 and  7 for sentiment analysis, Tables 8 and 9 for NER and Tables 10 and 11 for DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Looking at the results, we can clearly say that XLM-R outperformed mBERT in  all of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' It can be noted from the results  that the highest transfer scores usually belong to the languages in the same language family as the source language  (English, German, Danish - Germanic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' : Preprint submitted to Elsevier  Page 11 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Table 5  Tasks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='953  tasks, models and linguistic similarity metrics used in the experiments are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The models were fine-tuned  by using PyTorch and the Huggingface Transformers library [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The hardware used was an Nvidia GTX 1080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Results  Both of the multilingual transformer models (mBERT, XLM-R) were fine-tuned with all of the proposed languages  for each task (sentiment analysis, NER, DEP) we introduced earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The models were fine-tuned using only the training  dataset from a single language before the evaluation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The model evaluation scores are presented in Tables 6 and  7 for sentiment analysis, Tables 8 and 9 for NER and Tables 10 and 11 for DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Looking at the results, we can clearly say that XLM-R outperformed mBERT in all of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The only exception  to this was the sentiment analysis task, where mBERT slightly outperformed XLM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' It can be noted from the results  that the highest transfer scores usually belong to the languages in the same language family as the source language  (English, German, Danish - Germanic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Japanese, Korean - Koreano-Japonic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also,  most of the time there is a clear difference in the scores when evaluating with the same language as the source compared  to zero-shot cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The exceptions to this are the sentiment analysis task for both models and the NER  task for XLM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' : Preprint submitted to Elsevier  Page 11 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Table 8  NER: Fi-scores for mBERT  TARGET  Danish  English  German  Croatian  Polish  Russian  Japanese  Korean  Danish  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='293  In dependency parsing, XLM-R slightly outperformed mBERT as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, in the sentiment analysis  task mBERT scored slightly higher than XLM-R overall, with both models scoring high across all language pairs  Some language pairs even achieving zero-shot cross-lingual transfer F-score of over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' In this task, there seems not  to be a clear pattern what kind of language pairs tend to yield higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example, Slavic languages seem  to work better as sources for Danish compared to German languages in the case of both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The scores are also  similarly high across the board for the NER task with XLM-R, with the model being able to achieve very high scores  with zero-shot transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The performance difference between mBERT and XLM-R is also more noticeable in the case  of NER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  As can be seen from Table 12, both Japanese and Korean worked decently well as cross-lingual transfer sources  for both sentiment analysis and NER tasks, even though being very different from the other languages used in the  experiments as they are the only non Indo-European languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, in the case of DEP their performance  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 12 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Table 11  DEP: LAS-scores for XLM-R  TARGET  Danish  English  German  Croatian  Polish  Russian  Japanese  Korean  Danish  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Except for this case with DEP, all of the proposed languages seem to be quite equal as cross-  lingual transfer sources in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Interestingly, German, Croatian and Russian seem to perform slightly better overall  compared to the other languages, especially with mBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' A similar phenomenon was also experienced by Turc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  [40] and in our previous research [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Effect of Linguistic Similarity  We calculated the correlation between the zero-shot cross-lingual transfer results of the two models and each of the  four proposed linguistic similarity metrics (EzGlot, eLinguistics, WALS and lang2vec) in all proposed NLP tasks using  both Pearson’s and Spearman’s correlation coefficients (p-value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We were forced to ignore some of the language pairs  when calculating the correlations with the EzGlot metric as some of the similarity values were missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The correlation  analysis results are shown in Table 13 for sentiment analysis, Table 14 for NER, Table 15 for DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Looking at the results, one can say that there is mostly a strong correlation between lang2vec, WALS and  eLinguistics metrics and the cross-lingual zero-shot transfer score, and a strong-moderate correlation between the  EzGlot metric and the transfer scores for NER and DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In the case of sentiment analysis, the correlation is noticeably  lower with XLM-R, staying at a moderate level with all of the linguistic similarity metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The correlation is strongest  with the dependency parsing task with XLM-R, with the highest absolute Spearman’s correlation being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Also, the results show p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='  For both of the models, the correlation for lang2vec, WALS and eLinguistics metrics are generally higher than  EzGlot, except in the case of sentiment analysis, where EzGlot’s correlation is slightly higher than WALS and  eLinguistics using both Pearson’s and Spearman’s correlation coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='  [40] and in our previous research [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" Effect of Linguistic Similarity  We calculated the correlation between the zero-shot cross-lingual transfer results of the two models and each of the  four proposed linguistic similarity metrics (EzGlot, eLinguistics, WALS and lang2vec) in all proposed NLP tasks using  both Pearson's and Spearman's correlation coefficients (p-value)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We were forced to ignore some of the language pairs  when calculating the correlations with the EzGlot metric as some of the similarity values were missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The correlation  analysis results are shown in Table 13 for sentiment analysis, Table 14 for NER, Table 15 for DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Looking at the results, one can say that there is mostly a strong correlation between lang2vec, WALS and  eLinguistics metrics and the cross-lingual zero-shot transfer score, and a strong-moderate correlation between the  EzGlot metric and the transfer scores for NER and DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In the case of sentiment analysis, the correlation is noticeably  lower with XLM-R, staying at a moderate level with all of the linguistic similarity metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" The correlation is strongest  with the dependency parsing task with XLM-R, with the highest absolute Spearman's correlation being O." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='897 with  eLinguistics metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, the results show p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='05 for all of the tasks, models and metrics, indicating statistical  significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content="  For both of the models, the correlation for lang2vec, WALS and eLinguistics metrics are generally higher than  EzGlot, except in the case of sentiment analysis, where EzGlot's correlation is slightly higher than WALS and  eLinguistics using both Pearson's and Spearman's correlation coefficients." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, the correlations were generally slightly  stronger with mBERT in sentiment analysis, while XLM-R had higher correlations in both NER and DEP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  However, the results changed drastically for all tasks except dependency parsing, when we removed the anchor  points of same source-target language pairs (monolingual scenarios), leaving only the zero-shot transfer results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This  was necessary to do in order remove the bias brought by the monolingual data points, as the scores are higher and the  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" : Preprint submitted to Elsevier  Page 13 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Table 13  Sentiment analysis: Pearson’s and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics  Pearson  Spearman  XLM-R  mBERT  XLM-R  mBERT  p  p-value  p  p-value  p  p-value  p  p-value  WALS  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='000  EzGlot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='000  lang2vec  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content="000  Table 14  NER: Pearson's and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics  Pearson  Spearman  XLM-R  mBERT  XLM-R  mBERT  p  p-value  p  p-value  p  p-value  p  p-value  WALS  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='514  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='004  eLinguistics  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='000  lang2vec  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content="000  Table 15  DEP: Pearson’s and Spearman's correlation coefficients for model LAS scores and linguistic similarity metrics  Pearson  Spearman  XLM-R  mBERT  XLM-R  mBERT  p  p-value  p  p-value  p  p-value  p  p-value  WALS  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='000  lang2vec  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='000  languages would also be the most similar (same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The results after removing the same source-target language pairs are  shown in Table 16 for sentiment analysis, Table 17 for NER, Table 18 for DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  First of all, for eLinguistics and WALS similarity metrics, the correlations generally dropped from strong to  moderate for the NER task while EzGlot’s correlation fell close to zero and also lost all statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='  Interestingly, for sentiment analysis, the result looks completely opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The correlations of EzGlot and lang2vec  only fell slightly while WALS’ and eLinguistics’ correlation plummeted down and lost statistical significance for  XLM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The correlations in the dependency parsing task only dropped slightly for all of the linguistic similarity  metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, after removing the same source-target language pairs, the strongest correlations are still found in DEP,  followed by NER, while sentiment analysis still has the weakest correlations overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, as linguistic similarity  still correlates with cross-lingual transfer performance, we can get an improved model performance by using linguistic  similarity for transfer language selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" : Preprint submitted to Elsevier  Page 14 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Table 13  Sentiment analysis: Pearson's and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics  Pearson  Spearman  XLM-R  mBERT  XLM-R  mBERT  p-value  p-value  p  p-value  p-value  p  p  p  WALS  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='000  languages would also be the most similar (same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The results after removing the same source-target language pairs are  shown in Table 16 for sentiment analysis, Table 17 for NER, Table 18 for DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content="  First of all, for eLinguistics and WALS similarity metrics, the correlations generally dropped from strong to  moderate for the NER task while EzGlot's correlation fell close to zero and also lost all statistical significance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" The  correlation of lang2vec also plummeted and mostly lost statistical significance, but remained higher than EzGlot's  only fell slightly while WALS' and eLinguistics' correlation plummeted down and lost statistical significance for  XLM-R." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The correlations in the dependency parsing task only dropped slightly for all of the linguistic similarity  followed by NER, while sentiment analysis still has the weakest correlations overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, as linguistic similarity  still correlates with cross-lingual transfer performance, we can get an improved model performance by using linguistic  similarity for transfer language selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" : Preprint submitted to Elsevier  Page 14 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Table 16  Sentiment analysis: Pearson’s and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics  for zero-shot only  Pearson  Spearman  XLM-R  mBERT  XLM-R  mBERT  p  p-value  p  p-value  p  p-value  p  p-value  WALS  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Transfer Language Performance  XLM-R outperforming mBERT generally matches our expectations, as it also did so on a variety of benchmark  tasks [12, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason behind this most likely is the fact that XLM-R uses a vastly larger amount of data for  pretraining compared to mBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The performance difference between the two models is the most clear in the NER  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  According to the results, simply choosing English as the transfer source did not yield top results most of the time,  sometimes even as a source language to other languages in the Germanic language group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For example, it had a lower  than average performance in sentiment analysis and an average performance in the other two tasks probably due to  its simplicity when compared to both Danish and German.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' It was also slightly outperformed by Slavic languages in  some cases when used as a source for other Germanic languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Another reason could be the influence of French  [88, 89, 90], which might further distance it from the other Germanic languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, the differences in morphology  could be a factor here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Danish and German probably work better with each other due to a great amount of historic  mutual influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" : Preprint submitted to Elsevier  Page 15 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Table 16  Sentiment analysis: Pearson's and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics  for zero-shot only  Pearson  Spearman  XLM-R  mBERT  XLM-R  mBERT  p  p-value  p  p-value  p  p-value  p-value  p  WALS  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content="001  Table 17  NER: Pearson's and Spearman's correlation coefficients for model F1 scores and linguistic similarity metrics for zero-shot  only  Pearson  Spearman  XLM-R  mBERT  XLM-R  mBERT  p  p-value  p-value  p-value  p-value  p  p  p  WALS  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Discussion  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Transfer Language Performance  XLM-R outperforming mBERT generally matches our expectations, as it also did so on a variety of benchmark  tasks [12, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason behind this most likely is the fact that XLM-R uses a vastly larger amount of data for  pretraining compared to mBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The performance difference between the two models is the most clear in the NER  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  According to the results, simply choosing English as the transfer source did not yield top results most of the time,  than average performance in sentiment analysis and an average performance in the other two tasks probably due to  its simplicity when compared to both Danish and German.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' It was also slightly outperformed by Slavic languages in  some cases when used as a source for other Germanic languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Another reason could be the influence of French  [88, 89, 90], which might further distance it from the other Germanic languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, the differences in morphology  could be a factor here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Danish and German probably work better with each other due to a great amount of historic  mutual influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 15 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  In sentiment analysis, the models achieved slightly better scores with English and it was on-par with other Germanic  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, all of the Slavic languages still tended to work slightly better as transfer sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' These results  show that other languages should also be considered over English as the cross-lingual transfer source if available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For  the other two tasks, NER and DEP, English performed well and showed to be a good transfer source for the other two  Germanic languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In most cases, using languages from the same language family as the source language yielded the highest cross-  lingual transfer scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This matches with the typical intuition-based selection process used to select source language for  cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, relying only on intuition and looking purely at language families when when selecting  the transfer language will lead to diminished results in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  One example would be taking Polish as the target language for NER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' One could expect that in this case, the  best transfer languages would be Croatian and Russian, but looking at the results (Tables 8 and 9) German had a  higher cross-lingual transfer score even though it is from the Germanic language family, not Slavic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This could be, for  example, due to mutual influence of these two languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The grammar of both Danish and English is relatively simple  compared to German, which could aid them in generalizing better with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Looking at the scores, it can be  noted that German is a good source for both Germanic and Slavic languages, which could mean that the historical  mutual influence between the Germans and Slavs could be a factor here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Furthermore, German, in addition to having a  higher average performance on most tasks, tended to also work exceptionally well also as a source language for other  Slavic languages, most likely because of the reasons discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In addition, Japanese and Korean did not achieve comparably better scores with one another, contrary to our  expectations, and were even slightly outperformed by the other languages approximately half of the time, even though  being more similar with each other compared to any of the other proposed languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Here the reason could be, for  example, the differences in the writing systems, as neither of these two languages use alphabets and their systems also  greatly differ from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Also, both Russian and Croatian had a higher than average performance on most of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This was similar to  our previous research [26] where Russian performed exceptionally well as a transfer source for offensive language  identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, unlike in our previous research, Russian did not perform noticeably well as the transfer  language source for Korean and Japanese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Thus the phenomenon experienced previously is most likely related to  the topic of offensive language identification itself or to the properties of these specific datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We will investigate  this in later research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, Japanese and Korean had a satisfying performance as source languages for the Germanic  and Slavic languages in both sentiment analysis and NER tasks, even though Japanese and Korean are fundamentally  different from the languages of these two families, as they are the only non Indo-European languages in the proposed  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This demonstrates that multilingual transformer models are also able to leverage knowledge even from very distant  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Analysis of Linguistic Similarity Metrics  The correlation between cross-lingual transfer performance and the similarity metrics were strong or moderate  with all of the proposed metrics, which would suggest that using even a single feature such as lexical information or  by comparing phonetic consonants is still effective to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' EzGlot  However, when considering only the zero-shot transfer results, EzGlot’s similarity metric’s correlation dropped  drastically and out of statistical significance in the NER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This shows that it does not necessarily rely on lexical  features and that other linguistic features need to be considered when choosing the source language for NER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' On  the other hand, the same happened with the lang2vec metric despite it being created using different features from  multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, the opposite happened in the sentiment analysis task, as both WALS and eLinguistics metrics’  correlation dropped drastically and out of statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This hints the importance of lexical similarity when  choosing the source language for sentiment analysis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' eLinguistics  Surprisingly, even though using only a predefined set of phonetic consonants for its calculation, the correlation of  eLinguistics’ similarity metric was stronger in all tasks compared to the the correlation of the WALS metric, which we  quantified from the WALS database using linguistic features from multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The correlation of eLinguistics  was also higher than the averaged lang2vec metric in both NER and DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason behind this could be that including  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 16 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  In sentiment analysis, the models achieved slightly better scores with English and it was on-par with other Germanic  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, all of the Slavic languages still tended to work slightly better as transfer sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' These results  show that other languages should also be considered over English as the cross-lingual transfer source if available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' For  the other two tasks, NER and DEP, English performed well and showed to be a good transfer source for the other two  Germanic languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In most cases, using languages from the same language family as the source language yielded the highest cross-  lingual transfer scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This matches with the typicalintuition-based selection process used to select source language for  cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, relying only on intuition and looking purely at language families when when selecting  the transfer language will lead to diminished results in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  One example would be taking Polish as the target language for NER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' One could expect that in this case, the  best transfer languages would be Croatian and Russian, but looking at the results (Tables 8 and 9) German had a  example, due to mutual influence of these two languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The grammar of both Danish and English is relatively simple  compared to German, which could aid them in generalizing better with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Looking at the scores, it can be  noted that German is a good source for both Germanic and Slavic languages, which could mean that the historical  mutual influence between the Germans and Slavs could be a factor here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Furthermore, German, in addition to having a  higher average performance on most tasks, tended to also work exceptionally well also as a source language for other  Slavic languages, most likely because of the reasons discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In addition, Japanese and Korean did not achieve comparably better scores with one another, contrary to our  expectations, and were even slightly outperformed by the other languages approximately half of the time, even though  being more similar with each other compared to any of the other proposed languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Here the reason could be, for  example, the differences in the writing systems, as neither of these two languages use alphabets and their systems also  greatly differ from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Also, both Russian and Croatian had a higher than average performance on most of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This was similar to  our previous research [26] where Russian performed exceptionally well as a transfer source for offensive language  identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, unlike in our previous research, Russian did not perform noticeably well as the transfer  the topic of offensive language identification itself or to the properties of these specific datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We will investigate  this in later research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, Japanese and Korean had a satisfying performance as source languages for the Germanic  and Slavic languages in both sentiment analysis and NER tasks, even though Japanese and Korean are fundamentally  different from the languages of these two families, as they are the only non Indo-European languages in the proposed  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This demonstrates that multilingual transformer models are also able to leverage knowledge even from very distant  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Analysis of Linguistic Similarity Metrics  The correlation between cross-lingual transfer performance and the similarity metrics were strong or moderate  with all of the proposed metrics, which would suggest that using even a single feature such as lexical information or  by comparing phonetic consonants is still effective to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" EzGlot  However, when considering only the zero-shot transfer results, EzGlot's similarity metric's correlation dropped  drastically and out of statistical significance in the NER task." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This shows that it does not necessarily rely on lexical  features and that other linguistic features need to be considered when choosing the source language for NER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' On  multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" Also, the opposite happened in the sentiment analysis task, as both WALS and eLinguistics metrics'  correlation dropped drastically and out of statistical significance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This hints the importance of lexical similarity when  choosing the source language for sentiment analysis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" eLinguistics  Surprisingly, even though using only a predefined set of phonetic consonants for its calculation, the correlation of  eLinguistics' similarity metric was stronger in all tasks compared to the the correlation of the WALS metric, which we  quantified from the WALS database using linguistic features from multiple domains." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The correlation of eLinguistics  was also higher than the averaged lang2vec metric in both NER and DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason behind this could be that including  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 16 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  all of possible features between each language pair could have caused too many irrelevant features to be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This  can cause a possible bias the metric calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  However, the eLinguistics metric also has its weak points as it is based on only a single aspect of language, even  though its correlation being the strongest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' One can see from Table  1 that eLinguistics shows Japanese being very  distant from Korean, being at the same level as Polish and Russian, with Croatian being seemingly closer to Korean  than Japanese, which is not true due to the similarities in the vocabulary and grammar of Japanese and Korean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Taking  a look at Tables 4 and 3, it is clear that the WALS and lang2vec metrics are a lot more robust to this kind of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The reason most likely is that instead of using only a single linguistic feature like the eLinguistics metric, the WALS  and lang2vec metrics are based on a large amount of features spanning over multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Averaged lang2vec and Quantified WALS  Both lang2vec and WALS had a strong correlation with the DEP task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Although both metrics are based on a large  number of linguistic features spanning over multiple domains, their correlations varied greatly with the other two  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Specifically, in NER, the correlation of lang2vec was noticeably lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In sentiment analysis, the correlation of  WALS plummeted while lang2vec stayed at a moderate level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason is probably in the calculation of the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In the quantified WALS metric, features are treated as continuous whereas lang2vec uses one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, in  quantified WALS, every single feature has the same weight whereas in averaged lang2vec every category of features has  the same weight but might contain a different amount of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Additionally, the features in lang2vec are collected  from multiple sources, which increases the amount of data while possibly introducing incoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Lastly, the number  of features used in the similarity calculations with the quantified WALS metric varies slightly between language pairs  due to missing values in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Lang2vec tries to counter this by using a model to predict the missing values,  although this might introduce more errors in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In the future, we will take another glance at the WALS database, aiming for a better quantification by looking at  the importance of each feature group (syntactic, lexical, phonetic, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=') and weighing accordingly while filtering out  redundant features in order to develop an even more effective and comprehensive similarity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We will also take a  look at lang2vec, aiming to filter out redundant features and weigh the categories accordingly instead of simply taking  an average in order to make it better suited for transfer language selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' After all, both WALS and lang2vec metrics  are more robust thanks to being based on a large amount of features spanning over multiple domains instead of using  only a single linguistic feature like eLinguistics or EzGlot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Task-Specific Analysis  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Sentiment Analysis  Looking at the f-scores of the sentiment analysis task, it is clear that the results are very high across the board and  the score differences between language groups are also very small, with sometimes languages from other language  groups than the target emerging as the best performers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is the case for example with Danish, as Croatian achieved  the highest zero-shot transfer scores for both mBERT and XLM-R instead of another Germanic language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  A trait only observed in this task was that lang2vec and EzGlot were the only metrics keeping a moderate correlation  in the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The fact that Ezglot’s correlation stayed moderate hints the importance of lexical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' One  could argue that the reason behind the overall high scores might be due to the task being too easy, as it simply required  the classification of the entries into positive and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This has also been shown in other research [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, this  also shows that it could be possible to achieve at least close to state-of-the-art results with multilingual transformer  models in a zero-shot cross-lingual setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This raises questions about how to improve the cross-lingual models to  better utilize cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In the future, it would be useful to further investigate the models’ behaviour in  zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This could also be useful in the further development of measures to support low-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Named Entity Recognition  For mBERT, the zero-shot results of the NER task look clearly lower than with same language pairs and quite  even across all of the proposed languages and the languages belonging to the same group having generally a slightly  higher score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, the results of XLM-R closely resemble those of the sentiment analysis task as the results are  considerably high across all language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This further shows the potential these models have in relieving the issues  with low-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, there is a moderate correlation between the zero-shot transfer performance and  linguistic similarity for both WALS and eLinguistics metrics, and a weak/moderate correlation with lang2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 17 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  all of possible features between each language pair could have caused too many irrelevant features to be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This  can cause a possible bias the metric calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  However, the eLinguistics metric also has its weak points as it is based on only a single aspect of language, even  though its correlation being the strongest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' One can see from Table 1 that eLinguistics shows Japanese being very  distant from Korean, being at the same level as Polish and Russian, with Croatian being seemingly closer to Korean  than Japanese, which is not true due to the similarities in the vocabulary and grammar of Japanese and Korean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Taking  a look at Tables 4 and 3, it is clear that the WALS and lang2vec metrics are a lot more robust to this kind of errors  The reason most likely is that instead of using only a single linguistic feature like the eLinguistics metric, the WALS  and lang2vec metrics are based on a large amount of features spanning over multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Averaged lang2vec and Quantified WALS  Both lang2vec and WALS had a strong correlation with the DEP task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Although both metrics are based on a large  number of linguistic features spanning over multiple domains, their correlations varied greatly with the other two  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Specifically, in NER, the correlation of lang2vec was noticeably lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In sentiment analysis, the correlation of  WALS plummeted while lang2vec stayed at a moderate level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason is probably in the calculation of the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In the quantified WALS metric, features are treated as continuous whereas lang2vec uses one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, in  quantified WALS, every single feature has the same weight whereas in averaged lang2vec every category of features has  the same weight but might contain a different amount of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Additionally, the features in lang2vec are collected  from multiple sources, which increases the amount of data while possibly introducing incoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Lastly, the number  of features used in the similarity calculations with the quantified WALS metric varies slightly between language pairs  due to missing values in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Lang2vec tries to counter this by using a model to predict the missing values,  although this might introduce more errors in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In the future, we will take another glance at the WALS database, aiming for a better quantification by looking at  the importance of each feature group (syntactic, lexical, phonetic, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=') and weighing accordingly while filtering out  redundant features in order to develop an even more effective and comprehensive similarity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We will also take a  look at lang2vec, aiming to filter out redundant features and weigh the categories accordingly instead of simply taking  an average in order to make it better suited for transfer language selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' After all, both WALS and lang2vec metrics  are more robust thanks to being based on a large amount of features spanning over multiple domains instead of using  only a single linguistic feature like eLinguistics or EzGlot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Task-Specific Analysis  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Sentiment Analysis  Looking at the f-scores of the sentiment analysis task, it is clear that the results are very high across the board and  the score differences between language groups are also very small, with sometimes languages from other language  groups than the target emerging as the best performers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This is the case for example with Danish, as Croatian achieved  the highest zero-shot transfer scores for both mBERT and XLM-R instead of another Germanic language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  A trait only observed in this task was that lang2vec and EzGlot were the only metrics keeping a moderate correlation  in the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" The fact that Ezglot's correlation stayed moderate hints the importance of lexical features." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' One  could argue that the reason behind the overall high scores might be due to the task being too easy, as it simply required  the classification of the entries into positive and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This has also been shown in other research [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, this  also shows that it could be possible to achieve at least close to state-of-the-art results with multilingual transformer  models in a zero-shot cross-lingual setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This raises questions about how to improve the cross-lingual models to  better utilize cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" In the future, it would be useful to further investigate the models' behaviour in  zero-shot setting." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This could also be useful in the further development of measures to support low-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Named Entity Recognition  For mBERT, the zero-shot results of the NER task look clearly lower than with same language pairs and quite  higher score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, the results of XLM-R closely resemble those of the sentiment analysis task as the results are  considerably high across all language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This further shows the potential these models have in relieving the issues  with low-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, there is a moderate correlation between the zero-shot transfer performance and  linguistic similarity for both WALS and eLinguistics metrics, and a weak/moderate correlation with lang2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 17 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Dependency Parsing  The zero-shot results in DEP also seem clearly lower than with same language pairs and somewhat even across the  board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The languages belonging to the same language family also generally have a slightly higher score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' As the task  requires the understanding of syntax and grammar and the scores are still reasonably high overall, the results could also  support studies claiming that cross-lingual transformer models are able to learn grammar without explicit information  [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  On the other hand, Japanese and Korean had very poor performance as source languages while also being very  difficult target languages in the DEP task, unlike in the sentiment analysis and NER tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This could mean that the  model is unable to generalize to the syntax and grammar of Indo-European languages with Japonic-Koreanic languages  and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason might be due to the differences in writing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In the DEP task, both XLM-R and  mBERT keep a strong correlation between the zero-shot transfer performance and linguistic similarity with WALS,  eLinguistics and lang2vec metrics and a moderate correlation with EzGlot in a zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Impact  As there is a correlation with linguistic similarity and cross-lingual transfer performance for all of the tasks,  including the abusive language identification task used in our previous research [26], it is possible to use linguistic  similarity for transfer language selection, at least for these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, the correlation varied greatly from task  to task, which means there is a lot of room for improvement in developing an optimal similarity metric for transfer  language selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In order to confirm the efficacy of choosing another language over English as the cross-lingual transfer source, we  performed a z-test between the results of using English as transfer source and using the language with the highest score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The test showed Z = —3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='18 < —1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='96 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='001 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='05 meaning that there is a significant difference between  using English and the optimal language as the cross-lingual transfer source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Based on these results, as there is a significant difference between using English and the optimal language as the  cross-lingual transfer source, it is better to look for high-resource languages that have proper data available and are as  close as possible to the target language based on a similarity metric instead of making a decision based on intuition  or simply relying on English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This allows one to make a more informed and effective decision and makes model  development more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Future Research  In the future, we are planning to analyze, what kind of linguistic features are the most important from the  point of view of cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' A solution could be grouping the features presented in WALS into syntactic,  lexical, phonetic, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=', and calculating, which feature group has the strongest correlation with the cross-lingual transfer  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We could then re-quantify the WALS database using this information in order to develop an even more  effective and comprehensive similarity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' It would also be beneficial if the WALS project received more attention  and the feature matrix became more densely populated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, instead of taking an average of lang2vec’s categories,  they should be weighed by importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  As shown by the DEP task, the models might be able to learn syntax and grammar without any explicit information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  This could mean that adding explicit syntactic and grammatical information to the pre-training process of the models  might also improve their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We will take a look at this in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, as the models achieved zero-shot  transfer scores rivaling those of the monolingual settings, especially in sentiment analysis, it would be useful to perform  an in-depth investigation about the models’ behaviour in a zero-shot transfer learning setting to possibly find insights  on how to improve their transfer learning capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Conclusions  In this research we studied cross-lingual transfer language selection for zero-shot learning using three different  NLP tasks, namely, sentiment analysis, NER, and dependency parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We showed the effectiveness of cross-lingual  zero-shot transfer learning with a total of eight languages from three language families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In this way, existing data from  higher-resource languages may be used to improve the performance of languages that lack sufficient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  We found a strong correlation between the similarity of the used languages and cross-lingual transfer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The transfer performance declines when the distance between the languages increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This allows for the selection  of a more suitable transfer language by assessing linguistic similarities rather than simply depending on intuition  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 18 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Dependency Parsing  board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The languages belonging to the same language family also generally have a slightly higher score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' As the task  requires the understanding of syntax and grammar and the scores are still reasonably high overall, the results could also  support studies claiming that cross-lingual transformer models are able to learn grammar without explicit information  [92]  On the other hand, Japanese and Korean had very poor performance as source languages while also being very  difficult target languages in the DEP task, unlike in the sentiment analysis and NER tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This could mean that the  model is unable to generalize to the syntax and grammar of Indo-European languages with Japonic-Koreanic languages  and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The reason might be due to the differences in writing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In the DEP task, both XLM-R and  mBERT keep a strong correlation between the zero-shot transfer performance and linguistic similarity with WALS,  eLinguistics and lang2vec metrics and a moderate correlation with EzGlot in a zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Impact  As there is a correlation with linguistic similarity and cross-lingual transfer performance for all of the tasks,  including the abusive language identification task used in our previous research [26], it is possible to use linguistic  similarity for transfer language selection, at least for these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' However, the correlation varied greatly from task  to task, which means there is a lot of room for improvement in developing an optimal similarity metric for transfer  language selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In order to confirm the efficacy of choosing another language over English as the cross-lingual transfer source, we  performed a z-test between the results of using English as transfer source and using the language with the highest score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The test showed Z = -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='18 < -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='96 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='001 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='05 meaning that there is a significant difference between  using English and the optimal language as the cross-lingual transfer source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Based on these results, as there is a significant difference between using English and the optimal language as the  cross-lingual transfer source, it is better to look for high-resource languages that have proper data available and are as  close as possible to the target language based on a similarity metric instead of making a decision based on intuition  or simply relying on English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This allows one to make a more informed and effective decision and makes model  development more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Future Research  In the future, we are planning to analyze, what kind of linguistic features are the most important from the  point of view of cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' A solution could be grouping the features presented in WALS into syntactic,  lexical, phonetic, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=', and calculating, which feature group has the strongest correlation with the cross-lingual transfer  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We could then re-quantify the WALS database using this information in order to develop an even more  effective and comprehensive similarity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' It would also be beneficial if the WALS project received more attention  and the feature matrix became more densely populated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" Also, instead of taking an average of lang2vec's categories,  they should be weighed by importance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  As shown by the DEP task, the models might be able to learn syntax and grammar without any explicit information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  This could mean that adding explicit syntactic and grammatical information to the pre-training process of the models  might also improve their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We will take a look at this in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" Also, as the models achieved zero-shot  transfer scores rivaling those of the monolingual settings, especially in sentiment analysis, it would be useful to perform  an in-depth investigation about the models' behaviour in a zero-shot transfer learning setting to possibly find insights  on how to improve their transfer learning capabilities." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Conclusions  In this research we studied cross-lingual transfer language selection for zero-shot learning using three different  NLP tasks, namely, sentiment analysis, NER, and dependency parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We showed the effectiveness of cross-lingual  zero-shot transfer learning with a total of eight languages from three language families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In this way, existing data from  higher-resource languages may be used to improve the performance of languages that lack sufficient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  We found a strong correlation between the similarity of the used languages and cross-lingual transfer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The transfer performance declines when the distance between the languages increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This allows for the selection  of a more suitable transfer language by assessing linguistic similarities rather than simply depending on intuition  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 18 of 23Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  or defaulting to English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' As our experiments have demonstrated, there is a significant difference in choosing the  optimal transfer language over defaulting to English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' As there is a correlation between linguistic similarity and transfer  performance and a significant difference between using English and the optimal language as the cross-lingual transfer  source, one should instead choose the source language based on a linguistic similarity measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Our experiments also  demonstrated that lexical information alone is insufficient to determine the optimal transfer languages at least for the  tasks of NER and DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  it is better to look for high-resource languages that have proper data available and are as close as possible to the target  language based on a similarity metric instead of making a decision based on intuition or simply relying on English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  This allows one to make a more informed and effective decision and makes model development more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The results showed that the proposed method for cross-lingual transfer language selection could also be useful as a  general method for other Natural Language Processing tasks, at least based on these tasks and our previous research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We  also showed that it is possible to achieve good performance on the target language in a zero-shot cross-lingual transfer  setting with multiple NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This helps in developing better systems, especially when dealing with low-resource  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  We improved a novel linguistic similarity metric consisting of various linguistic features by using the WALS  database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Our proposed method did not show the strongest correlation with the transfer performance, but it still showed  potential as a metric that could be useful for the selection process, especially if given a more refined or inclusive feature  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In the future, we will reassess the importances of the linguistic features used in the similarity metric calculation in  order to have a more refined feature set, aiming to create an even more effective and comprehensive linguistic similarity  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Lastly, even though the overall high scores in the sentiment analysis task might be caused by the task being too  easy, it also shows that it could be possible to achieve results close to those of a monolingual fine-tuning in a zero-shot  cross-lingual transfer setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This means it could be useful to thoroughly investigate the models’ behaviour in zero-shot  setting in order to find insights to improving their transfer capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Also, as the DEP task demonstrated that the  models might have a capability to understand grammar, adding explicit syntactic and grammatical information to the  models’ pre-training could also increase performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  CRediT authorship contribution statement  Juuso Eronen: Conceptualization, Methodology, Software, Writing - Original Draft, Writing - Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Michal Ptaszynski: Conceptualization, Methodology, Supervision, Data Curation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Fumito Masui: Conceptualization,  Methodology, Supervision, Data Curation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='  Modeling language variation and universals: A survey on typological linguistics for natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='  [6]  Pratik Joshi, Sebastin Santy, Amar Budhiraja, Kalika Bali, and Monojit Choudhury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' The state and fate of linguistic diversity and inclusion in  the NLP world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  [7]  Long Duong, Trevor Cohn, Steven Bird, and Paul Cook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Low resource dependency parsing: Cross-lingual parameter sharing in a neural  network parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In Proceedings of the 53rd annual meeting of the Association for Computational Linguistics and the 7th international joint  conference on natural language processing (volume 2: short papers), pages 845-850, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  [8]  Rouzbeh Ghasemi, Seyed Arad Ashrafi Asli, and Saeedeh Momtazi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Deep persian sentiment analysis: Cross-lingual training for low-resource  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Journal of Information Science, page 0165551520962781, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  [9]  Raj Dabre, Chenhui Chu, and Anoop Kunchukuttan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  A survey of multilingual neural machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=', 53(5),  September 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  [10]  Saurabh Gaikwad, Tharindu Ranasinghe, Marcos Zampieri, and Christopher M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Homan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Cross-lingual offensive language identification for  low resource languages: The case of marathi, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' : Preprint submitted to Elsevier  Page 19 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  or defaulting to English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' As our experiments have demonstrated, there is a significant difference in choosing the  optimal transfer language over defaulting to English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' As there is a correlation between linguistic similarity and transfer  performance and a significant difference between using English and the optimal language as the cross-lingual transfer  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' one should instead choose the source language based on a linguistic similarity measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Our experiments also  demonstrated that lexical information alone is insufficient to determine the optimal transfer languages at least for the  tasks of NER and DEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  it is better to look for high-resource languages that have proper data available and are as close as possible to the target  language based on a similarity metric instead of making a decision based on intuition or simply relying on English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  This allows one to make a more informed and effective decision and makes model development more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  The results showed that the proposed method for cross-lingual transfer language selection could also be useful as a  general method for other Natural Language Processing tasks, at least based on these tasks and our previous research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' We  also showed that it is possible to achieve good performance on the target language in a zero-shot cross-lingual transfer  setting with multiple NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' This helps in developing better systems, especially when dealing with low-resource  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  We improved a novel linguistic similarity metric consisting of various linguistic features by using the WALS  database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Our proposed method did not show the strongest correlation with the transfer performance, but it still showed  potential as a metric that could be useful for the selection process, especially if given a more refined or inclusive feature  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In the future, we will reassess the importances of the linguistic features used in the similarity metric calculation in  order to have a more refined feature set, aiming to create an even more effective and comprehensive linguistic similarity  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Lastly, even though the overall high scores in the sentiment analysis task might be caused by the task being too  easy, it also shows that it could be possible to achieve results close to those of a monolingual fine-tuning in a zero-shot  cross-lingual transfer setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" This means it could be useful to thoroughly investigate the models' behaviour in zero-shot  setting in order to find insights to improving their transfer capabilities." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" Also, as the DEP task demonstrated that the  models might have a capability to understand grammar, adding explicit syntactic and grammatical information to the  models' pre-training could also increase performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  CRediT authorship contribution statement  Juuso Eronen: Conceptualization, Methodology, Software, Writing - Original Draft, Writing - Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Michal Ptaszynski: Conceptualization, Methodology, Supervision, Data Curation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Applied Sciences, 10(17), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Sergey Smetanin and Michail Komarov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' JEEE Transactions on Knowledge  and Data Engineering, 34(1):50-70, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='  Named entity recognition using deep learning: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' IEEE, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Few-shot classification in named entity recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Association for Computing  Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' DaNE: A named  entity resource for Danish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' European Language Resources Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Association for Computational  Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Applied Sciences, 10(17), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' The multilingual amazon reviews corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='  t analysis of PolEmo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='0: Extended corpus of multi-domain  Kong, China, November 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  21st Conference on Business Informatics (CBI), volume 01, pages 482-486, July 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' A survey on recent advances in named entity recognition from deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' arXiv preprint  arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' A survey on deep learning for named entity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' IEEE Transactions on Knowledge  ind Data Engineering, 34(1):50-70, 2022  [61] Sajid Ali, Khalid Masood, Anas Riaz, and Amna Saud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Named entity recognition using deep learning: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' IEEE, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Few-shot classification in named entity recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=" In Proceedings of the  34th ACM/SIGAPP Symposium on Applied Computing, SAC '19, page 993-1000, New York, NY, USA, 2019." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Association for Computing  Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  [64] Rasmus Hvingelby, Amalie Brogaard Pauli, Maria Barrett, Christina Rosted, Lasse Malm Lidegaard, and Anders Sogaard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' DaNE: A named  entity resource for Danish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Proceedings of the 12th Language Resources and Evaluation Conference, pages 4597-4604, Marseille, France,  May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' European Language Resources Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Eronen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Lipton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Entity projection via machine translation for cross-lingual NER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Proceedings of  the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language  Processing (EMNLP-IJCNLP), pages 1083-1092, Hong Kong, China, November 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Bing Li, Yujie He, and Wenjin Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='  arXiv preprint arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='11112, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Sabine Weber and Mark Steedman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Zero-shot cross-lingual transfer is a hard baseline to beat in German fine-grained entity typing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In  Proceedings of the Second Workshop on Insights from Negative Results in NLP, pages 42-48, Online and Punta Cana, Dominican Republic,  November 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Xiaoman Pan, Boliang Zhang, Jonathan May, Joel Nothman, Kevin Knight, and Heng Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Cross-lingual name tagging and linking for 282  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Afshin Rahimi, Yuan Li, and Trevor Cohn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Massively multilingual transfer for NER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='  Jérg Tiedemann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Cross-lingual dependency parsing with universal dependencies and predicted pos labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='  Jiang Guo, Wanxiang Che, David Yarowsky, Haifeng Wang, and Ting Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Cross-lingual dependency parsing based on distributed  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International  Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 1234-1244, Beijing, China, July 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Association for  Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Ophélie Lacroix, Lauriane Aufrant, Guillaume Wisniewski, and Frangois Yvon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Frustratingly easy cross-lingual transfer for transition-based  dependency parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics:  Human Language Technologies, pages 1058-1063, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Mohit Bansal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Dependency link embeddings: Continuous representations of syntactic substructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  In Proceedings of the Ist Workshop  on Vector Space Modeling for Natural Language Processing, pages 102-108, Denver, Colorado, June 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Association for Computational  Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Dan Kondratyuk and Milan Straka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' 75 languages, 1 model: Parsing Universal Dependencies universally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Proceedings of the 2019 Conference  on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-  IJCNLP), pages 2779-2795, Hong Kong, China, November 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Matej Uléar and Marko Robnik-Sikonja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Finest bert and crosloengual bert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In International Conference on Text, Speech, and Dialogue, pages  104-111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Joakim Nivre, Marie-Catherine de Marneffe, Filip Ginter, Jan Haji¢, Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Manning, Sampo Pyysalo, Sebastian Schuster, Francis  Tyers, and Daniel Zeman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Universal Dependencies v2: An evergrowing multilingual treebank collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' In Proceedings of the 12th Language  Resources and Evaluation Conference, pages 4034-4043, Marseille, France, May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' European Language Resources Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Rishi Bommasani, Drew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Hudson, Ehsan Adeli, Russ Altman, Simran Arora, Sydney von Arx, Michael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Bernstein, Jeannette Bohg,  Antoine Bosselut, Emma Brunskill, Erik Brynjolfsson, Shyamal Buch, Dallas Card, Rodrigo Castellon, Niladri Chatterji, Annie Chen, Kathleen  Creel, Jared Quincy Davis, Dora Demszky, Chris Donahue, Moussa Doumbouya, Esin Durmus, Stefano Ermon, John Etchemendy, Kawin  Ethayarajh, Li Fei-Fei, Chelsea Finn, Trevor Gale, Lauren Gillespie, Karan Goel, Noah Goodman, Shelby Grossman, Neel Guha, Tatsunori  Hashimoto, Peter Henderson, John Hewitt, Daniel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Ho, Jenny Hong, Kyle Hsu, Jing Huang, Thomas Icard, Saahil Jain, Dan Jurafsky,  Pratyusha Kalluri, Siddharth Karamcheti, Geoff Keeling, Fereshte Khani, Omar Khattab, Pang Wei Koh, Mark Krass, Ranjay Krishna, Rohith  Kuditipudi, Ananya Kumar, Faisal Ladhak, Mina Lee, Tony Lee, Jure Leskovec, Isabelle Levent, Xiang Lisa Li, Xuechen Li, Tengyu Ma,  Ali Malik, Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Manning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Suvir Mirchandani,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Eric Mitchell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Zanele Munyikwa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Suraj Nair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Avanika Narayan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Deepak Narayanan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  Ben Newman,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Allen Nie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Juan Carlos Niebles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Hamed Nilforoshan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Julian Nyarko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Joon Sung  Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Chris Piech,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Rob Reich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Hongyu Ren,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Frieda Rong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Yusuf Roohani,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content='  Jack Ryan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Christopher Ré,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Dorsa Sadigh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Shiori Sagawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Andy Shih,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Krishnan Srinivasan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Rohan Taori,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Wang, William Wang, Bohan Wu, Jiajun Wu, Yuhuai Wu, Sang Michael Xie, Michihiro Yasunaga,  Jiaxuan You, Matei Zaharia, Michael Zhang, Tianyi Zhang, Xikun Zhang, Yuhui Zhang, Lucia Zheng, Kaitlyn Zhou, and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' On  the opportunities and risks of foundation models, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Attention  is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Advances in neural information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Advances in Neural Information Processing Systems,  32:7059-7069, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Sound correspondences in the world’s languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Language, 89:4—29, 03 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' : Preprint submitted to Elsevier  Page 23 of 23  Zero-Shot Cross-Lingual Transfer Language Selection Using Linguistic Similarity  Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content='  [86] Benedikt Szmrecsanyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Dialectological and folk dialectological concepts of space, pages 215-231, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Improving classifier  training efficiency for automatic cyberbullying detection with feature density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
+page_content=' Information Processing & Management, 58(5):102616, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+page_content=' Association for Computational  Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFST4oBgHgl3EQfGDhd/content/2301.13720v1.pdf'}
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+Actor-Director-Critic: A Novel Deep Reinforcement Learning Framework
+Zongwei Liu
+Xi’an Jiaotong University
+China
+Ting Haode@stu.xjtu.edu.cn
+Yonghong Song
+Xi’an Jiaotong University
+China
+songyh@xjtu.edu.cn
+Yuanlin Zhang
+Xi’an Jiaotong University
+China
+ylzhangxian@xjtu.edu.cn
+Abstract
+In this paper, we propose actor-director-critic, a new
+framework for deep reinforcement learning. Compared with
+the actor-critic framework, the director role is added, and
+action classification and action evaluation are applied si-
+multaneously to improve the decision-making performance
+of the agent. Firstly, the actions of the agent are divided
+into high quality actions and low quality actions according
+to the rewards returned from the environment. Then, the di-
+rector network is trained to have the ability to discriminate
+high and low quality actions and guide the actor network
+to reduce the repetitive exploration of low quality actions
+in the early stage of training. In addition, we propose an
+improved double estimator method to better solve the prob-
+lem of overestimation in the field of reinforcement learning.
+For the two critic networks used, we design two target critic
+networks for each critic network instead of one. In this way,
+the target value of each critic network can be calculated
+by taking the average of the outputs of the two target critic
+networks, which is more stable and accurate than using only
+one target critic network to obtain the target value. In order
+to verify the performance of the actor-director-critic frame-
+work and the improved double estimator method, we ap-
+plied them to the TD3 algorithm to improve the TD3 algo-
+rithm. Then, we carried out experiments in multiple envi-
+ronments in MuJoCo and compared the experimental data
+before and after the algorithm improvement. The final ex-
+perimental results show that the improved algorithm can
+achieve faster convergence speed and higher total return.
+1. Introduction
+Deep reinforcement learning is a trial-and-error ap-
+proach to learning, which takes the data obtained from the
+interaction between the agent and the environment as sam-
+ples for the agent’s policy learning. In the model-free re-
+inforcement learning algorithm, the agent’s cognition of
+the environment comes entirely from the sample data ob-
+tained during the interaction. It is this iterative trial-and-
+error learning method that makes a high percentage of low-
+quality data in a large amount of sample data, resulting in
+a poor utilization of the samples. Therefore, if there is a
+way to improve the sample utilization, the learning speed of
+reinforcement learning can be improved. There are many
+excellent existing model-free reinforcement learning algo-
+rithms that combine policy functions and value functions
+based on the actor-critic framework, which are applicable
+to a variety of action type problems with improved sam-
+ple training efficiency. In the actor-critic framework, actor
+learns parameterized policy function and critic learns pa-
+rameterized value function. The learned value function can
+provide more feedback information to the policy function
+than the environmental reward to guide the policy function
+in action decision making. Therefore, the learning of the
+policy function depends on the quality of the estimates of
+the value function that undergoes learning simultaneously.
+Because of this, it leads to the problem that until the value
+function can provide a reasonable signal to the policy func-
+tion, it is difficult for the agent to rely on the policy function
+to make good actions.
+In order to provide more guidance to the policy function
+before the value function can provide reasonable guidance
+signals for the policy function and speed up the learning
+of the policy function, we propose the actor-director-critic
+framework. Compared with actor-critic framework, actor-
+director-critic framework adds a director network with dis-
+criminative function to discriminate between low quality
+and high quality actions. Director can provide guidance
+other than critic to actor at the early stage of actor training,
+so that agent can reduce the repetitive exploration of low
+quality actions in the initial trial-and-error learning process,
+and further improve the training efficiency of samples. In
+the process of interacting with the environment, the agent
+receives a reward back from the environment for each ac-
+tion it performs. We simply classify the actions into two
+categories based on the reward returned, one is low quality
+actions with relatively low reward value, and the other is
+high quality actions with relatively high reward value. Be-
+arXiv:2301.03887v1  [cs.LG]  10 Jan 2023
+
+Figure 1. Implementation process of CTD3.
+fore training the director network, we divided the empirical
+dataset into two categories, a low quality dataset consisting
+of low quality actions and their corresponding rewards and
+states, and a high quality dataset consisting of high qual-
+ity actions and their corresponding rewards and states. We
+then use these two datasets above to train the ability of the
+director network to discriminate between high and low qual-
+ity actions, so that the director network can help the actor
+network to reduce the repetitive exploration of low quality
+actions during policy optimization. We set attenuation fac-
+tor so that the director focuses on the early stages of actor
+training.
+In addition, there is the problem of overestimation in re-
+inforcement learning. The overestimation problem is usu-
+ally caused by two reasons. One is the idea of maximiza-
+tion of the objective policy, where the maximum value of
+the estimated value function is used as the estimate of the
+true value, which leads to maximization bias and thus over-
+estimation. The second is the bootstrap phenomenon, which
+means using the predicted value of the value function at the
+future moment to update the estimated value of the value
+function at the present moment, which will also bring about
+the overestimation problem. The double estimator method
+is a common method to solve the overestimation problem.
+And existing method learns two independent Q networks,
+each of which has a target Q network. The target Q net-
+work copies its parameters from the Q network at regular
+intervals, and the output of the target Q network is then con-
+sidered as the target value to optimize the Q network. The
+problem is that the output of a single target Q is often un-
+stable, and this instability will reduce the accuracy of the
+target value and lower the training efficiency.
+In order to better solve the overestimation problem, we
+propose an improved double estimator method, which is to
+design two target Q networks for each Q network separately,
+and these two target Q networks update the parameters al-
+ternately according to a fixed time interval. The average
+of the outputs of the two target Q networks is used as the
+target value when optimizing the Q network. This method
+will obtain more stable and accurate target values than us-
+ing a single target Q network, and thus can better solve the
+overestimation problem.
+The TD3 [6] algorithm is an excellent performing al-
+gorithm based on the actor-critic framework. To test the
+performance of the above actor-director-critic framework
+and the improved double estimator method, we improve
+the TD3 algorithm using the actor-director-critic framework
+and the improved double estimator method, and refer to the
+improved algorithm as CTD3 algorithm for short. We con-
+ducted comparison experiments and ablation experiments
+in several environments in the MuJoCo, and all of them
+achieved good results.
+2. Related work
+2.1. Actor-Critic framework
+Model-free reinforcement learning algorithms can be
+classified as policy-based algorithms, value-based algo-
+rithms, and algorithms that combine policy and value. The
+
+agent forms a trajectory τ during its interaction with the
+environment, and the policy-based algorithm is to learn an
+action-generating function called policy function π, such
+that the agent is able to maximize the cumulative reward
+when making decisions under the guidance of this policy
+function. The cumulative reward J (τ) can be expressed as
+follows:
+J (τ) =
+K
+�
+k=0
+γkrt+k+1
+(1)
+The discount factor γ represents the effect of future re-
+wards on the present and takes a value between 0 and 1.
+At different moments, the agent is in different states, so the
+policy function π is meant to produce an action a = π(s)
+with state s as input, indicating the probability of taking ac-
+tion a in the case of state s. The policy-based algorithm
+directly optimizes the cumulative reward J (τ), which is of
+most interest to the agent. And policy-based algorithm can
+be applied to problems of either discrete or continuous ac-
+tion types, with the disadvantage that the sample training
+is relatively inefficient. The value function can provide in-
+formation about J (τ). The value function has two forms,
+one is the state value function, which indicates the expected
+value of the subsequent cumulative reward that can be ob-
+tained in a certain state s. The state value function V(s) can
+be expressed as follows:
+V(s) = E(
+K
+�
+k=0
+γkrt+k+1|St = s)
+(2)
+The other is the action value function, which represents
+the expected value of the cumulative reward that can be ob-
+tained after performing action a in some state s. The action
+value function Q(s, a) can be expressed as follows:
+Q(s, a) = E(
+K
+�
+k=0
+γkrt+k+1|St = s, At = a)
+(3)
+The value-based algorithm firstly learns a value function,
+and either the state value function or the action value func-
+tion ultimately uses the computed (s,a) to generate the opti-
+mal policy. The contribution of each action is measured by
+calculating and comparing the output of the function cor-
+responding to all possible actions in the current state. The
+agent should be in a state with high value as far as possible
+and choose actions with high value as far as possible, so as
+to achieve the purpose of maximizing cumulative rewards.
+The advantage of the value-based algorithm is that the sam-
+ple training is more efficient, and the disadvantage is that it
+is often limited to problems with discrete action types and
+is not suitable for problems where the target policy is a ran-
+dom one.
+The algorithm based on the actor-critic framework is an
+algorithm that combines the policy gradient method and the
+value function method, which can not only solve the prob-
+lem of multiple action types, but also the sample training
+efficiency is relatively high. Two parameterized functions
+need to be learned in the actor-critic framework, in which
+actor learns the parameterized policy function and critic
+learns the parameterized value function. The policy func-
+tion guides the agent to make action decisions, and the value
+function is responsible for evaluating the actions. Actor is a
+network representing the policy function, which selects the
+actions to be performed based on the current state informa-
+tion of the environment, and learns a better policy using the
+policy gradient under the guidance of critic. Actor’s opti-
+mization objective can be expressed as follows:
+maxJ = E[Q(s, a)]
+(4)
+Critic is a network representing the value function that
+learns a value function based on the data collected during
+the actor’s interaction with the environment, and helps the
+actor to make policy updates by evaluating the quality of the
+actor’s actions. The optimization objective of critic can be
+expressed as follows:
+minL = E[(Q(s, a) − (r + γQ(s′, a′)))2]
+(5)
+The learned value function can provide more feedback
+information to the policy function than the environmental
+reward, so the policy function can achieve better results
+when it learns based on the information provided by the
+value function that has completed learning.
+In the field
+of single-agent reinforcement learning, many existing ex-
+cellent algorithms use the actor-critic framework, such as
+DDPG [12], TRPO [17], A3C [14], PPO [18], ACER [20],
+SAC [8], D4PG [1], TD3 [6], DAC [24], TAAC [23], DCAC
+[2], ESAC [19], and so on. In the field of multi-agent rein-
+forcement learning, there are also many multi-agent algo-
+rithms that continue to use the actor-critic framework, such
+as MADDPG [13], COMA [5], IPPO [4], SEAC [3], CM3
+[21], MACAAC [15], MAPPO [22], HAPPO [11], and so
+on.
+In addition, the QMIX [16] algorithm also borrows
+ideas from the actor-critic framework, where the agent RNN
+network is equivalent to the actor network and the mix-
+ing network is equivalent to the critic network. Of course,
+there are also many algorithms that improve on the actor-
+critic framework, such as the multi-agent algorithm MAAC
+[10], which introduces the attention mechanism based on
+the actor-critic framework and proposes the actor-attention-
+critic framework, in the hope that the agent can selectively
+focus on the information that is more conducive to gaining
+greater returns when learning other agent policies.
+As stated in the introduction, reinforcement learning suf-
+fers from relatively low sample utilization and, for the actor-
+critic framework, it is difficult for an agent to rely on the
+
+policy function to make good actions until the value func-
+tion can provide a reasonable signal for the policy func-
+tion.
+Regarding the two roles in the actor-critic frame-
+work, we can regard the actor as the student and the critic
+as the teacher.
+In real life, a student often needs more
+teachers to teach him to learn more, and the more teach-
+ers a student has, the higher the probability of getting good
+grades. So the concept of multiple teachers can be intro-
+duced on the basis of the actor-critic framework. In the
+actor-director-critic framework proposed in this paper, the
+director is equivalent to another teacher for the actor, pro-
+viding guidance to the actor other than the critic and helping
+the actor learn better policies faster.
+2.2. Double estimator method
+Double estimator is a common method to solve the over-
+estimation problem, and two methods of double estimator
+will be described below. One is a method that has been
+used in some algorithms such as Double Q-Learning [9],
+which learns two independent Q networks, and each Q net-
+work uses the output value of the other during training as an
+estimate of the action value function of the optimal action
+sought by itself. This can be described as follows:
+y1 = r + γQ2(s′, a′)
+(6)
+y2 = r + γQ1(s′, a′)
+(7)
+Although two Q networks are used, only one Q network
+is updated at each update, and they both have a 50% prob-
+ability of being selected randomly. Another method, which
+has been used in some algorithms such as TD3 [6], also
+uses two Q networks that are independent of each other.
+The difference is that both networks are updated simulta-
+neously during the training process, using the minimum of
+their outputs as the estimate of the value function. More-
+over, when updating the network parameters, the estimate
+of the value function is shared between the two Q networks
+in the calculation of the loss function. This can be described
+as follows:
+y = r + γminQ1,2(s′, a′)
+(8)
+In the existing actor-critic framework, two relatively in-
+dependent critic networks need to be trained, and then the
+risk of overestimation is diluted by randomly selecting the
+output value of one of the critic networks or by taking the
+minimum value of the output of the two critic networks. In
+the two double estimator methods mentioned above, each
+critic network has only one target critic network. In order
+to better solve the overestimation problem, this paper pro-
+poses an improved double estimator method by designing
+two target critic networks for each critic network.
+Algorithm 1 CTD3
+Initialize the director network Dφ, critical network Qω1,
+Qω2 and actor network µθ with random parameters φ,
+ω1, ω2, θ
+Initialize target networks ω′
+1-1 ← ω1, ω′
+1-2 ← ω1,
+ω′
+2-1 ← ω2, ω′
+2-2 ← ω2, θ′ ← θ
+Initialize three replay buffers B, B1, B2
+for t = 1 to T do
+Select action with exploration noise
+�at = µθ(st) + ϵt, ϵt ← Nt(0, σ)
+Execute action �at to get reward rt and new state st+1
+Store transition tuple (st, at, rt, st+1) in B
+Choose a reward value cut-off of the right size, R
+if rt > R then
+at ∈ ah, Store (st, at, rt, st+1) in B1
+else
+at ∈ al, Store (st, at, rt, st+1) in B2
+Sample a random minibatch of N transitions
+(sj, aj, rj, sj+1) from B1
+Sample a random minibatch of N transitions
+(sk, ak, rk, sk+1) from B2
+Use (sj, aj, rj, sj+1), (sk, ak, rk, sk+1) to update
+director network Dφ:
+∇φV(φ) =
+1
+N
+N
+�
+i=1
+(∇φDφ(si, ahi) − ∇φDφ(si, ali))
+Sample a random minibatch of N transitions
+(si, ai, ri, si+1) from B
+�ai = µθ(si) + ϵi, ϵi ← Nt(0, σ)
+�
+ai+1 = µθ′(si+1) + ϵ′
+i, ϵ′
+i ← clip(Nt(0, σ′), -c, c)
+yi = ri+ γQmin(
+Qω′
+1-1
+(si+1, �
+ai+1)+Qω′
+1-2(si+1, �
+ai+1)
+2
+,
+Qω′
+2-1(si+1, �
+ai+1)+Qω′
+2-2(si+1, �
+ai+1)
+2
+)
+Use (si, ai, ri, si+1) to update critic network
+Qω1 or Qω2:
+∇ωL(ω) = 1
+N
+N
+�
+i=1
+(Qω(si, �ai) − yi)∇ωQω(si, �ai)
+if t mod d then
+Use (si, ai, ri, si+1) to update actor network µθ:
+∇θJ (θ) = 1
+N
+N
+�
+i=1
+(γD∇�aDφ(si, �ai)∇θµθ(si)+
+∇�aQω(si, �ai)∇θµθ(si))
+θ′ ← τθ + (1 − τ)θ′
+end if
+When t is odd, update target networks:
+ω′
+1-1 ← τω1 + (1 − τ)ω′
+1-1
+ω′
+1-2 ← τω1 + (1 − τ)ω′
+1-2
+When t is even, update target networks:
+ω′
+2-1 ← τω2 + (1 − τ)ω′
+2-1
+ω′
+2-2 ← τω2 + (1 − τ)ω′
+2-2
+end for
+
+2.3. Policy network delay update and target policy
+smoothing regularization
+Bias is introduced in the process of function approx-
+imation using neural networks, and the problem of error
+accumulation will come to the fore as the number of it-
+erations increases. In the existing actor-critic framework,
+if the critic’ assessments are inaccurate, the actor will not
+learn an accurate policy. In order to make the critic’s eval-
+uation more accurate, the frequency of critic updates can
+be increased and the frequency of actor updates can be de-
+creased, which is the method of delaying the update of the
+policy network used in the TD3 [6] algorithm. In our pro-
+posed actor-director-critic framework, we also adopt this
+approach, and the director network and the critic network
+converge earlier than the actor network, so that the actor
+network can be instructed to reduce unnecessary and ineffi-
+cient update operations when updating. Ultimately, training
+efficiency can be improved.
+In addition, the TD3 [6] algorithm also uses the process-
+ing of adding clipped noise to the target action to solve the
+problem of error accumulation, with the aim of smoothing
+the values of a small area around the target action enough
+so that those actions that are similar to the target action can
+have similar values to the target action. This smoothing op-
+eration based on noise processing can reduce the impact of
+certain wrong value estimates on the whole policy learning
+and reduce the generation of errors, so we retain this noise
+processing operation when improving the TD3 algorithm.
+This can be described as follows:
+ϵ ← clip(N(0, σ), -c, c)
+(9)
+a = π(s) + ϵ
+(10)
+3. Proposed method
+3.1. Actor-Director-Critic framework
+There are three roles in actor-director-critic framework:
+actor, director, and critic. As in actor-critic framework, the
+critic in the actor-director-critic framework is still a network
+representing a value function that learns a value function
+based on the data collected from the actor’s interaction with
+the environment. Critic plays a role in evaluating the long-
+term benefits of the actor’s actions by comparing the mag-
+nitude of the value function. Critic’s optimization goal can
+also be expressed as follows:
+minL = E[(Q(s, a) − (r + γQ(s′, a′)))2]
+(11)
+Director is a network that represents the discriminative
+function. In specific experiments, we classify the actions
+based on the feedback rewards as simple superiority or in-
+feriority. We obtain a low quality dataset and a high quality
+dataset, and then train the discriminative ability of the direc-
+tor network on the actions. We introduced a decay factor to
+make the director work mainly in the initial phase of train-
+ing. In the initial stage of training, the actor can quickly
+recognize which actions are worth exploring and which ac-
+tions should be avoided with the help of the director, and
+then quickly learn to reduce the exploration of those poorer
+actions and improve the training speed. The optimization
+objective of the director is as follows:
+maxV = Eah∼pdata(ah)[D(s, ah)]+
+Eal∼pdata(al)[1 − D(s, al)]
+(12)
+The actor is still a network that represents the policy
+function and chooses the actions to be performed based on
+the state information of the current environment. At this
+point, it no longer receives guidance only from the critic, but
+also from the director. The critic is the lifelong teacher who
+teaches the actor to make action decisions in the long-term
+interest, while the director, by design, is the stage teacher
+who primarily guides the actor in the early stages of learn-
+ing to quickly distinguish between the low quality actions
+and high quality actions. Under the joint guidance of both,
+the actor learns the optimal policy much faster. γ is a decay
+factor, and actor’s optimization goals can be described as
+follows:
+maxJ = γE[D(s, a)] + E[Q(s, a)]
+(13)
+3.2. Improved double estimator method
+In the actor-director-critic framework, we design two
+critic networks and two target critic networks for each critic
+network. Target critic networks are independent of each
+other and do not need to be trained. We only need to copy
+the parameters of the critic network to the target critic net-
+work in a soft update manner periodically. It should be
+noted that for each critic network, the two target critic net-
+works are updated alternately, and only one of the target
+networks is updated at a time. When optimizing the two
+critic networks used, they share the same target value. The
+target value is calculated by first averaging the outputs of
+the two target critic networks for each critic network and
+then taking the minimum of the two averages, as shown in
+the following equation:
+Q′ = min(Q1-1(s′, a′) + Q1-2(s′, a′)
+2
+,
+Q2-1(s′, a′) + Q2-2(s′, a′)
+2
+)
+(14)
+The advantage of this design is that precisely because the
+alternate updates of the target critic networks make them
+represent the training results of the critic network in differ-
+ent historical periods, the average of the target values cal-
+culated by the training results in different historical periods
+
+can better dilute some prediction biases and thus improve
+the stability and accuracy of the target values.
+3.3. Classified twin delayed deep deterministic pol-
+icy gradient algorithm
+As mentioned in the introduction, in order to test the per-
+formance of the above actor-director-critic framework and
+the improved double estimator method, we apply them to
+the TD3 algorithm to improve the TD3 algorithm. The TD3
+algorithm is chosen because it is a well-performing algo-
+rithm based on the actor-critic framework. The full name of
+the TD3 algorithm is twin delayed deep deterministic policy
+gradient algorithm, and we refer to the improved algorithm
+as classified twin delayed deep deterministic policy gradi-
+ent algorithm, abbreviated as CTD3. The implementation
+process of CTD3 is shown in Figure 1.
+The actor makes action choices based on the state of the
+environment it observes. There are two critic networks, and
+each critic network has two target critic networks. The pa-
+rameters of the target critic networks are copied from the
+critic network in the way of soft updating alternately. To
+solve the overestimation problem, the minimum value of
+the output of the two critic networks is chosen as a refer-
+ence to evaluate the actor’s action in the long term interest.
+In addition, we select appropriate reward value cut-off to
+classify actions into low quality actions and high quality
+actions based on the reward values returned from the envi-
+ronment, and use low-quality and high-quality datasets to
+train the discriminative ability of the director. Then director
+can guide the actor to reduce repetitive exploration of low
+quality actions during training.
+In the CTD3 algorithm we keep the two methods used
+in the TD3 algorithm to solve the error accumulation prob-
+lem. One is to delay the update of the policy network, that
+is, to reduce the frequency of actor updates, and the other
+is the smoothing operation of the target policy, that is, to
+add clipped noise. The way to add clipped noise can be
+described as follows:
+ϵ ← N(0, σ)
+(15)
+�a = µθ(s) + ϵ
+(16)
+ϵ′ ← clip(N(0, σ′), -c, c)
+(17)
+�a′ = µθ′(s′) + ϵ′
+(18)
+The target action of adding truncated noise will be used
+for the calculation of the target value of the value function.
+Two target critic networks are designed for each critic net-
+work to solve the overestimation problem.
+Q′ = min(Qω′
+1-1(s′, �a′) + Qω′
+1-2(s′, �a′)
+2
+,
+Qω′
+2-1(s′, �a′) + Qω′
+2-2(s′, �a′)
+2
+)
+(19)
+y = r + γQQ′
+(20)
+The optimization objective of the critic network is to
+minimize the loss between the predicted target value y and
+the output value Qω(s, �a).
+L(ω) = E[(Qω(s, �a) − y)2]
+(21)
+∇ωL(ω) = 1
+N
+N
+�
+i=1
+(Qω(si, �ai) − yi)∇ωQω(si, �ai)
+(22)
+yi = ri + γQmin(
+Qω′
+1-1(s′
+i, �a′
+i) + Qω′
+1-2(s′
+i, �a′
+i)
+2
+,
+Qω′
+2-1(s′
+i, �a′
+i) + Qω′
+2-2(s′
+i, �a′
+i)
+2
+)
+(23)
+The director network learns to discriminate between high
+and low quality actions through different datasets, with the
+output being 1 when the input is a high quality action and 0
+when the input is a low quality action. Director network is
+designed with reference to the concept of GAN [7], and its
+goal is to perform the learning of the discriminant function
+V(φ).
+V(φ) = Eah∼pdata(ah)[Dφ(s, ah)]+
+Eal∼pdata(al)[1 − Dφ(s, al)]
+(24)
+∇φV(φ) = 1
+N
+N
+�
+i=1
+(∇φDφ(si, ahi) − ∇φDφ(si, ali)) (25)
+The actor network performs policy optimization with the
+joint help of the critic network and the director network,
+and the optimization goal is to maximize the value of the
+value function output by the critic network as well as to
+maximize the value of the discriminant function output by
+the director network. Here the decay factor γD is used to
+make the director focus on the early stage of training and
+speed up the training.
+J (θ) = γDE[Dφ(s, �a)] + E[Qω(s, �a)]
+(26)
+∇θJ (θ) = 1
+N
+N
+�
+i=1
+(γD∇�aDφ(si, �ai)∇θµθ(si)+
+∇�aQω(si, �ai)∇θµθ(si))
+(27)
+CTD3 is summarized in Algorithm 1.
+
+4. Experiment
+4.1. Experimental environments
+Figure 2. Six experimental environments in MuJoCo.
+We conducted experiments in MuJoCo simulation envi-
+ronment. The six experimental environments selected in
+Figure 2 are all continuous action environments and are rich
+in reward value settings.
+4.2. Comparative evaluation and ablation study
+As already mentioned, in order to test the effectiveness
+of the proposed actor-director-critic framework and the im-
+proved double estimator method, we apply them to the TD3
+algorithm. CTD3 algorithm is obtained by improving TD3.
+First of all, we would like to explain why we chose TD3
+algorithm for comparison, because TD3 is a very good re-
+inforcement learning algorithm based on actor-critic frame-
+work. In the original paper of the TD3 [6] algorithm pub-
+lished in ICML, this method can completely beat DDPG,
+PPO, TRPO, ACKTR, and SAC. Therefore, we choose the
+TD3 algorithm for comparison experiments.
+We conduct comparative experiments on CTD3 and TD3
+in six environments under the MuJoCo platform. For each
+environment a specific random seed is set and the maxi-
+mum number of experimental steps is limited. The data are
+smoothed at the end of the experiments, using the sliding
+average filtering method. A suitable sliding window size is
+set for each data of different environments, and the data in
+the window are averaged. All data are smoothed when the
+window is sliding from the beginning to the end. On the one
+hand, the smoothed data are written to Table 1, and it should
+be noted that the data of the six environments written to Ta-
+ble 1 are the data at the end of the last training round run
+by agent, both for CTD3 and TD3. On the other hand, all
+the smoothed data of CTD3 and TD3 are plotted as curves
+to obtain Figure 3, where the training process of CTD3 and
+TD3 in each environment can be visually compared.
+CTD3 is a new algorithm that improves TD3 by apply-
+ing the two ideas of actor-director-critic framework and the
+improved double estimator method proposed in this paper
+to the TD3 algorithm. We can compare the performance of
+CTD3 and TD3 by comparing the data in Table 1 and the
+curves in Figure 3. In all the six environments, the total
+return of CTD3 is higher than that of TD3, and the conver-
+gence rate of CTD3 is also faster than that of TD3. Since
+the performance of the CTD3 algorithm is better than TD3
+in terms of convergence speed and final score, it can prove
+that the actor-director-critic framework and the improved
+double estimator method proposed in this paper are feasi-
+ble.
+TD3+ADCF means that the ADC framework is ap-
+plied to the TD3 algorithm to improve the TD3 algo-
+rithm. TD3+IDEM means that the improved double esti-
+mator method is applied to the TD3 algorithm to improve
+the TD3 algorithm. CTD3(TD3+ADCF+IDEM) means that
+the ADC framework and the improved double estimator
+method are applied to the TD3 algorithm at the same time
+to improve the TD3 algorithm and obtain the CTD3 algo-
+rithm. For ablation experiments, we can analyze the per-
+formance of the ADC framework and the improved double
+estimator method proposed in this paper by comparing the
+data of TD3, TD3+ADCF, TD3+IDEM and CTD3 in Ta-
+ble 1 and the curves in Figure 3. As shown in Table 1, in
+all six experimental environments, no matter TD3+ADCF
+or TD3+IDEM, their total returns are higher than those
+of TD3.
+The curve in Figure 3 also shows that in all
+six experimental environments, no matter TD3+ADCF or
+TD3+IDEM, their convergence speed is faster than that of
+TD3. In most cases, CTD3 outperforms TD3+ADCF and
+TD3+IDEM, but sometimes it is outperformed by one or
+the other.
+In conclusion, the improved performance compared to
+TD3, either using the actor-director-critic framework alone
+or the improved double estimator method alone, or use both
+of them, are sufficient to demonstrate the feasibility of our
+proposed methods in this paper.
+5. Conclusion
+Reinforcement learning is a kind of trial-and-error learn-
+ing, which has the problem of low sample utilization. More-
+over, in existing deep reinforcement learning algorithms
+based on the actor-critic framework, there is also a problem
+that it is difficult for an agent to rely on the policy function
+to make good actions until the value function is learned to
+the extent that it can provide reasonable guidance to the pol-
+icy function. Therefore, in order to alleviate these two prob-
+lems, this paper proposes a new deep reinforcement learn-
+ing framework, actor-director-critic framework. The direc-
+tor role is added to the actor-critic framework, and a decay
+factor is set for the director. By training the director to dis-
+
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+Figure 3. Performance curves in six environments.
+Environment
+TD3
+TD3+ADCF
+TD3+IDEM
+CTD3(TD3+ADCF+IDEM)
+InvertedDoublePendulum
+9341.302
+9342.752
+9350.902
+9342.690
+HalfCheetah
+8214.770
+9460.794
+9156.181
+10324.831
+Humanoid
+4755.148
+5034.893
+5085.446
+5057.048
+Ant
+2949.150
+4861.062
+4779.486
+5171.217
+Walker2d
+3818.958
+4902.341
+4208.518
+4468.348
+Hopper
+2395.742
+3169.502
+3118.340
+3252.941
+Table 1. Experimental results in six environments.
+criminate between low quality and high quality actions, it
+can help the actor reduce repetitive attempts to perform low
+quality actions during exploration, thus improving sample
+utilization and speeding up training.
+In solving the problem of overestimation in reinforce-
+ment learning, for the method called double estimator
+method that requires training two Q networks and design-
+ing one target network for each Q network, this paper tries
+to propose an improved double estimator method to design
+two target networks for each Q network. For each Q net-
+work, its two target networks are alternately soft updated.
+The outputs of these target networks are comprehensively
+used to calculate the target value of Q network. First, the
+output average of the two target networks owned by each Q
+network is calculated, and then the smaller of the two aver-
+age values is taken as the final target value of the Q network,
+and this target value is shared by both Q networks. so as to
+alleviate the problem of overestimation. So that the problem
+of overvaluation can be mitigated much better than before.
+To test the feasibility of both the actor-director-critic
+framework and the improved double estimator method, we
+applied them to the TD3 algorithm and improved the TD3
+algorithm to obtain the CTD3 algorithm. We conducted
+comparative and ablation experiments in several environ-
+ments in the MuJoCo. Experimental results show that the
+CTD3 algorithm generally has faster convergence speed and
+higher total return than TD3, proving the feasibility of the
+methods proposed in this paper.
+
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+page_content='Actor-Director-Critic: A Novel Deep Reinforcement Learning Framework  Zongwei Liu  Xi’an Jiaotong University  China  Ting Haode@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content='cn  Yonghong Song  Xi’an Jiaotong University  China  songyh@xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='cn  Yuanlin Zhang  Xi’an Jiaotong University  China  ylzhangxian@xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='cn  Abstract  In this paper, we propose actor-director-critic, a new  framework for deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Compared with  the actor-critic framework, the director role is added, and  action classification and action evaluation are applied si-  multaneously to improve the decision-making performance  of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Firstly, the actions of the agent are divided  into high quality actions and low quality actions according  to the rewards returned from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Then, the di-  rector network is trained to have the ability to discriminate  high and low quality actions and guide the actor network  to reduce the repetitive exploration of low quality actions  in the early stage of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In addition, we propose an  improved double estimator method to better solve the prob-  lem of overestimation in the field of reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  For the two critic networks used, we design two target critic  networks for each critic network instead of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In this way,  the target value of each critic network can be calculated  by taking the average of the outputs of the two target critic  networks, which is more stable and accurate than using only  one target critic network to obtain the target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In order  to verify the performance of the actor-director-critic frame-  work and the improved double estimator method, we ap-  plied them to the TD3 algorithm to improve the TD3 algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Then, we carried out experiments in multiple envi-  ronments in MuJoCo and compared the experimental data  before and after the algorithm improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The final ex-  perimental results show that the improved algorithm can  achieve faster convergence speed and higher total return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Introduction  Deep reinforcement learning is a trial-and-error ap-  proach to learning, which takes the data obtained from the  interaction between the agent and the environment as sam-  ples for the agent’s policy learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In the model-free re-  inforcement learning algorithm, the agent’s cognition of  the environment comes entirely from the sample data ob-  tained during the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' It is this iterative trial-and-  error learning method that makes a high percentage of low-  quality data in a large amount of sample data, resulting in  a poor utilization of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Therefore, if there is a  way to improve the sample utilization, the learning speed of  reinforcement learning can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' There are many  excellent existing model-free reinforcement learning algo-  rithms that combine policy functions and value functions  based on the actor-critic framework, which are applicable  to a variety of action type problems with improved sam-  ple training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In the actor-critic framework, actor  learns parameterized policy function and critic learns pa-  rameterized value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The learned value function can  provide more feedback information to the policy function  than the environmental reward to guide the policy function  in action decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Therefore, the learning of the  policy function depends on the quality of the estimates of  the value function that undergoes learning simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  Because of this, it leads to the problem that until the value  function can provide a reasonable signal to the policy func-  tion, it is difficult for the agent to rely on the policy function  to make good actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  In order to provide more guidance to the policy function  before the value function can provide reasonable guidance  signals for the policy function and speed up the learning  of the policy function, we propose the actor-director-critic  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Compared with actor-critic framework, actor-  director-critic framework adds a director network with dis-  criminative function to discriminate between low quality  and high quality actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Director can provide guidance  other than critic to actor at the early stage of actor training,  so that agent can reduce the repetitive exploration of low  quality actions in the initial trial-and-error learning process,  and further improve the training efficiency of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In  the process of interacting with the environment, the agent  receives a reward back from the environment for each ac-  tion it performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' We simply classify the actions into two  categories based on the reward returned, one is low quality  actions with relatively low reward value, and the other is  high quality actions with relatively high reward value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Be-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='03887v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='LG]  10 Jan 2023  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Implementation process of CTD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  fore training the director network, we divided the empirical  dataset into two categories, a low quality dataset consisting  of low quality actions and their corresponding rewards and  states, and a high quality dataset consisting of high qual-  ity actions and their corresponding rewards and states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' We  then use these two datasets above to train the ability of the  director network to discriminate between high and low qual-  ity actions, so that the director network can help the actor  network to reduce the repetitive exploration of low quality  actions during policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' We set attenuation fac-  tor so that the director focuses on the early stages of actor  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  In addition, there is the problem of overestimation in re-  inforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The overestimation problem is usu-  ally caused by two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' One is the idea of maximiza-  tion of the objective policy, where the maximum value of  the estimated value function is used as the estimate of the  true value, which leads to maximization bias and thus over-  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The second is the bootstrap phenomenon, which  means using the predicted value of the value function at the  future moment to update the estimated value of the value  function at the present moment, which will also bring about  the overestimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The double estimator method  is a common method to solve the overestimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  And existing method learns two independent Q networks,  each of which has a target Q network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The target Q net-  work copies its parameters from the Q network at regular  intervals, and the output of the target Q network is then con-  sidered as the target value to optimize the Q network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The  problem is that the output of a single target Q is often un-  stable, and this instability will reduce the accuracy of the  target value and lower the training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  In order to better solve the overestimation problem, we  propose an improved double estimator method, which is to  design two target Q networks for each Q network separately,  and these two target Q networks update the parameters al-  ternately according to a fixed time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The average  of the outputs of the two target Q networks is used as the  target value when optimizing the Q network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' This method  will obtain more stable and accurate target values than us-  ing a single target Q network, and thus can better solve the  overestimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  The TD3 [6] algorithm is an excellent performing al-  gorithm based on the actor-critic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' To test the  performance of the above actor-director-critic framework  and the improved double estimator method, we improve  the TD3 algorithm using the actor-director-critic framework  and the improved double estimator method, and refer to the  improved algorithm as CTD3 algorithm for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' We con-  ducted comparison experiments and ablation experiments  in several environments in the MuJoCo, and all of them  achieved good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Related work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Actor-Critic framework  Model-free reinforcement learning algorithms can be  classified as policy-based algorithms, value-based algo-  rithms, and algorithms that combine policy and value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The  agent forms a trajectory τ during its interaction with the  environment, and the policy-based algorithm is to learn an  action-generating function called policy function π, such  that the agent is able to maximize the cumulative reward  when making decisions under the guidance of this policy  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The cumulative reward J (τ) can be expressed as  follows:  J (τ) =  K  �  k=0  γkrt+k+1  (1)  The discount factor γ represents the effect of future re-  wards on the present and takes a value between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  At different moments, the agent is in different states, so the  policy function π is meant to produce an action a = π(s)  with state s as input, indicating the probability of taking ac-  tion a in the case of state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The policy-based algorithm  directly optimizes the cumulative reward J (τ), which is of  most interest to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' And policy-based algorithm can  be applied to problems of either discrete or continuous ac-  tion types, with the disadvantage that the sample training  is relatively inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The value function can provide in-  formation about J (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The value function has two forms,  one is the state value function, which indicates the expected  value of the subsequent cumulative reward that can be ob-  tained in a certain state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The state value function V(s) can  be expressed as follows:  V(s) = E(  K  �  k=0  γkrt+k+1|St = s)  (2)  The other is the action value function, which represents  the expected value of the cumulative reward that can be ob-  tained after performing action a in some state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The action  value function Q(s, a) can be expressed as follows:  Q(s, a) = E(  K  �  k=0  γkrt+k+1|St = s, At = a)  (3)  The value-based algorithm firstly learns a value function,  and either the state value function or the action value func-  tion ultimately uses the computed (s,a) to generate the opti-  mal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The contribution of each action is measured by  calculating and comparing the output of the function cor-  responding to all possible actions in the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The  agent should be in a state with high value as far as possible  and choose actions with high value as far as possible, so as  to achieve the purpose of maximizing cumulative rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  The advantage of the value-based algorithm is that the sam-  ple training is more efficient, and the disadvantage is that it  is often limited to problems with discrete action types and  is not suitable for problems where the target policy is a ran-  dom one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  The algorithm based on the actor-critic framework is an  algorithm that combines the policy gradient method and the  value function method, which can not only solve the prob-  lem of multiple action types, but also the sample training  efficiency is relatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Two parameterized functions  need to be learned in the actor-critic framework, in which  actor learns the parameterized policy function and critic  learns the parameterized value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The policy func-  tion guides the agent to make action decisions, and the value  function is responsible for evaluating the actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Actor is a  network representing the policy function, which selects the  actions to be performed based on the current state informa-  tion of the environment, and learns a better policy using the  policy gradient under the guidance of critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Actor’s opti-  mization objective can be expressed as follows:  maxJ = E[Q(s, a)]  (4)  Critic is a network representing the value function that  learns a value function based on the data collected during  the actor’s interaction with the environment, and helps the  actor to make policy updates by evaluating the quality of the  actor’s actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The optimization objective of critic can be  expressed as follows:  minL = E[(Q(s, a) − (r + γQ(s′, a′)))2]  (5)  The learned value function can provide more feedback  information to the policy function than the environmental  reward, so the policy function can achieve better results  when it learns based on the information provided by the  value function that has completed learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  In the field  of single-agent reinforcement learning, many existing ex-  cellent algorithms use the actor-critic framework, such as  DDPG [12], TRPO [17], A3C [14], PPO [18], ACER [20],  SAC [8], D4PG [1], TD3 [6], DAC [24], TAAC [23], DCAC  [2], ESAC [19], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In the field of multi-agent rein-  forcement learning, there are also many multi-agent algo-  rithms that continue to use the actor-critic framework, such  as MADDPG [13], COMA [5], IPPO [4], SEAC [3], CM3  [21], MACAAC [15], MAPPO [22], HAPPO [11], and so  on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  In addition, the QMIX [16] algorithm also borrows  ideas from the actor-critic framework, where the agent RNN  network is equivalent to the actor network and the mix-  ing network is equivalent to the critic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Of course,  there are also many algorithms that improve on the actor-  critic framework, such as the multi-agent algorithm MAAC  [10], which introduces the attention mechanism based on  the actor-critic framework and proposes the actor-attention-  critic framework, in the hope that the agent can selectively  focus on the information that is more conducive to gaining  greater returns when learning other agent policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  As stated in the introduction, reinforcement learning suf-  fers from relatively low sample utilization and, for the actor-  critic framework, it is difficult for an agent to rely on the  policy function to make good actions until the value func-  tion can provide a reasonable signal for the policy func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  Regarding the two roles in the actor-critic frame-  work, we can regard the actor as the student and the critic  as the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  In real life, a student often needs more  teachers to teach him to learn more, and the more teach-  ers a student has, the higher the probability of getting good  grades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' So the concept of multiple teachers can be intro-  duced on the basis of the actor-critic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In the  actor-director-critic framework proposed in this paper, the  director is equivalent to another teacher for the actor, pro-  viding guidance to the actor other than the critic and helping  the actor learn better policies faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Double estimator method  Double estimator is a common method to solve the over-  estimation problem, and two methods of double estimator  will be described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' One is a method that has been  used in some algorithms such as Double Q-Learning [9],  which learns two independent Q networks, and each Q net-  work uses the output value of the other during training as an  estimate of the action value function of the optimal action  sought by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' This can be described as follows:  y1 = r + γQ2(s′, a′)  (6)  y2 = r + γQ1(s′, a′)  (7)  Although two Q networks are used, only one Q network  is updated at each update, and they both have a 50% prob-  ability of being selected randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Another method, which  has been used in some algorithms such as TD3 [6], also  uses two Q networks that are independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  The difference is that both networks are updated simulta-  neously during the training process, using the minimum of  their outputs as the estimate of the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' More-  over, when updating the network parameters, the estimate  of the value function is shared between the two Q networks  in the calculation of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' This can be described  as follows:  y = r + γminQ1,2(s′, a′)  (8)  In the existing actor-critic framework, two relatively in-  dependent critic networks need to be trained, and then the  risk of overestimation is diluted by randomly selecting the  output value of one of the critic networks or by taking the  minimum value of the output of the two critic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In  the two double estimator methods mentioned above, each  critic network has only one target critic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In order  to better solve the overestimation problem, this paper pro-  poses an improved double estimator method by designing  two target critic networks for each critic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  Algorithm 1 CTD3  Initialize the director network Dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' critical network Qω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  Qω2 and actor network µθ with random parameters φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' θ  Initialize target networks ω′  1-1 ← ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ω′  1-2 ← ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  ω′  2-1 ← ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ω′  2-2 ← ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' θ′ ← θ  Initialize three replay buffers B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' B1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' B2  for t = 1 to T do  Select action with exploration noise  �at = µθ(st) + ϵt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ϵt ← Nt(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' σ)  Execute action �at to get reward rt and new state st+1  Store transition tuple (st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' rt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' st+1) in B  Choose a reward value cut-off of the right size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' R  if rt > R then  at ∈ ah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Store (st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' rt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' st+1) in B1  else  at ∈ al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Store (st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' rt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' st+1) in B2  Sample a random minibatch of N transitions  (sj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' rj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' sj+1) from B1  Sample a random minibatch of N transitions  (sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' rk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' sk+1) from B2  Use (sj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' rj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' sj+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' (sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' rk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' sk+1) to update  director network Dφ:  ∇φV(φ) =  1  N  N  �  i=1  (∇φDφ(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ahi) − ∇φDφ(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ali))  Sample a random minibatch of N transitions  (si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' si+1) from B  �ai = µθ(si) + ϵi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ϵi ← Nt(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' σ)  �  ai+1 = µθ′(si+1) + ϵ′  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ϵ′  i ← clip(Nt(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' σ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' -c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' c)  yi = ri+ γQmin(  Qω′  1-1  (si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' �  ai+1)+Qω′  1-2(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' �  ai+1)  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  Qω′  2-1(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' �  ai+1)+Qω′  2-2(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' �  ai+1)  2  )  Use (si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' si+1) to update critic network  Qω1 or Qω2:  ∇ωL(ω) = 1  N  N  �  i=1  (Qω(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' �ai) − yi)∇ωQω(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' �ai)  if t mod d then  Use (si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' si+1) to update actor network µθ:  ∇θJ (θ) = 1  N  N  �  i=1  (γD∇�aDφ(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' �ai)∇θµθ(si)+  ∇�aQω(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' �ai)∇θµθ(si))  θ′ ← τθ + (1 − τ)θ′  end if  When t is odd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' update target networks:  ω′  1-1 ← τω1 + (1 − τ)ω′  1-1  ω′  1-2 ← τω1 + (1 − τ)ω′  1-2  When t is even,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' update target networks:  ω′  2-1 ← τω2 + (1 − τ)ω′  2-1  ω′  2-2 ← τω2 + (1 − τ)ω′  2-2  end for  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Policy network delay update and target policy  smoothing regularization  Bias is introduced in the process of function approx-  imation using neural networks, and the problem of error  accumulation will come to the fore as the number of it-  erations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In the existing actor-critic framework,  if the critic’ assessments are inaccurate, the actor will not  learn an accurate policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In order to make the critic’s eval-  uation more accurate, the frequency of critic updates can  be increased and the frequency of actor updates can be de-  creased, which is the method of delaying the update of the  policy network used in the TD3 [6] algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In our pro-  posed actor-director-critic framework, we also adopt this  approach, and the director network and the critic network  converge earlier than the actor network, so that the actor  network can be instructed to reduce unnecessary and ineffi-  cient update operations when updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Ultimately, training  efficiency can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  In addition, the TD3 [6] algorithm also uses the process-  ing of adding clipped noise to the target action to solve the  problem of error accumulation, with the aim of smoothing  the values of a small area around the target action enough  so that those actions that are similar to the target action can  have similar values to the target action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' This smoothing op-  eration based on noise processing can reduce the impact of  certain wrong value estimates on the whole policy learning  and reduce the generation of errors, so we retain this noise  processing operation when improving the TD3 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  This can be described as follows:  ϵ ← clip(N(0, σ), -c, c)  (9)  a = π(s) + ϵ  (10)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Proposed method  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Actor-Director-Critic framework  There are three roles in actor-director-critic framework:  actor, director, and critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' As in actor-critic framework, the  critic in the actor-director-critic framework is still a network  representing a value function that learns a value function  based on the data collected from the actor’s interaction with  the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Critic plays a role in evaluating the long-  term benefits of the actor’s actions by comparing the mag-  nitude of the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Critic’s optimization goal can  also be expressed as follows:  minL = E[(Q(s, a) − (r + γQ(s′, a′)))2]  (11)  Director is a network that represents the discriminative  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In specific experiments, we classify the actions  based on the feedback rewards as simple superiority or in-  feriority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' We obtain a low quality dataset and a high quality  dataset, and then train the discriminative ability of the direc-  tor network on the actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' We introduced a decay factor to  make the director work mainly in the initial phase of train-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In the initial stage of training, the actor can quickly  recognize which actions are worth exploring and which ac-  tions should be avoided with the help of the director, and  then quickly learn to reduce the exploration of those poorer  actions and improve the training speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The optimization  objective of the director is as follows:  maxV = Eah∼pdata(ah)[D(s, ah)]+  Eal∼pdata(al)[1 − D(s, al)]  (12)  The actor is still a network that represents the policy  function and chooses the actions to be performed based on  the state information of the current environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' At this  point, it no longer receives guidance only from the critic, but  also from the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The critic is the lifelong teacher who  teaches the actor to make action decisions in the long-term  interest, while the director, by design, is the stage teacher  who primarily guides the actor in the early stages of learn-  ing to quickly distinguish between the low quality actions  and high quality actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Under the joint guidance of both,  the actor learns the optimal policy much faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' γ is a decay  factor, and actor’s optimization goals can be described as  follows:  maxJ = γE[D(s, a)] + E[Q(s, a)]  (13)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Improved double estimator method  In the actor-director-critic framework, we design two  critic networks and two target critic networks for each critic  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Target critic networks are independent of each  other and do not need to be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' We only need to copy  the parameters of the critic network to the target critic net-  work in a soft update manner periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' It should be  noted that for each critic network, the two target critic net-  works are updated alternately, and only one of the target  networks is updated at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' When optimizing the two  critic networks used, they share the same target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The  target value is calculated by first averaging the outputs of  the two target critic networks for each critic network and  then taking the minimum of the two averages,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' as shown in  the following equation:  Q′ = min(Q1-1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' a′) + Q1-2(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' a′)  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  Q2-1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' a′) + Q2-2(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' a′)  2  )  (14)  The advantage of this design is that precisely because the  alternate updates of the target critic networks make them  represent the training results of the critic network in differ-  ent historical periods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' the average of the target values cal-  culated by the training results in different historical periods  can better dilute some prediction biases and thus improve  the stability and accuracy of the target values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Classified twin delayed deep deterministic pol-  icy gradient algorithm  As mentioned in the introduction, in order to test the per-  formance of the above actor-director-critic framework and  the improved double estimator method, we apply them to  the TD3 algorithm to improve the TD3 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The TD3  algorithm is chosen because it is a well-performing algo-  rithm based on the actor-critic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The full name of  the TD3 algorithm is twin delayed deep deterministic policy  gradient algorithm, and we refer to the improved algorithm  as classified twin delayed deep deterministic policy gradi-  ent algorithm, abbreviated as CTD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The implementation  process of CTD3 is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  The actor makes action choices based on the state of the  environment it observes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' There are two critic networks, and  each critic network has two target critic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The pa-  rameters of the target critic networks are copied from the  critic network in the way of soft updating alternately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' To  solve the overestimation problem, the minimum value of  the output of the two critic networks is chosen as a refer-  ence to evaluate the actor’s action in the long term interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  In addition, we select appropriate reward value cut-off to  classify actions into low quality actions and high quality  actions based on the reward values returned from the envi-  ronment, and use low-quality and high-quality datasets to  train the discriminative ability of the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Then director  can guide the actor to reduce repetitive exploration of low  quality actions during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  In the CTD3 algorithm we keep the two methods used  in the TD3 algorithm to solve the error accumulation prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' One is to delay the update of the policy network, that  is, to reduce the frequency of actor updates, and the other  is the smoothing operation of the target policy, that is, to  add clipped noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The way to add clipped noise can be  described as follows:  ϵ ← N(0, σ)  (15)  �a = µθ(s) + ϵ  (16)  ϵ′ ← clip(N(0, σ′), -c, c)  (17)  �a′ = µθ′(s′) + ϵ′  (18)  The target action of adding truncated noise will be used  for the calculation of the target value of the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  Two target critic networks are designed for each critic net-  work to solve the overestimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  Q′ = min(Qω′  1-1(s′, �a′) + Qω′  1-2(s′, �a′)  2  ,  Qω′  2-1(s′, �a′) + Qω′  2-2(s′, �a′)  2  )  (19)  y = r + γQQ′  (20)  The optimization objective of the critic network is to  minimize the loss between the predicted target value y and  the output value Qω(s, �a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  L(ω) = E[(Qω(s, �a) − y)2]  (21)  ∇ωL(ω) = 1  N  N  �  i=1  (Qω(si, �ai) − yi)∇ωQω(si, �ai)  (22)  yi = ri + γQmin(  Qω′  1-1(s′  i, �a′  i) + Qω′  1-2(s′  i, �a′  i)  2  ,  Qω′  2-1(s′  i, �a′  i) + Qω′  2-2(s′  i, �a′  i)  2  )  (23)  The director network learns to discriminate between high  and low quality actions through different datasets, with the  output being 1 when the input is a high quality action and 0  when the input is a low quality action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Director network is  designed with reference to the concept of GAN [7], and its  goal is to perform the learning of the discriminant function  V(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  V(φ) = Eah∼pdata(ah)[Dφ(s, ah)]+  Eal∼pdata(al)[1 − Dφ(s, al)]  (24)  ∇φV(φ) = 1  N  N  �  i=1  (∇φDφ(si, ahi) − ∇φDφ(si, ali)) (25)  The actor network performs policy optimization with the  joint help of the critic network and the director network,  and the optimization goal is to maximize the value of the  value function output by the critic network as well as to  maximize the value of the discriminant function output by  the director network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Here the decay factor γD is used to  make the director focus on the early stage of training and  speed up the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  J (θ) = γDE[Dφ(s, �a)] + E[Qω(s, �a)]  (26)  ∇θJ (θ) = 1  N  N  �  i=1  (γD∇�aDφ(si, �ai)∇θµθ(si)+  ∇�aQω(si, �ai)∇θµθ(si))  (27)  CTD3 is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Experiment  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Experimental environments  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Six experimental environments in MuJoCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  We conducted experiments in MuJoCo simulation envi-  ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The six experimental environments selected in  Figure 2 are all continuous action environments and are rich  in reward value settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Comparative evaluation and ablation study  As already mentioned, in order to test the effectiveness  of the proposed actor-director-critic framework and the im-  proved double estimator method, we apply them to the TD3  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' CTD3 algorithm is obtained by improving TD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  First of all, we would like to explain why we chose TD3  algorithm for comparison, because TD3 is a very good re-  inforcement learning algorithm based on actor-critic frame-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In the original paper of the TD3 [6] algorithm pub-  lished in ICML, this method can completely beat DDPG,  PPO, TRPO, ACKTR, and SAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Therefore, we choose the  TD3 algorithm for comparison experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  We conduct comparative experiments on CTD3 and TD3  in six environments under the MuJoCo platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' For each  environment a specific random seed is set and the maxi-  mum number of experimental steps is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The data are  smoothed at the end of the experiments, using the sliding  average filtering method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' A suitable sliding window size is  set for each data of different environments, and the data in  the window are averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' All data are smoothed when the  window is sliding from the beginning to the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' On the one  hand, the smoothed data are written to Table 1, and it should  be noted that the data of the six environments written to Ta-  ble 1 are the data at the end of the last training round run  by agent, both for CTD3 and TD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' On the other hand, all  the smoothed data of CTD3 and TD3 are plotted as curves  to obtain Figure 3, where the training process of CTD3 and  TD3 in each environment can be visually compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  CTD3 is a new algorithm that improves TD3 by apply-  ing the two ideas of actor-director-critic framework and the  improved double estimator method proposed in this paper  to the TD3 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' We can compare the performance of  CTD3 and TD3 by comparing the data in Table 1 and the  curves in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In all the six environments, the total  return of CTD3 is higher than that of TD3, and the conver-  gence rate of CTD3 is also faster than that of TD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Since  the performance of the CTD3 algorithm is better than TD3  in terms of convergence speed and final score, it can prove  that the actor-director-critic framework and the improved  double estimator method proposed in this paper are feasi-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  TD3+ADCF means that the ADC framework is ap-  plied to the TD3 algorithm to improve the TD3 algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' TD3+IDEM means that the improved double esti-  mator method is applied to the TD3 algorithm to improve  the TD3 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' CTD3(TD3+ADCF+IDEM) means that  the ADC framework and the improved double estimator  method are applied to the TD3 algorithm at the same time  to improve the TD3 algorithm and obtain the CTD3 algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' For ablation experiments, we can analyze the per-  formance of the ADC framework and the improved double  estimator method proposed in this paper by comparing the  data of TD3, TD3+ADCF, TD3+IDEM and CTD3 in Ta-  ble 1 and the curves in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' As shown in Table 1, in  all six experimental environments, no matter TD3+ADCF  or TD3+IDEM, their total returns are higher than those  of TD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  The curve in Figure 3 also shows that in all  six experimental environments, no matter TD3+ADCF or  TD3+IDEM, their convergence speed is faster than that of  TD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In most cases, CTD3 outperforms TD3+ADCF and  TD3+IDEM, but sometimes it is outperformed by one or  the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  In conclusion, the improved performance compared to  TD3, either using the actor-director-critic framework alone  or the improved double estimator method alone, or use both  of them, are sufficient to demonstrate the feasibility of our  proposed methods in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Conclusion  Reinforcement learning is a kind of trial-and-error learn-  ing, which has the problem of low sample utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' More-  over, in existing deep reinforcement learning algorithms  based on the actor-critic framework, there is also a problem  that it is difficult for an agent to rely on the policy function  to make good actions until the value function is learned to  the extent that it can provide reasonable guidance to the pol-  icy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Therefore, in order to alleviate these two prob-  lems, this paper proposes a new deep reinforcement learn-  ing framework, actor-director-critic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' The direc-  tor role is added to the actor-critic framework, and a decay  factor is set for the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' By training the director to dis-  �  �  �  �  �  �  �#� "������  �  ����  ����  ����  ����  ��&�!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='�  ��%����%�!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='#����$�������$�$�  ���  ��������  ��������  �������������������  �  ��  ��  ��  ��  ���  �"��!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='������  �  ����  ����  ����  ����  ����  ��$� �  ��#����������"��  ���  ��������  ��������  �������������������  �  ��  ��  ��  ��  ���  �"��!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='������  �  ����  ����  ����  ����  ����  ��%� �  ��$���#������  ���  ��������  ��������  �������������������  �  ��  ��  ��  ��  ���  �����������  �  ����  ����  ����  ����  ����  �� ���  ��������  ���  ��������  ��������  �������������������  �  ��  ��  ��  ��  ���  �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='�� ������  �  ����  ����  ����  ����  ����  ��#���  ��"����������  ���  ��������  ��������  �������������������  �  ��  ��  ��  ��  ���  � ���������  �  ���  ����  ����  ����  ����  ����  ��"���  ��!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='��������  ���  ��������  ��������  �������������������  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Performance curves in six environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  Environment  TD3  TD3+ADCF  TD3+IDEM  CTD3(TD3+ADCF+IDEM)  InvertedDoublePendulum  9341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content='690  HalfCheetah  8214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content='217  Walker2d  3818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content='348  Hopper  2395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='742  3169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='502  3118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='340  3252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='941  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Experimental results in six environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  criminate between low quality and high quality actions, it  can help the actor reduce repetitive attempts to perform low  quality actions during exploration, thus improving sample  utilization and speeding up training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  In solving the problem of overestimation in reinforce-  ment learning, for the method called double estimator  method that requires training two Q networks and design-  ing one target network for each Q network, this paper tries  to propose an improved double estimator method to design  two target networks for each Q network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' For each Q net-  work, its two target networks are alternately soft updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  The outputs of these target networks are comprehensively  used to calculate the target value of Q network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' First, the  output average of the two target networks owned by each Q  network is calculated, and then the smaller of the two aver-  age values is taken as the final target value of the Q network,  and this target value is shared by both Q networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' so as to  alleviate the problem of overestimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' So that the problem  of overvaluation can be mitigated much better than before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  To test the feasibility of both the actor-director-critic  framework and the improved double estimator method, we  applied them to the TD3 algorithm and improved the TD3  algorithm to obtain the CTD3 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' We conducted  comparative and ablation experiments in several environ-  ments in the MuJoCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Experimental results show that the  CTD3 algorithm generally has faster convergence speed and  higher total return than TD3, proving the feasibility of the  methods proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  References  [1] Gabriel Barth-Maron, Matthew W Hoffman, David Bud-  den, Will Dabney, Dan Horgan, TB Dhruva, Alistair Muldal,  Nicolas Heess, and Timothy Lillicrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Distributed distribu-  tional deterministic policy gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content=' Reinforcement learn-  ing with random delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content='  Shared experience actor-critic for multi-agent reinforcement  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content=' Multi-agent actor-critic  for mixed cooperative-competitive environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content=' Attention actor-critic algo-  rithm for multi-agent constrained co-operative reinforcement  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content=' Off-policy evolutionary reinforcement learning  with maximum mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content=' arXiv  preprint arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='01224, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' 3  [21] Jiachen Yang, Alireza Nakhaei, David Isele, Kikuo Fu-  jimura, and Hongyuan Zha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' Cm3: Cooperative multi-goal  multi-stage multi-agent reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' In Interna-  tional Conference on Learning Representations, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content='  arXiv preprint  arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content=' Advances in Neu-  ral Information Processing Systems, 34:29021–29033, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+page_content=' Dac: The double  actor-critic architecture for learning options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content='  Advances in  Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
+page_content=' 3' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfbgfz/content/2301.03887v1.pdf'}
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+Learning to Exploit Temporal Structure for
+Biomedical Vision–Language Processing
+Shruthi Bannur∗, Stephanie Hyland∗†, Qianchu Liu, Fernando P´erez-Garc´ıa, Maximilian Ilse,
+Daniel C. Castro, Benedikt Boecking, Harshita Sharma, Kenza Bouzid, Anja Thieme,
+Anton Schwaighofer, Maria Wetscherek, Matthew P. Lungren, Aditya Nori
+Javier Alvarez-Valle, and Ozan Oktay
+Microsoft Health Futures
+Abstract
+Self-supervised learning in vision–language processing
+(VLP) exploits semantic alignment between imaging and
+text modalities. Prior work in biomedical VLP has mostly
+relied on the alignment of single image and report pairs
+even though clinical notes commonly refer to prior im-
+ages.
+This does not only introduce poor alignment be-
+tween the modalities but also a missed opportunity to ex-
+ploit rich self-supervision through existing temporal con-
+tent in the data.
+In this work, we explicitly account for
+prior images and reports when available during both train-
+ing and fine-tuning. Our approach, named BioViL-T, uses
+a CNN–Transformer hybrid multi-image encoder trained
+jointly with a text model.
+It is designed to be versatile
+to arising challenges such as pose variations and miss-
+ing input images across time. The resulting model excels
+on downstream tasks both in single- and multi-image se-
+tups, achieving state-of-the-art (SOTA) performance on (I)
+progression classification, (II) phrase grounding, and (III)
+report generation, whilst offering consistent improvements
+on disease classification and sentence-similarity tasks. We
+release a novel multi-modal temporal benchmark dataset,
+MS-CXR-T, to quantify the quality of vision–language rep-
+resentations in terms of temporal semantics. Our experi-
+mental results show the advantages of incorporating prior
+images and reports to make most use of the data.
+1. Introduction
+Self-supervision from image–text pairs has enabled the
+development of flexible general-purpose vision–language
+models both in the general domain [39, 52, 76] and for
+specialised domains such as biomedicine and radiology
+∗These authors contributed equally.
+†Corresponding author: stephanie.hyland@microsoft.com
+"pleural fluid in the right base"
+"lung nodule remains unchanged"
+"pleural effusion is worsening"
+Image 
+encoder
+Text 
+encoder
+Image 
+encoder
+Text 
+encoder
+✓
+×
+✓
+×
+?
+✓
+×
+×
+✓
+×
+?
+×
+×
+×
+×
+?
+?
+?
+?
+×
+Prior image
+Current image
+Spatiotemporal modelling
+Spatial modelling
+Current image 
+ 
+InfoNCE 
+affinity matrix 
+Current image 
+Prior image
+ 
+(if available)
+Clinical report 
+Existing methods
+Proposed method
+InfoNCE 
+affinity matrix 
+Clinical report 
+(b)
+(a)
+(c)
+(d)
+Figure 1. (a) Existing visual–language pre-training approaches
+[9, 31, 80] often use only a single image for contrastive learning
+(e.g., InfoNCE [48]). (b) In such settings, discarding the temporal
+connectivity of images limits the alignment of image–text pairs as
+shown with the affinity matrix, leading to suboptimal pre-training
+and missed opportunity to create additional model supervision for
+free. (c, d) Our approach exploits this domain knowledge by learn-
+ing to incorporate a series of images and correlate them to reports,
+leading to pre-trained models that can generalise to a wider range
+of downstream tasks whilst achieving SOTA performance.
+[9, 31, 80]. Vision–language processing (VLP) has shown
+that cross-modal supervision can provide a richer signal for
+training both image [18] and text [9] models. However, the
+success of VLP relies on paired samples sharing semantics,
+i.e., given an image and text pair, the text should describe
+the image with minimal extraneous detail [15,16,34].
+In this regard, VLP in biomedicine and radiology poses
+a distinctive challenge, as reports routinely include compar-
+isons to prior imaging studies [3, 46, 56]. Without knowl-
+1
+arXiv:2301.04558v1  [cs.CV]  11 Jan 2023
+
+edge of this prior image1, temporal information in the text
+modality, e.g. “Pneumonia is improving”, could pertain to
+any image containing “Pneumonia”, producing ambiguity
+during contrastive training (Figure 1). Despite this, the ex-
+isting VLP work to date considers alignment between only
+single images and reports [9,31,45,80], going so far as to re-
+move temporal content from reports in training data to pre-
+vent ‘hallucinations’ in downstream report generation [53].
+However, temporal information can provide complementary
+self-supervision, solely by exploiting existing structure, and
+without requiring any additional data.
+In this work, we neither ignore nor remove temporal in-
+formation in the text modality, but explicitly account for it
+during pre-training. Rather than treating all image–report
+pairs in the dataset as independent, we exploit temporal cor-
+relations by making prior images available for comparison
+to a given report. To learn from this structure, we develop
+a temporal VLP pre-training framework named BioViL-T.
+A core component is its new multi-image encoder that can
+handle the absence of prior images and potential spatial
+misalignment between images across time. BioViL-T takes
+into account prior images where available, removing cross-
+modal ambiguity as illustrated in Fig. 1. Linking multi-
+ple images during pre-training proves beneficial to both im-
+age and text models: we report state-of-the-art (SOTA) per-
+formance on both temporal image classification and report
+generation. In the latter case, we show that prefixing the
+prior report substantially increases performance, again re-
+flecting the value of prior information. We emphasise that
+the benefit is not restricted to temporal downstream tasks:
+our approach also achieves SOTA on non-temporal tasks of
+pneumonia detection [59] and phrase grounding [10], un-
+derscoring the value of a cleaner learning signal during VLP
+without needing to modify or add to the training dataset.
+Our contributions can be summarised as follows:
+• We introduce a novel pre-training framework, called
+BioViL-T, which leverages the temporal relationship of
+samples to self-supervise VLP models. BioViL-T ex-
+tends the applicability of biomedical pre-trained models
+to a new set of downstream tasks without compromising
+performance on existing benchmarks.
+• We develop a generic multi-image encoder that handles
+missing image inputs and incorporates longitudinal in-
+formation without requiring explicit image registration.
+• We achieve SOTA results in biomedical report genera-
+tion, temporal image classification, and phrase ground-
+ing downstream benchmarks by accounting for prior
+context in self-supervised training and fine-tuning.
+• We release a new multimodal benchmark dataset,
+MS-CXR-T, curated by an expert radiologist. It enables
+1In the MIMIC-CXR v2 dataset [35], around 40% of reports explicitly
+reference a previous image. See Appendix B for details.
+benchmarking of biomedical VLP models in terms of
+temporal semantics extracted from image and text data.
+2. Related work
+Vision–language processing
+Self-supervised VLP can
+significantly reduce the need for manual labels required for
+the training of image encoders [18, 52]. The availability of
+large-scale paired image–text datasets has thus led to rapid
+development of general-purpose VLP models. Objectives
+include contrastive and discriminative image–text matching
+[39,52,68] including local variants [31,75], auto-regressive
+(AR) captioning [4, 38, 76] and multi-modal masked mod-
+elling objectives [13,39,60].
+Biomedical vision–language processing
+Paired medical
+image–report datasets were originally used for supervised
+learning via (typically) automated label extraction from
+clinical reports [32, 62, 69]. Using such datasets, advances
+in general-domain self-supervised VLP have been demon-
+strated to benefit biomedical imaging applications [9, 31,
+80].
+Work has incorporated ideas from general-domain
+VLP such as the original CLIP-style cross-modal con-
+trastive objective [80], multi-modal masking with merged
+co-attention on image–text representations [45], and adap-
+tations to the data of the domain. For example, a radiology
+report may have sparse image-specific details, prompting
+a local modification to the contrastive loss enabling align-
+ment between text tokens and image patches [31]. Domain-
+specific pre-training of the text model is shown to benefit
+biomedical VLP [9], and preferential masking of medical
+terms during masked language modelling (MLM) was ex-
+plored [74]. Here we use a local loss and domain-specific
+pre-training of the text model, but did not find a benefit to
+preferential masking. Similarly, cross-attention [21] is used
+rather than merged co-attention for image-guided MLM.
+Longitudinal modelling of medical images
+While prior
+images are used in unimodal supervised longitudinal analy-
+sis of medical images [36,57,67,73], temporal information
+has not directly been employed for self-supervision. The
+closest work exploits patient metadata to select positive or
+negative examples in unimodal contrastive learning [66,78].
+Existing models typically employ either late fusion of
+global image representations [57,63,67,73], which can miss
+fine-grained localised changes [31], or explicit spatial cor-
+respondence of features, using fixed spatial grids [47] or
+object detection [36]. Registering image pairs is commonly
+used for change detection in other contexts [17,51,58], and
+has been applied to medical imaging [5, 22].
+For chest
+X-rays (CXRs) however, registration entails the ill-posed
+problem of aligning 2D projections of 3D geometry, which
+inevitably results in residual misalignment. Our approach
+does not rely on bounding boxes or explicit graph construc-
+2
+
+tion as it uses self-attention of visual tokens across time to
+handle any spatial misalignment.
+Self-supervision across time
+Self-supervision has found
+applications on densely-sampled time series data (e.g.,
+video) to capture temporal information [29,54,77,79]. Our
+problem setting involves sparsely and sporadically sampled
+data where temporal pretext tasks are less applicable [2].
+Similarly, it requires text supervision to enable both static
+and temporal learning, when temporal structure is present.
+3. BioViL-T training framework
+Our approach comprises a multi-image encoder designed
+to extract spatio-temporal features from sequences of im-
+ages (Section 3.1) and a text encoder incorporating optional
+cross-attention on image features. The models are trained
+jointly with image-guided MLM and cross-modal global
+and local contrastive objectives (Section 3.2). The resulting
+image and text models are later adapted for uni- or multi-
+modal downstream tasks as described in Section 3.3. Im-
+plementation details are presented in Appendices E and F.
+For a given image and report pair (xcurr
+img ,xcurr
+txt ), the re-
+port xcurr
+txt describes the current image content and changes
+in reference to prior images. Our proposed formulation fo-
+cuses on a single prior image; however, it can be gener-
+alised to multiple prior images depending on the applica-
+tion. Hence, we construct datasets by including the prior
+image whenever it exists2: (xcurr
+img ,xprior
+img ,xcurr
+txt ) ∈ Dm or
+(xcurr
+img ,∅,xcurr
+txt ) ∈ Ds with the resulting dataset being a
+union of single and multi-image examples: D = Dm ∪ Ds.
+3.1. Extracting spatio-temporal image features
+Clinical findings are often observed across different im-
+age regions and co-occur simultaneously, which requires
+dense level visual reasoning across time to capture both
+static and temporal features. In contrast to late global fusion
+[63] and bounding-box based approaches [36], BioViL-T
+leverages local correspondences between image regions
+across time using transformer self-attention blocks [20].
+Thus our method does not require an explicit image reg-
+istration step between time points.
+We propose a hybrid CNN–Transformer encoder model
+due to its data efficiency and spatial flexibility of
+cross-attention across time points:
+Eimg
+∶ RW ×H
+→
+RW ′×H′×Dimg and Aimg ∶ RT ×L×Dimg → RL×Dimg, where
+W, H, and T correspond to spatiotemporal dimensions,
+L = W ′H′ is the number of visual tokens per image,
+and Dimg is the embedding dimension. Here Eimg (e.g.,
+ResNet-50 [30]) serves as a stem network [50] to provide
+visual token features of individual images. The CNN’s in-
+ductive biases [23, 50] ensure data efficiency of our hy-
+2The prior report is not included during pre-training as it may further
+reference an earlier study, reintroducing temporal ambiguity.
+brid model, making it ideal for smaller scale biomedical
+datasets. Eimg is initialised with BioViL weights [9]. The
+main purpose of Aimg is to capture patch embedding inter-
+actions across time when a prior image xprior
+img is available
+and to aggregate them into a fixed-length token represen-
+tation. Input visual tokens, Hcurr
+0
+= Pcurr ∶= Eimg(xcurr
+img ),
+Hprior
+0
+∶= Eimg(xprior
+img ) are augmented with spatial and tem-
+poral positional encodings and flattened across the spatial
+dimensions. They are then processed by K transformer en-
+coder [65] layers A as follows:
+[Hcurr
+k
+Hprior
+k
+] = Ak([ Hcurr
+k−1 + S + 1L ⊗ tcurr
+Hprior
+k−1 + S + 1L ⊗ tprior]),
+(1)
+for k = 1,...,K, where S ∈ RL×Dimg denotes 2D sinusoidal
+positional encodings [12] and T = [tcurr;tprior] ∈ R2×Dimg
+is its temporal counterpart, which is learnt (Fig. 2) [4]. The
+layer-normalised (LN) [6] output of the final transformer
+encoder block Pdiff ∶= LN(Hcurr
+K
+) is an ‘aggregated’ repre-
+sentation of patch-level progression information anchored
+on the current image. Figure 3 shows attention roll-out [1]
+applied to Pdiff after pre-training, showing how the prior
+image contributes to the fused representation. Figure A.3
+further highlights the robustness to variations in pose un-
+derlining that registration is not necessary for this encoder.
+Static-temporal feature decomposition
+When a prior
+image is available the final image representation V ∶=
+Pcurr ⊕ Pdiff ∈ RW ′×H′×2Dimg is formed by concatenating
+two sets of features (similar to [7]): those from the current
+image alone (Pcurr) and the temporal features from cur-
+rent and prior images (Pdiff). In this way, self-attention
+is mainly required to cope with pose variations and patch
+comparisons across time in extracting temporal content, re-
+moving the need for registration or explicit spatial feature
+alignment.
+When no prior scan is available (x ∈ Ds),
+Aimg is not used and Pdiff is replaced by a learnable to-
+ken pmiss ∈ RDimg, replicated across the spatial dimen-
+sions. Section 4.5 later demonstrates that Aimg highlights
+the value of feature decomposition for tasks such as phrase
+grounding which require well-localised features [10].
+Hereafter, downstream tasks that require solely single
+image features, Pcurr, are referred to as static tasks, and the
+ones that benefit from additional progression information,
+Pdiff, as temporal tasks, e.g., report decoding.
+3.2. Text-supervision for spatio-temporal learning
+Let w = (w1,...,wM) denote a vector of M tokens of
+a report xtxt after tokenisation. We first obtain contextu-
+alised token features Etxt(w) ∈ RM×Dtxt by passing a se-
+quence of text tokens w = (w1,...,wM) through a BERT
+encoder Etxt [19]. The input sequence is prepended with
+either a [CLS] or [MLM] token associated with a down-
+stream training objective, conditioning the output features
+3
+
+(If available)
+Transformer
+encoder 
+blocks 
+CXR-BERT
+MLM loss
+CNN
+CNN
+Global + local
+contrastive loss
+[CLS] increased right pleural  
+effusion [SEP] left lower lobe  
+pneumonia is improving [SEP]
+[MLM] [MASK] right pleural 
+effusion [SEP] [MASK] lower lobe  
+pneumonia is [MASK] [SEP]
+CXR-BERT
+Increased right pleural 
+effusion. Left lower lobe  
+pneumonia is improving.
+(self-attention 
++ feed-forward network)
+Temporal
+encoding
+Spatial
+encoding
+Flatten and
+concatenate
+If prior available
+Otherwise
+"Missing image"
+embedding
+Cross-attention
+For masked language modelling
+For contrastive loss
+[CLS]
+[MLM]
+Shared
+weights
+Shared
+weights
+"Difference"
+embedding
+Figure 2. The proposed self-supervised VLP training framework BioViL-T: Image representations V are extracted from single and
+multiple input scans (whenever available) using a hybrid CNN and transformer encoder [23,50]. This design choice is to increase the data-
+efficiency and enable the fusion of temporal content without requiring image registration. They are later matched with their corresponding
+text representations obtained with CXR-BERT [9] using local [31] and global InfoNCE [48] training objectives. As an additional model
+supervision, multi-modal fused representations, obtained with cross-attention, are used for image-guided masked language modelling.
+similar to [38, 41].
+During training, we do two forward
+passes through Etxt: once with masking at 45% probabil-
+ity (for the MLM objective) and once without masking for
+contrastive learning, as shown in Figure 2. The text encoder
+is initialised with the weights of CXR-BERT3 [9], a model
+pre-trained on domain-specific vocabulary and corpora.
+Both text and image features are later projected into a
+joint latent space with φtxt ∶ RDtxt → RD, and similarly
+vproj
+w,h ∶= φimg(vw,h) where φimg ∶ RDimg → RD, with φ
+being a two-layer perceptron in our experiments.
+Contrastive objectives
+Let r ∶= [Etxt(w)][CLS] denote
+the global representation of w, with rproj ∶= φtxt(r) its pro-
+jected version. Given projected patch embeddings vproj
+w,h , we
+can compute a global cosine similarity SC(¯vproj,rproj) and
+a local similarity using weighted pairwise cosine similari-
+ties across text tokens and projected patch embeddings [31,
+75]. These similarities are used in both global and local
+contrastive objectives with the InfoNCE loss [48, 52]. The
+local loss proves crucial both for static phrase-grounding
+and temporal image classification (see Table 4), highlight-
+ing the importance of localised self-supervision.
+Image-guided masked language modelling
+Prior work
+[9,45] has shown that biomedical visual-language learning
+benefits from an auxiliary task such as MLM since captur-
+ing the joint distribution of tokens can stabilise and improve
+3https : / / huggingface . co / microsoft / BiomedVLP -
+CXR-BERT-general
+language understanding during joint learning. Given a batch
+B of token vectors w, it is often defined as the cross-entropy
+for predicting the randomly sampled masked tokens, m ⊂
+{1,...,M}, LMLM = − 1
+∣B∣ ∑w∈B log pθ(wm ∣w/m), where
+θ are the weights of the text encoder Etxt.
+In the absence of image information, however, certain
+masked findings and attributes are not readily predicted,
+e.g., “[MASK] is worsening”. As shown in the general do-
+main [13], visual information can help disambiguate such
+masked predictions and provide additional cross-modal su-
+pervision. Thus, we use cross-attention [21,65] to the image
+features vproj
+w,h during this task. Specifically, for our image-
+guided MLM objective we model pθ(wm ∣w/m,vproj
+w,h ).
+3.3. Adaptations to downstream tasks
+BioViL-T can be adapted to various downstream tasks.
+For phrase-grounding and zero-shot inference, we rely
+on SC(rproj, vproj
+w,h ) similar to [9, 31]. For multiple-text
+prompts, projected text embeddings are marginalised prior
+to ℓ2-normalisation [52].
+To enable language decoding,
+vproj
+w,h inputs are cross-attended by text queries w, and
+causal-attention is utilised between text tokens [38,65]. Dif-
+fering from [9,31,80], we show that report generation tasks
+can greatly benefit from temporal joint latent space.
+Conditioning on prior reports
+In contrast to existing
+work, we incorporate the prior report as a prompt to contex-
+tualise the report generation task: pΦ(wcurr
+txt ∣wprior
+txt , vproj
+w,h ),
+4
+
+where Φ are the multi-modal encoder–decoder network’s
+weights, and wcurr
+txt , wprior
+txt
+denote text tokens for current
+and prior reports respectively. This is analogous to fine-
+tuning GPT-3 [11] with prompts and instructions [70], but
+conditioning on both images and the previous report. A
+dedicated separation token [SEP] is added into the input
+sequence [wprior
+txt ,[SEP],wcurr
+txt ].
+Curation of imaging datasets
+CXR datasets [35] often
+contain multiple image acquisitions Z = {ximg
+1
+,...,ximg
+Z }
+in a single visit due to data quality issues such as a lim-
+ited field-of-view or scanning the wrong body part (Fig-
+ure A.4). Unlike [9,31,80], we conduct curation to choose
+higher quality images among the potential candidates in-
+stead of performing a random selection. For this step, a
+separate BioViL-T is trained on ‘clean’ studies with sin-
+gle acquisitions and later used in a zero-shot setting to de-
+tect out-of-distribution samples [25,26] arising from the re-
+imaging process. The candidate ˆz is selected as follows:
+ˆz = arg maxz∈Z SC(¯vproj
+z
+, rproj) s.t. ∣sˆz − sZ/ˆz∣ > δ for a
+margin δ. This approach is applied to enhance the quality
+of the temporal classification dataset given its limited size.
+4. Datasets & experiments
+Here, we demonstrate BioViL-T’s data efficiency and
+adaptability to a wide range of applications, and show how
+the model achieves SOTA performance on various down-
+stream tasks by learning from data instances linked across
+time, making effective use of domain priors and the avail-
+able training data. Specifically, our model is evaluated on
+a diverse set of downstream tasks including zero- and few-
+shot static and temporal image classification, language de-
+coding (report generation), phrase-grounding [10], and sen-
+tence similarity.
+MS-CXR-T benchmark
+We release a new multi-modal
+benchmark dataset4, MS-CXR-T, to evaluate biomedical
+VLP models on two distinct temporal tasks: image clas-
+sification and sentence similarity. The former comprises
+multi-image and ground-truth label pairs (N = 1326) across
+5 findings, with classes corresponding to 3 states of disease
+progression for each finding:
+{Improving, Stable,
+Worsening}. The latter quantifies the temporal-semantic
+similarity of text embeddings extracted from pairs of sen-
+tences (N = 361). The pairs can be either paraphrases or
+contradictions in terms of disease progression. The data for
+both tasks was manually annotated and reviewed by a board
+certified radiologist. Appendix C provides further details on
+its data distribution and annotation protocol.
+Datasets
+For pre-training, we use the MIMIC-CXR v2
+[27, 35] chest X-ray dataset, which contains longitudinal
+4MS-CXR-T benchmark dataset will soon be released at: https://
+aka.ms/ms-cxr-t
+Prior image
+Current image
+Prior image
+Current image
+Figure 3.
+Attention rollout maps [1] from the reference patch
+(marked with ★) to the current and prior images. The bounding
+boxes, annotated by a radiologist, show the extent of consolida-
+tion. Note that the reference patch attends to its anatomical neigh-
+bourhood in the prior image despite the misalignment between
+prior and current images. The grid (14 × 14) represents the patch
+tokens processed in the transformer encoder blocks.
+imaging studies with corresponding radiological reports,
+see Fig. B.1 for the distribution of studies per patient. We
+only use frontal view (AP/PA) scans and discard samples
+where reports do not contain an IMPRESSION section. From
+this data, we gather 174.1k and 4.9k text-image pairs for
+model training and validation respectively, with a major-
+ity of text-image pairs including a prior image: ∣Dtrain
+m
+∣ =
+118.8k, ∣Dtrain
+s
+∣ = 55.3k. The text consists of the IMPRES-
+SION section and, for MLM additionally the FINDINGS sec-
+tion if available. Note that no manual labels are used during
+pre-training and no additional data is used for the methods
+that leverage the link between current and prior images. For
+early stopping we track the validation loss - see Appendix E
+for implementation details.
+Downstream evaluations are performed on a disjoint
+held-out test set shared across all tasks, ∣Dtest∣ = 2971.
+For report generation, we extend this test set with samples
+from healthy subjects (N = 815) to match the prevalence
+of pathological studies used in prior work [14, 24, 44]. For
+fine-tuning on temporal image classification, we use labels
+from the Chest ImaGenome dataset [71] as in [36] (statis-
+tics in Table F.2). In detail, we use the following bench-
+mark datasets: (I) MS-CXR [10] for phrase grounding, (II)
+the RSNA Pneumonia dataset [59,69] to test zero-shot and
+fine-tuned classification, (III) MS-CXR-T for temporal sen-
+tence similarity and temporal image classification.
+Comparison approaches
+We compare our approach to
+other domain-specific SOTA pre-training frameworks [9,
+31] specifically on phrase-grounding and zero-shot predic-
+tive performance. The non-temporal BioViL framework [9]
+is most similar to our approach and provides insight into
+non-temporal pre-training. Our ResNet model is initialised
+with BioViL weights and architecturally have only added
+the transformer encoder block to support multiple images,
+and cross-attention is utilised for image-guided MLM. We
+additionally compare to internal ablations such as remov-
+ing the past report during report generation and masking
+prior images during phrase grounding.
+For SOTA per-
+5
+
+357Table 1.
+Results for report generation task:
+Predictions are
+evaluated in terms of lexical (BLEU-4, ROUGE) and factual-
+ity metrics (CHEXBERT, TEM). Approaches are grouped into
+two broad categories: nearest-neighbour (NN) and auto-regressive
+(AR). BioViL-T pre-training consistently yields improved decod-
+ing. Further, the consistent performance gains of using prior im-
+age and report demonstrate the importance of such domain priors.
+‘PI / PR’ indicate usage of prior image and report, respectively.
+Method
+Pre-training PI / PR BLEU-4
+ROUGE
+CHEXBERT
+TEM
+NN
+CXR-RePaiR-2 [24] BioViL
+ / 
+2.1
+14.3
+28.1
+12.5
+Baseline (NN) [9]
+BioViL
+ / 
+3.7
+20.0
+28.3
+11.1
+Proposed (NN)
+BioViL-T
+/ 
+4.5
+20.5
+29.0
+13.0
+AR
+Baseline (AR) [9]
+BioViL
+ / 
+7.5 ± 0.1
+27.9 ± 0.1
+29.3 ± 0.3
+13.8 ± 0.1
+Proposed
+BioViL-T
+/ 
+8.2 ± 0.1
+28.7 ± 0.1
+30.2 ± 0.7
+16.0 ± 0.3
+Proposed
+BioViL-T
+/ 
+9.2 ± 0.3
+29.6 ± 0.1
+31.7 ± 1.0
+17.5 ± 0.1
+formance comparison, various AR and nearest-neighbour
+(NN) based language decoding approaches are used as base-
+lines: IFCC [44], R2Gen [14], CXR-RePaiR-2 [24], and
+CXR-RePaiR-Select [24].
+For the temporal classification task, we compare against
+a baseline exploiting the BioViL image encoder [9], and an
+approach that makes use of graph convolutions across re-
+gions of interest extracted from bounding boxes [36]. For
+BioViL, we perform affine image registration (with 4 DoF)
+for each pair of scans to cope with pose variations, and the
+encoded images are concatenated along the feature dimen-
+sion and classified via a multilayer perceptron. For [36],
+we compare to the three-class setting. Lastly, we bench-
+mark our final text model in isolation against domain spe-
+cific SOTA models in a temporal sentence similarity task:
+CXR-BERT [9] and PubMedBert [28].
+Metrics
+Due to class imbalance,
+we report macro-
+accuracy for temporal image classification.
+For phrase
+grounding, we use mean Intersection-Over-Union (mIoU)
+and Contrast-to-Noise-Ratio (CNR) [9]. The latter mea-
+sures the discrepancies between cosine similarities inside
+and out of the bounding box region without requiring hard
+thresholds.
+To evaluate the quality of generated reports,
+we use both the standard lexical metrics, e.g., BLEU [49],
+ROUGE-L [40], and also domain-specific factuality metric:
+CheXbert5 [61]. To directly probe the generation of change-
+related information, we introduce a new metric called tem-
+poral entity matching (TEM) to compute the match score of
+a fixed set of temporal entities (see Appendix D).
+4.1. Temporal pre-training yields data efficiency
+Downstream tasks are enabled with minimal labels.
+The sections ‘NN’ and ‘Z&F’ on Tables 1 and 2 report
+zero- and few-shot performance on tasks benefitting from
+temporal information: temporal image classification and re-
+5The average of the weighted-F1 score across 14 pathological observa-
+tions labelled by CheXbert.
+Table 2.
+Temporal image classification results (repeated for 4
+random seeds) on the MS-CXR-T benchmark for fully-supervised
+and zero-/few-shot (Z&F) learning settings, in terms of macro-
+accuracy across the three classes for each finding. Affine regis-
+tration is performed for the baseline method (denoted with suffix
+‘w/reg’), to partially address the pose variations across scans.
+Method (% of labels) Pre-train Consolidation Pl. effusion Pneumonia Pneumothorax Edema
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+Z&F
+BioViL-T prompt (0%) Temporal
+53.6 ± 1.9
+59.7 ± 2.1
+58.0 ± 3.9
+34.9 ± 1.0
+64.2 ± 1.5
+BioViL-T (10%)
+Temporal
+59.7 ± 2.4
+62.4 ± 1.4
+60.1 ± 2.1
+35.3 ± 2.6
+62.6 ± 1.7
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+Supervised
+CNN + Transformer
+ImageNet
+44.0 ± 2.0
+61.3 ± 1.6
+45.1 ± 3.5
+31.5 ± 3.1
+65.5 ± 1.1
+CheXRelNet [36]
+ImageNet
+47
+47
+47
+36
+49
+BioViL [9]
+Static
+56.1 ± 1.5
+62.3 ± 1.1
+59.4 ± 1.0
+41.7 ± 2.8
+67.5 ± 0.8
+BioViL w/reg [9]
+Static
+56.0 ± 1.5
+63.0 ± 0.9
+60.2 ± 0.7
+42.5 ± 2.7
+67.5 ± 0.9
+BioViL-T
+Temporal
+61.1 ± 2.4
+67.0 ± 0.8
+61.9 ± 1.9
+42.6 ± 1.6
+68.5 ± 0.8
+Table 3. Report generation results obtained with SOTA baseline
+methods using the same train/test splits. For comparison, we re-
+port the same lexical (BLEU-2) and factuality (CHEXBERT) met-
+rics from [24], and the specific sections decoded by each method.
+Method
+Decoded sections
+BLEU-2
+CHEXBERT
+R2gen [14]
+Findings & Impression
+21.20 ± 0.10
+14.80 ± 0.30
+IFCC [44]
+Findings
+21.70 ± 0.10
+27.00 ± 0.40
+CXR-RePaiR-Sel. [24]
+Impression
+5.00 ± 0.10
+27.40 ± 0.30
+BioViL-T
+Impression
+15.86 ± 0.14
+34.83 ± 0.73
+BioViL-T
+Findings & Impression
+21.31 ± 0.19
+35.86 ± 0.35
+port generation. Here we measure the quality of the learnt
+joint latent space and the extent to which BioViL-T enables
+efficient use of raw data. For zero-shot classification we
+prompt the AR fine-tuned model with prefix: “[FINDING]
+is” and compare the next-token probability of words mean-
+ing ‘improving’, ‘stable’, and ‘worsening’ (Appendix F.4).
+Without using any labelled data, Table 2 shows that the
+proposed AR-based approach already yields performance
+superior to prior fully-supervised work [36] on temporal
+image classification.
+With only 10% of labels, classifi-
+cation fine-tuning provides a further boost, indicating that
+BioViL-T produces a multi-image encoder readily adapted
+to temporal tasks. Similarly, in a zero-shot report-retrieval
+setting, the findings show that compared to temporally-
+agnostic pre-training, BioViL-T leveraging prior images
+improves across all metrics.
+Consistent with prior work
+[24], the retrieved reports already preserve factuality with
+high CheXbert scores, more-so than the other metrics which
+measure fine-grained specifics of phrasing. This demon-
+strates that the latent space captures the high-level seman-
+tics of the clinical features. Fine-grained phrasing however
+will be substantially improved by AR fine-tuning.
+4.2. Achieving SOTA performance with BioViL-T
+A wide range of downstream tasks benefit substantially
+from temporally-aware pre-training.
+Through downstream adaptations and fine-tuning our
+model, we report SOTA performance on report generation
+and temporal image classification tasks. For the former, us-
+6
+
+ing both prior images and reports during fine-tuning sub-
+stantially improves across metrics (Table 1). In particular,
+TEM metric results show that temporal context is key for
+accurately describing change in the generated report while
+avoiding hallucinations (see Table A.1 for examples). Com-
+paring to published results on a comparable test split and
+metrics (Sec. 4.1), we conclude that BioViL-T with fine-
+tuning achieves SOTA on report generation, producing re-
+ports that are lexically on par with prior work but substan-
+tially more factually accurate. Note that we do ‘vanilla’
+AR fine-tuning to focus on the impact of the pre-trained
+encoders, so application-specific supervision [44] could be
+used in conjunction to further boost performance.
+In temporal image classification (Tab. 2), BioViL-T pre-
+training outperforms the non-temporal baseline (BioViL)
+and improves on previously-reported results [36] by up to
+20 percentage points (pp).
+Furthermore, baseline meth-
+ods that rely on image registration (BioViL w/reg), under-
+perform compared to the proposed approach. Further anal-
+ysis reveals that errors tend to be in cases with disagreement
+between radiologists (Appendix A.2). We also note that pre-
+training is critical for a hybrid CNN-transformer model on
+this task, likely due to the small labelled dataset. Lastly, cu-
+ration of temporal training data is observed to improve the
+classification results by .68 pp aggregated across the find-
+ings, see Appendix A.4 for details.
+4.3. Static tasks benefit from temporal learning
+BioViL-T broadens the range of applicable downstream
+tasks whilst contributing to performance on static tasks.
+In this section, we demonstrate that performance im-
+provements afforded by BioViL-T are not restricted to tem-
+poral tasks – static tasks also benefit.
+Below, we re-
+port results on zero- and few-shot pneumonia classification
+from single images [59], where BioViL-T establishes a new
+SOTA compared to prior work [9,31].
+RSNA Pneumonia Detection Benchmark [59]
+Classification results for train & test split of 70% – 30% respectively
+Method
+% of Labels
+Supervision
+Acc.
+F1
+AUROC
+GLoRIA [31]
+
+Zero-shot
+0.70
+0.58
+-
+BioViL [9]
+
+Zero-shot
+0.732
+0.665
+0.831
+BioViL-T
+
+Zero-shot
+0.805
+0.706
+0.871
+BioViL [9]
+1%
+Few-shot
+0.805
+0.723
+0.881
+BioViL-T
+1%
+Few-shot
+0.814
+0.730
+0.890
+We see a similar trend on the MS-CXR phrase grounding
+benchmark (below). This task can be solved with single
+images, however we show that the inclusion of the prior
+image (where available) does not impair the performance
+of BioViL-T: feature decomposition effectively preserves
+localised information from the current image.
+MS-CXR benchmark results [10] (5-runs with different seeds)
+“Multi-image” column indicates the input images used at test time.
+Method
+Multi-Image
+Avg. CNR
+Avg. mIoU
+BioViL [9]
+
+1.07 ± 0.04
+0.229 ± 0.005
++ Local loss [9,31]
+
+1.21 ± 0.05
+0.202 ± 0.010
+BioViL-T
+
+1.33 ± 0.04
+0.243 ± 0.005
+BioViL-T
+
+1.32 ± 0.04
+0.240 ± 0.005
+4.4. Towards better sentence embedding quality
+Language models acquire increased temporal sensitivity.
+We hypothesise that text encoders learn temporal seman-
+tics through supervision from longitudinal image series. To
+verify this, RadNLI [44] and MS-CXR-T datasets are used
+in a zero-shot binary classification setting. Cosine similarity
+of sentence pair embeddings [55] are treated as class-logits
+to label each pair either as paraphrase or contradiction. See
+Appendix F.6 for further details.
+Our text model is benchmarked against SOTA domain-
+specific BERT models.
+The results below (MS-CXR-T),
+collected over 4 runs, show that the proposed framework
+greatly increases the sensitivity of sentence embeddings to
+temporal content. At same time, it learns to better capture
+the static content (RadNLI). Here, CXR-BERT-Specialised
+[9] is a text-model learnt through single-image based pre-
+training. Note that our text model is initialised with CXR-
+BERT-General, illustrating the substantial increase in tem-
+poral and static sensitivity due to BioViL-T pre-training.
+MS-CXR-T (361 pairs)
+RadNLI (145 pairs)
+Text Model
+Accuracy
+ROC-AUC
+Accuracy
+ROC-AUC
+PubMedBERT [28]
+60.39
+.542
+81.38
+.727
+CXR-BERT-G [9]
+62.60
+.601
+87.59
+.902
+CXR-BERT-S [9]
+78.12
+.837
+89.66
+.932
+BioViL-T
+87.77 ± 0.5
+.933 ± .003
+90.52 ± 1.0
+.947 ± .003
+4.5. Ablation experiments
+In Table 4 we report extensive ablations across the multi-
+image encoder architecture, pre-training choices, and AR
+fine-tuning for report generation.
+Image encoder
+Table 4 shows that decomposition of
+static and progression features is essential to ensure good
+performance on single-image tasks, such as phrase ground-
+ing. For temporal representations, on the other hand, posi-
+tional encodings are essential to disambiguate the order of
+scans, i.e., permutation variance across time.
+Model pre-training
+The corresponding results are shown
+in the middle section of Table 4. The local contrastive loss
+proves crucial to ensure meaningful language supervision
+during pre-training, followed by the image-guided MLM
+objective. Lastly, use of the FINDINGS section results in
+only minor performance gains as the key findings are al-
+ready captured in the IMPRESSION section.
+7
+
+Table 4. Ablation experiments on image encoder, pre-training set-
+tings, and report generation (repeated for 4 random seeds). Note
+that for temporal classification, linear probing is applied to frozen
+image embeddings. In report generation, the baseline method is
+fine-tuned with both prior image and report.
+Ablation
+Avg. CNR (mIoU)
+Pl. Effusion Acc.
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Encoder
+Baseline
+1.33 ± 0.02 (.248)
+64.8 ± 0.6
+No temporal encoding
+1.32 ± 0.02 (.242)
+62.9 ± 1.0
+No decomposition
+1.11 ± 0.08 (.203)
+64.0 ± 0.6
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Pre-training
+Baseline
+1.33 ± 0.02 (.248)
+64.8 ± 0.6
+Findings section discarded
+1.32 ± 0.01 (.246)
+63.8 ± 0.8
+No MLM loss
+1.28 ± 0.02 (.238)
+63.2 ± 0.7
+No local contrastive loss
+1.18 ± 0.02 (.236)
+60.2 ± 0.6
+Ablation
+ROUGE
+TEM
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Report gen.
+Baseline
+29.64 ± 0.08
+17.54 ± 0.11
+No prior image
+29.35 ± 0.25
+16.30 ± 0.40
+No prior report
+28.67 ± 0.12
+16.00 ± 0.30
+No prior image & report
+27.78 ± 0.09
+13.65 ± 0.48
+No separation token
+26.00 ± 0.40
+15.50 ± 1.06
+Report generation
+The importance of prior image and re-
+port is demonstrated by the substantial drop in the “no prior
+image & report” ablation especially on TEM metric, con-
+firming our hypothesis that temporal context is crucial for
+improving report quality. While both inputs are crucial for
+optimal performance, the prior report proves to be more im-
+portant. This is expected as the prior report is a summary of
+the image and thus provides a stronger and clearer signal.
+The prior image however cannot be dismissed entirely as it
+provides granular details which may not always be docu-
+mented in a report. Finally, we found the separation token
+is crucial in differentiating between the predicted tokens for
+the current report and tokens from the prior report.
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Progression
+Support devices
+Other
+Stop word
+Positional
+Meta
+Anatomy
+Descriptive
+Size or degree
+Finding
+Uncertain
+∆prior
+img (w)
+Figure 4. Mean token-level increase in image-guided MLM loss
+when prior image is discarded, grouped by token category. The
+prior image is excluded during inference to measure its impact
+on masked token predictions. Progression tokens are significantly
+better predicted when prior images are incorporated into image
+embeddings. The top five Progression tokens are ‘persist’, ‘im-
+proving’, ‘remains’, ‘unchanged’, and ‘residual’.
+4.6. Which tokens require a prior image in MLM?
+We leverage the MLM objective in an inference setting to
+analyse the influence of prior images in predicting masked
+tokens.
+Inspired by the ∆ image loss of [8], we define
+∆prior
+img as the change in loss by conditioning the estimation
+with a prior image for a given token w as follows:
+∆prior
+img (w) = l(w,xcurr
+img ,∅) − l(w,xcurr
+img ,xprior
+img )
+(2)
+where l(w,xcurr
+img ,xprior
+img ) is the cross-entropy of predicting
+the masked token w given visual features (MLM loss for a
+single token), averaged over sentences in which w appears.
+∆prior
+img is a measure of how much that token benefits from
+access to the prior image, as well as an assessment of the
+contribution of the prior image to the image representation.
+In Figure 4 we show the distribution of ∆prior
+img
+as a func-
+tion of token category (e.g., Anatomy, Positional; see F.5
+for annotation details).
+For Progression-type terms in particular, the model
+heavily relies on the prior image for image-guided MLM.
+We further observe that this effect is specific to temporal to-
+kens; as expected, those from other semantic categories do
+not consistently rely on the prior image.
+5. Conclusion
+In this paper, we introduced BioViL-T, a vision–
+language pre-training framework enabling alignment be-
+tween text and multiple images. BioViL-T makes use of
+a novel multi-image encoder and explicitly decomposes
+static–temporal features to augment the current image rep-
+resentation with information from prior images. This en-
+ables the grounding of temporal references in the text. To
+our knowledge, this is the first method capable of lever-
+aging the temporal content commonly present in biomed-
+ical text. It addresses an important limitation in existing
+VLP approaches, which simply discard such context. Also,
+incorporating such multi-modal temporal content provides
+strong learning signals to the model, resulting in richer rep-
+resentations and improved downstream performance.
+We demonstrate the value of this paradigm through ex-
+tensive experiments: BioViL-T excels on both static and
+temporal tasks, establishing new SOTA on report genera-
+tion, temporal image classification, few/zero-shot pneumo-
+nia detection, and phrase grounding. Furthermore, we re-
+lease a new multi-modal benchmark (MS-CXR-T) to mea-
+sure the quality of image and text representations in terms
+of temporal semantics, enabling more diverse evaluation of
+biomedical VLP models. The corresponding model weights
+and code will be made publicly available6.
+For applications requiring the encoding of more than
+2 images, the proposed model can scale linearly with the
+6Code will be released at: https://aka.ms/biovil-t-code
+8
+
+number of time points by switching to cross-attention of vi-
+sual tokens across time. Further exploration and evaluation
+are required on diverse datasets to characterise what kinds
+of tasks would benefit from a temporal modelling approach,
+and specifically from the proposed methodology.
+Acknowledgements:
+We would like to thank Hoifung
+Poon, Melanie Bernhardt, Melissa Bristow and Naoto
+Usuyama for their valuable feedback and contributions, and
+Hannah Richardson for helping with the compliance review
+of the datasets used in this study.
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+12
+
+A. Additional Results and Analyses
+A.1. Qualitative analysis of generated reports
+Table A.1 shows example reports generated with
+BioViL-T and BioViL models, which are compared to the
+reference radiologist’s reports. In comparison with BioViL
+which only models the current image, BioViL-T shows the
+benefit from incorporating prior study information and is
+able to provide factually more accurate reports especially
+in terms of describing temporal progression of the findings.
+This is showcased in the first two examples in the table: In
+the first row, BioViL-T is able to comment on not only the
+presence of the pleural effusion but also its improvement
+while BioViL fails to mention the change. In the second
+example, BioViL-T is able to correctly identify that there is
+no relevant change by comparing with the previous study,
+while BioViL wrongly hallucinates the tube in the current
+image as a new placement. BioViL-T can also avoid hallu-
+cination of the temporal information when there is no prior
+study. For instance, in the third example, BioViL-T cor-
+rectly acknowledges that there is no prior image and gen-
+erates the report based on information from the single cur-
+rent image, while BioViL hallucinates a non-exisistent prior
+study and wrongly generates temporal descriptions in the
+report.
+A.2. Further analysis on temporal classification
+A subset of the MS-CXR-T benchmark dataset is re-
+annotated by an expert radiologist by blinding them to the
+existing ground-truth labels and displaying only pairs of im-
+ages obtained from each subject. With the new set of labels,
+the analysis focuses on measuring the correlation between
+inter-rater agreement and image model’s prediction errors.
+Figure A.1 shows the dependency between the two where
+the x-axis corresponds to the cross entropy loss between the
+MS-CXR-T benchmark labels and model predictions. We
+observe lower model performance on cases with smaller
+inter-rater reliability for the three classes in the dataset, in-
+dicating that the model’s prediction errors occur more often
+for the cases where experts may disagree with each other.
+A.3. Self-attention visualisation
+In Figure A.2, we show examples of self-attention roll-
+out [1] maps for pleural effusion and consolidation, includ-
+ing radiologist-annotated bounding boxes surrounding the
+corresponding pathology in each prior and current image.
+To model the attention flow through the transformer en-
+coder block, we first average each attention weight matrix
+across all heads, subsequently we multiply the matrices be-
+tween every two layers. For every block we add the identity
+matrix in order to model the residual connections. Last, we
+only keep the top 10 % of attention weights per block to re-
+duce noise in the final rollout map. In contrast to [20], we
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Classification loss
+Improving
+Stable
+Worsening
+Agreement
+Disagreement
+Figure A.1.
+Cross entropy between model predictions and
+MS-CXR-T temporal classification labels. ‘Disagreement’ indi-
+cates cases for which annotations differed amongst radiologists.
+Model performance is higher for cases with with low ambiguity
+(‘Agreement’).
+do not visualize the rollout map with respect to a [CLS]
+token. Instead, we choose a reference image patch from the
+center of the radiologist-annotated bounding boxes, marked
+with ★ in Figure A.2.
+We find that the rollout maps in Figure A.2 are in good
+agreement with radiologist-annotated bounding boxes, i.e.,
+the reference patch attends to other patches within the
+bounding boxes in the prior and current image. In addition,
+we find that BioViL-T is robust to pose variations, e.g., in
+Figure A.2 (a) we show that despite the vertical shift be-
+tween prior and current image, the reference patch attends
+to the correct image patches in the prior image.
+To further assess the robustness of BioViL-T against
+pose variations between prior and current images, we per-
+formed multiple rotations to the prior image within a pair
+and computed rollout maps from the same reference patch
+in the current image. Figure A.3 shows that BioViL-T con-
+sistently attends to the corresponding anatomical region in-
+dependently of the spatial transformation applied, demon-
+strating that registration is not needed.
+A.4. Data curation of imaging datasets
+Large datasets often contain instances that are misla-
+belled or out of distribution [34]. We used BioViL-T to per-
+form pairwise ranking of instances in MIMIC-CXR (Sec-
+tion 3.3, δ = 0.2) and selected representative examples
+found in the dataset. Our method is able to select the most
+appropriate image for a range of different image-acquisition
+or image-processing issues (Figure A.4).
+We found that many lateral acquisitions in the dataset
+were unexpectedly labelled as frontal (Figure A.4a). Some
+images contained only noise (Figure A.4b), non-human
+samples (Figures A.4d and A.4e) or incorrect anatomy (Fig-
+ure A.4g). Often, acquisitions with an incomplete field of
+view (FOV) (i.e., the lungs are not completely visible) were
+repeated (Figure A.4c). Lastly, post-processed images were
+13
+
+Reference (by radiologist)
+BioViL
+BioViL-T
+No evidence of acute cardiopul-
+monary process.
+Decreased
+right pleural effusion.
+Small right pleural effusion.
+Small
+right
+pleural
+effusion,
+decreased since .
+No relevant change
+as
+com-
+pared to the previous image. No
+evidence of pneumonia or other
+parenchymal pathology. Normal
+size of the heart.
+No pleural
+effusions.
+In comparison with the study of , there has
+been placement of a right ij port - a - cath that
+extends to the lower svc. No evidence of post
+procedure pneumothorax. The cardiac silhouette
+is within normal limits and there is no vascular
+congestion or pleural effusion. There is some
+asymmetry of opacification at the bases, more
+prominent on the left. In the appropriate clinical
+setting, this could possibly represent a develop-
+ing consolidation.
+As compared to the previous ra-
+diograph, no relevant change is
+seen. The lung volumes are nor-
+mal.
+Normal size of the car-
+diac silhouette. Normal hilar and
+mediastinal structures. No pneu-
+monia, no pulmonary edema, no
+pleural effusions.
+No previous images .
+The car-
+diac silhouette is within normal
+limits and there is no evidence of
+vascular congestion, pleural effu-
+sion, or acute focal pneumonia.
+In comparison
+with the study of
+, there is
+little change
+and no evidence of acute car-
+diopulmonary disease. No pneumonia, vascular
+congestion, or pleural effusion.
+No previous images .
+The car-
+diac silhouette is within normal
+limits and there is no vascular
+congestion, pleural effusion, or
+acute focal pneumonia.
+Table A.1. Comparison between reports generated by radiologists, BioViL using only a single current image and BioViL-T using both
+the current and previous study. BioViL-T with access to longitudinal information can generate more accurate reports with more precise
+details on the progression of findings (as in the first and second example) while avoiding hallucination (in the third example). Blue box
+highlights the correct temporal information and brown box highlights incorrect temporal information including hallucination.
+Prior image
+Current image
+(a) Example of improving pleural effusion
+Prior image
+Current image
+(b) Example of stable pleural effusion
+Prior image
+Current image
+(c) Example of worsening pleural effusion
+Prior image
+Current image
+(d) Example of improving consolidation
+Prior image
+Current image
+(e) Example of stable consolidation
+Prior image
+Current image
+(f) Example of worsening consolidation
+Figure A.2. Self-attention rollout maps [1] from the reference patch (marked with ★) to the current and prior images, overlaid on example
+cases of (a) improving, (b) stable and (c) worsening pleural effusion (top row) and consolidation (bottom row). The bounding boxes,
+annotated by a radiologist, show the area corresponding to the pathology. The centre patch in the bounding box for the current image was
+selected as reference. The grid (14 × 14) represents the visual tokens processed in the transformer encoder blocks.
+14
+
+O★PORTABLEPORTABLE357Prior image
+Current image
+(a) Previous image rotated -30°
+Prior image
+Current image
+(b) Original pair
+Prior image
+Current image
+(c) Previous image rotated 30°
+Figure A.3. Comparison of roll-out maps computed after applying in-plane spatial rotations to the prior image. The reference visual token
+(★) attends to the corresponding anatomical region annotated by an expert independent of the underlying spatial transformation.
+detected by the algorithm such as contrast-enhanced scans
+(Figure A.4i) that are not often used for diagnostic purposes
+in clinical practice.
+B. Temporal aspects of the MIMIC-CXR v.2
+dataset
+Subjects in the MIMIC-CXR dataset often have multi-
+ple associated studies that happened at different times. A
+study, sometimes referred to as an ‘exam’ or ‘procedure’,
+refers to “one or more images taken on a single visit to a
+medical facility”7. To assess pathology progression, radi-
+ologists compare images (also referred to as ‘scans’ or ‘se-
+ries’) from different studies. In the MIMIC-CXR dataset,
+each study (with one or more images) is accompanied by the
+report written by the radiologist. Figure B.1 represents the
+distribution of studies per subject within MIMIC-CXR and
+the corresponding cumulative distribution function, show-
+ing that 67 % of the subjects have at least two different as-
+sociated studies (and therefore at least two images acquired
+at different stages of the disease).
+Another way to quantify temporal information in
+MIMIC-CXR is through the progression labels provided
+by the Chest ImaGenome dataset [71]. These progression
+labels are extracted from the reports and thus identify the
+cases when the radiologist explicitly describes changes. We
+found that in MIMIC, around 40 % of the reports are as-
+sociated with a progression label from any of the available
+findings defined by ImaGenome.
+C. MS-CXR-T benchmark
+C.1. Temporal image classification
+The MS-CXR-T temporal image classification contains
+progression labels for five findings (Consolidation, Edema,
+Pleural Effusion, Pneumonia and Pneumothorax) across
+three progression classes (Improving, Stable, and
+Worsening).
+This benchmark builds on the publicly
+available Chest ImaGenome gold and Chest ImaGenome
+7Adapted from https://ncithesaurus.nci.nih.gov/
+silver datasets [71] which provide progression labels auto-
+matically derived from radiology reports. We collected a set
+of studies that are part of the ImaGenome silver dataset, ex-
+cluding any studies that had been previously verified as part
+of the ImaGenome gold dataset. Additionally, we excluded
+studies where there are multiple progression labels for a sin-
+gle pathology (e.g. left pleural effusion has increased, right
+pleural effusion remains stable). We conducted a review
+process of the selected candidates, asking a board certified
+radiologist to either accept or reject the label. To inform
+their review of the labels, the radiologist was given access to
+the radiology report for the current image, and the sentence
+from which the auto generated label had been extracted.
+After collecting our curated labels and labels from the
+ImaGenome gold dataset, we matched the report-based la-
+bels to specific image pairs, performing a second data cu-
+ration step to create the image dataset. To ensure the di-
+agnostic quality of all images in the dataset, if a study had
+multiple frontal scans we performed a quality control step
+asking a radiologist to select the best image for each study.
+Fig. F.1 shows examples from the benchmark across differ-
+ent pathologies and progression labels.
+The class distribution for the image classification task in
+MS-CXR-T is shown in Tab. C.1. As seen in the table, the
+class distribution of the dataset skews towards the stable
+and worsening classes. This could be explained as pa-
+tients are more likely to get a chest X-ray scan when their
+condition is stable or deteriorating as opposed to when there
+is an improvement in patient condition.
+C.2. Temporal sentence similarity
+In this section, we describe the process of creating the
+MS-CXR-T temporal sentence similarity benchmark, which
+consists of pairs of paraphrase or contradiction sentences in
+terms of disease progression. We create this dataset using
+two different methods, RadGraph where paraphrase and
+contradiction sentence pairs are discovered by analysing
+graph representations of sentences and Swaps where para-
+phrases and contradictions are created by swapping out tem-
+poral keywords in the sentence.
+To create this dataset, we first collected a set of sentences
+15
+
+(a) Incorrect view
+(b) Invalid acquisition
+(c) Incomplete field of view
+(d) Non-human sample
+(e) Non-human sample
+(f) Inverted intensities
+(g) Non-chest sample
+(h) Image orientation
+(i) Post-processed image
+(j) Processing artefacts
+Figure A.4. Pairwise ranking of images performed by the proposed data curation method (see Section 3.3) on images from the MIMIC-
+CXR v2 dataset. Images highlighted with dashed green rectangles are automatically selected by our method and used for training to
+improve model’s downstream performance. The rejected image samples may not be appropriate for training due to image acquisition or
+image processing issues as shown in each subfigure above.
+0
+5
+10
+15
+20
+25
+Number of studies per subject
+0
+7000
+14000
+21000
+28000
+35000
+Number of subjects
+0%
+20%
+40%
+60%
+80%
+100%
+Cumulative percentage
+Figure B.1. Number of studies per subject in the MIMIC-CXR
+dataset. A study, sometimes referred to as an ‘exam’ or ‘proce-
+dure’, refers to “one or more images taken on a single visit to
+a medical facility” (adapted from https://ncithesaurus.
+nci.nih.gov/). Note that 67 % of subjects have at least two
+studies that happened at different times.
+Table C.1. MS-CXR-T temporal image classification benchmark:
+Showing the distribution of multi-image studies across different
+clinical findings, distribution of classes {Improving, Stable,
+Worsening} per finding, and number of subjects.
+Findings
+# of annotation pairs
+Class distribution
+# of subjects
+Consolidation
+201
+14% / 42% / 44%
+187
+Edema
+266
+31% / 26% / 43%
+241
+Pleural effusion
+411
+19% / 49% / 32%
+370
+Pneumonia
+237
+8% / 25% / 67%
+218
+Pneumothorax
+211
+15% / 55% / 30%
+148
+Total
+1326
+18% / 40% / 42%
+800
+from the MIMIC dataset, using the Stanza constituency
+parser [81] to extract individual sentences from reports. Us-
+ing the CheXbert labeller [62], we filtered this set to sen-
+tences that described one of seven pathologies - Atelecta-
+Subset
+# of paraphrase pairs
+# of contradiction pairs
+Total
+Radgraph
+42
+75
+117
+Swaps
+99
+145
+244
+Total
+141
+220
+361
+Table C.2. MS-CXR-T temporal sentence similarity benchmark:
+Number of paraphrase and contradiction examples in the full
+dataset and across the RadGraph and Swaps subsets (see Ap-
+pendix C.2).
+sis, Consolidation, Edema, Lung Opacity, Pleural Effusion,
+Pneumonia or Pneumothorax. We then filtered to sentences
+which contained at least one mention of a temporal key-
+word. Using this sentence pool, paraphrase and contradic-
+tion pairs were constructed in two ways. (I) We paired sen-
+tences from the sentence pool by matching on RadGraph
+[33] entities, relaxing the matching constraint only for tem-
+poral keywords and possible mentions of pathologies. (II)
+We swapped out temporal keywords in a sentence to cre-
+ate sentence pairs, choosing swap candidates from the top 5
+masked token predictions from CXR-BERT-Specialized [9]
+provided they were temporal keywords. After creating can-
+didate sentence pairs, we manually filtered out sentence
+pairs with ambiguous differences in terms of disease pro-
+gression. A board certified radiologist then annotated each
+sentence pair as either paraphrase or contradiction. Sen-
+tences were filtered out in the annotation process if (I) they
+were not clear paraphrases or contradictions (II) the sen-
+tences differed in meaning and this difference was not re-
+lated to any temporal information (III) they were not gram-
+matically correct.
+Examples from the benchmark are shown in Table C.3.
+The distribution of sentence pairs across the paraphrase and
+contradiction classes are described in Table C.2.
+16
+
+SEMI-EREC
+3013
+PORTABLE00.00SUPINE
+PORTABLPORTABLISEMI-ERECT
+PORTABLELabel
+Sentence 1
+Sentence 2
+Swaps
+Paraphrase
+“Unchanged small-to-moderate right pleural effusion.”
+“Stable small-to-moderate right pleural effusion.”
+Contradiction
+“Interval worsening of the right-sided pneumothorax.”
+“Interval resolution of the right-sided pneumothorax.”
+RadGraph
+Paraphrase
+“There has also been a slight increase in left basal consolidation.”
+“There is slight interval progression of left basal consolidation.”
+Contradiction
+“Right mid and lower lung consolidations are unchanged.”
+“There has been worsening of the consolidation involving the right
+mid and lower lung fields.”
+Table C.3. Examples of paraphrase and contradiction sentence pairs from the MS-CXR-T temporal sentence similarity benchmark. The
+examples are selected from the RadGraph and Swaps subsets (see Appendix C.2).
+D. Temporal entity matching
+To quantify how well the generated report describes
+progression-related information, we propose a new metric,
+namely temporal entity matching (TEM) score.
+D.1. Metric Formulation
+We first extract entities (tagged as “observation” or “ob-
+servation modifier”) from the text by running the named en-
+tity recognition model in the Stanza library [81]. Within the
+extracted entities, we manually curated a list of temporal
+entities that indicate progression (listed below). The list is
+reviewed by an expert radiologist. Given extracted tempo-
+ral entities E in N pairs of reference and generated reports,
+we calculate global precision (pE) and global recall (rE) as
+below:
+pE = ∑N
+i=1 ∣Ei
+gen ∩ Ei
+ref∣
+∑N
+i=1 ∣Eigen∣
+(3)
+rE = ∑N
+i=1 ∣Ei
+gen ∩ Ei
+ref∣
+∑N
+i=1 ∣Ei
+ref∣
+(4)
+The TEM score is the harmonic mean of precision and
+recall (also known as the F1 score).
+D.2. List of temporal keywords
+The list of temporal keywords used to compute the TEM
+score are as follows: {bigger, change, cleared, constant,
+decrease, decreased, decreasing, elevated, elevation, en-
+larged, enlargement, enlarging, expanded, greater, growing,
+improved, improvement, improving, increase, increased,
+increasing, larger, new, persistence, persistent, persisting,
+progression, progressive, reduced, removal, resolution, re-
+solved, resolving, smaller, stability, stable, stably, un-
+changed, unfolded, worse, worsen, worsened, worsening,
+unaltered}.
+E. Architecture and implementation details
+E.1. Hyper-parameters
+The models are trained in a distributed setting across 8
+GPU cards. For pre-training, we use a batch size of 240
+(30 * 8 GPUs) and the AdamW optimizer [42]. We use a
+linear learning rate scheduler with a warm-up proportion of
+0.03 and base learning rate of 2 × 10−5. We train for a max-
+imum of 50 epochs and use validation set loss for check-
+point selection. The overall loss is a sum of components
+with weighting factors: global contrastive (1.0), local con-
+trastive (0.5), and image-guided MLM (1.0) respectively,
+see Sec. 3.1 for further details on their formulation.
+Following [9] we use sentence permutation as text-based
+data augmentation. Similarly, spelling errors in the reports
+are corrected prior to tokenisation of the text data8. For
+image augmentations, note that we apply the same augmen-
+tation to current and prior images to prevent severe mis-
+alignment. We resize the shorter edge to 512 and centre-
+crop to (448, 448). We apply random affine transformations
+(rotation up to 30○ and shear up to 15○) and colour jitter
+(brightness and contrast).
+E.2. Training infrastructure
+We train with distributed data processing (DDP) on eight
+NVIDIA Tesla V100s with 32GB of memory each. To han-
+dle inconsistently-present prior images with DDP, we define
+a custom batch sampler. This sampler is a mixture of two
+samplers, in proportion to their dataset coverage: a sam-
+pler which produces batches with only multi-image exam-
+ples – (xcurr
+img ,xprior
+img ,xcurr
+txt ) ∈ Dm and one with only single-
+image examples – (xcurr
+img ,∅,xcurr
+txt ) ∈ Ds. Each GPU then
+processes a batch which is entirely single or multi-image,
+avoiding branching logic within the forward pass and en-
+abling an efficient single pass through the CNN to process
+all input images (current or prior) by concatenating them
+along the batch dimension.
+We confirmed that although the custom sampler theoret-
+ically impacts the order in which the dataset is traversed,
+it has a negligible effect on training metrics relative to fully
+random sampling. Since we train on eight GPUs and collect
+negatives across all GPUs during contrastive training, each
+update involves on average a representative mixture of both
+single-image and multi-image samples.
+8https://github.com/farrell236/mimic-cxr/blob/
+master/txt/section_parser.py
+17
+
+Finally, following [9] we use the DICOM images from
+MIMIC-CXR to avoid JPEG compression artefacts.
+F. Adaptation and experimentation details
+F.1. Fine-tuning BioViL-T for report generation
+During fine-tuning of BioViL-T for report generation,
+we minimise the cross entropy loss to maximise the log like-
+lihood of the report in an autoregressive manner given the
+input images. The model is initialised from the pretrained
+weights of the image encoder and the text encoder. Sim-
+ilar to the cross-modal masked language modelling task,
+we additionally train a linear projection layer to map the
+projected patch embeddings to the same hidden dimension
+of the text encoder, and we train cross-attention layers in
+each transformer block. The difference from the masked
+language modelling task is that we change the bidirectional
+self-attention to unidirectional causal attention that can only
+access the past tokens. If trained with prior report, we pass
+the prior report as prefix to condition the generation of the
+current report (the current and prior report are separated by
+[SEP]), and we only back-propagate the gradients from
+the loss on the tokens in the current report.
+For all experiments, we train the model for 100 epochs
+and we chose the best checkpoint according to metrics on
+the validation set. We performed grid search for learning
+rate in [10−5,2 × 10−5,5 × 10−5] and found 2 × 10−5 to be
+optimal. We ran each experiment with 3 random seeds and
+report mean and standard deviation.
+In addition to the metrics we reported in the main text,
+we also evaluate the generated reports by named entity met-
+ric (NEM). This metric was defined in [44] to measure the
+accuracy of reporting clinically relevant entities in the gen-
+erated reports (Similar to how TEM is computed to measure
+the match of temporal entities in our study). Following [44],
+we extract entities (tagged as “observation” or “observa-
+tion modifier”) from the text by running the named entity
+recognition model in the Stanza library [81]. The results are
+presented in Tab. F.1.
+F.2. Nearest-neighbour-based report retrieval
+The joint latent space learnt by BioViL-T can also be
+used to directly perform report retrieval without requiring
+task-specific model fine-tuning. Given the test image, we
+retrieve its semantically closest report from the training set
+in the joint latent space. Specifically, we encode each test
+image with the image model in BioViL-T and collect its
+projected image embeddings, and similarly we encode all
+the reports in the training data with their projected text em-
+beddings. For each test study, we compute cosine similarity
+between the test image embedding and all the text embed-
+dings from the training set in the joint latent space, and we
+retrieve the closest text embedding and use its correspond-
+Table F.1.
+Results for report generation task: Predictions are
+evaluated on NEM. The approaches are grouped into two broad
+categories: NN (Nearest Neighbour) and AR (Auto-Regressive).
+BioViL-T pre-training consistently yields superior decoding per-
+formance. Further, the use of prior image and report consistently
+yield performance gains demonstrating the importance of such do-
+main priors.
+Method
+Pre-training Prior Img/Report
+NEM
+NN
+CXR-RePaiR-2 [24]
+BioViL
+ / 
+13.36
+Baseline (NN) [9]
+BioViL
+ / 
+16.25
+Proposed (NN)
+BioViL-T
+/ 
+17.55
+AR
+Baseline (AR) [9]
+BioViL
+ / 
+24.27 ± 0.22
+Proposed
+BioViL-T
+/ 
+25.50 ± 0.04
+Proposed
+BioViL-T
+/ 
+26.95 ± 0.17
+ing report as the prediction. To evaluate the retrieval perfor-
+mance, we use the same decoding metrics on the retrieved
+reports and report results in the top section of Table 1.
+In a separate set of experiments, we also tried performing
+nearest neighbour search only within the image embedding
+space by retrieving the report associated with the closet im-
+age embedding, but this yielded sub-optimal performance
+compared with using the joint latent space.
+F.3. Fine-tuning for temporal image classification
+In this section, we describe the training dataset and fine-
+tuning procedure for the fully supervised and few-shot set-
+tings of the temporal image classification task. For this task,
+we finetune BioViL-T on a subset of the Chest ImaGenome
+silver dataset [71] to predict progression labels for 5 differ-
+ent pathologies. To create our training dataset, we filter out
+image pairs from this dataset where there are multiple di-
+rections of progression of a single pathology in the image-
+pair. We additionally perform an automatic data curation
+step to choose higher quality image pairs when possible, as
+described in 3.3. Table F.2 shows the number of training
+samples and label distribution for the training dataset.
+Table F.2. Statistics of the training dataset used for downstream
+fine-tuning on temporal image classification.
+Findings
+# labelled pairs
+Class distribution
+# of subjects
+Consolidation
+7012
+15% / 42% / 43%
+3308
+Edema
+14170
+28% / 33% / 39%
+4813
+Pleural effusion
+26320
+16% / 53% / 31%
+6838
+Pneumonia
+8471
+12% / 29% / 59%
+4197
+Pneumothorax
+3795
+21% / 57% / 22%
+1161
+For the fully supervised setting, we add a multilayer clas-
+sification head to the BioViL-T image encoder and fine-
+tune the model independently for each pathology. We use
+weighted cross entropy loss with a batch size of 128 and
+the AdamW optimizer [42]. During parameter optimisation,
+positional encodings and missing-image embeddings are
+18
+
+Target class
+Tokens
+Improving
+better, cleared, decreased, decreasing, im-
+proved, improving, reduced, resolved, re-
+solving, smaller
+Stable
+constant, stable, unchanged
+Worsening
+bigger, developing, enlarged, enlarging,
+greater, growing, increased, increasing,
+larger,
+new,
+progressing,
+progressive,
+worse, worsened, worsening
+Table F.3. Prompting the AR language decoding model for zero-
+shot image classification. The list above shows the mapping from
+decoded tokens to progression classes.
+exempt from weight decay penalty as in [72]. We train for
+30 epochs, with a linear learning rate schedule, a warmup
+proportion of 0.03 and a base learning rate of 1 × 10−5. For
+data augmentation, we first resize the shorter edge of the im-
+age to 512 and centre crop to (448, 448). We apply random
+horizontal flips, random cropping, random affine transfor-
+mations (rotation up to 30○, shear up to 15○), colour trans-
+forms (brightness and contrast) and Gaussian noise.
+For the few-shot setting we tune only a single-layer lin-
+ear head on the BioViL-T image encoder and freeze the rest
+of the encoder. We initialise the weight matrix of the linear
+head with values from encoded text prompts [9] for each
+of the three progression classes, and the bias matrix is ini-
+tialised with zeros. To train, we again use weighted cross
+entropy loss, with a batch size of 32 and the AdamW opti-
+mizer. We use a learning rate of 1 × 10−3 and train for 40
+epochs. For data augmentation, we resize the shorter edge
+of the image to 448 and center crop to (448, 488). We ap-
+ply random horizontal flips, random affine transformations
+(rotation up to 45○ and shear up to 25○), colour transforms
+(brightness and contrast). As in the pre-training step, we
+always synchronise image data augmentations to apply the
+identical transforms to the current and prior images.
+F.4. Auto-regressive prompting for zero-shot tem-
+poral image classification
+Following the GPT-3 style language prompting [11], we
+prompt the fine-tuned AR language decoding model with
+the template: “[FINDING] is” and infer the next token
+to perform temporal classification for each of the five find-
+ings. The mapping from the predicted next token to the
+three progression classes is characterised by a short list of
+tokens provided in Table F.3. After computing the posterior
+for each token in the list, the obtained values are normalised
+across the three classes, and the class with the highest score
+is selected as the prediction. The corresponding results are
+reported in Table 2.
+F.5. Further analysis of image-guided MLM
+In Section 4.6 we used a simplified notation for the
+computation of ∆prior
+img (m) for ease of exposition – here
+we provide further detail. Recall that w = (w1,...,wM)
+is a sequence of tokens and w/m is that sequence
+with token m masked.
+Let pθ(wm ∣w/m,xcurr
+img ,xprior
+img )
+be the text model’s predicted probability of token m
+given xcurr
+img ,xprior
+img , and w/m (θ are the weights of the
+model).
+Then, l(w,pθ(wm ∣w/m,xcurr
+img ,xprior
+img )) is the
+cross-entropy loss of predicting token m given those inputs.
+It is possible for different sentences in a report to refer
+to the same image finding. Since we mask single tokens
+at a time, to prevent information leakage from other sen-
+tences we consider each sentence in a report independently.
+Suppose report xcurr
+txt consists of S sentences, so we have
+xcurr
+txt = [w1,[SEP],...,[SEP],wS], where ws is the to-
+kens of sentence s and [SEP] separates sentences.
+For a given sample (xcurr
+img ,xprior
+img ,xcurr
+txt ) ∈ Dm in the test
+set indexed by i, we define
+δi(m) = ∑
+s∈S
+[l(m,pθ(ws
+m ∣ws
+/m,xcurr
+img ,∅))
+−l(m,pθ(ws
+m ∣ws
+/m,xcurr
+img ,xprior
+img ))]
+This is the MLM loss for predicting m given each sentence
+in the report with and without the prior image. Note that if
+m does not appear in a given sentence, its contribution to
+the sum is zero. The overall ∆prior
+img (m) is computed across
+all samples:
+∆prior
+img =
+1
+Nm
+⎛
+⎝ ∑
+i∈Dtest
+m
+δi(m)⎞
+⎠
+(5)
+where Nm is the number of sentences in reports in Dtest
+m
+in which token m appears. This estimate is subject to high
+variance when Nm is small. Hence, for Figure 4 we filter
+to tokens m with Nm ≥ 10. We collected 931 tokens with
+Nm ≥ 10 from the validation set for manual annotation by
+a board-certified radiologist. The categories, shown in Fig-
+ure 4 and described in Table F.4 are specific to the radiology
+domain.
+F.6. Sentence similarity experiment
+The text models are evaluated in isolation to observe if
+their encoding is sensitive to key clinical observations. To
+achieve this, we assess the quality of sentence represen-
+tations obtained from our text model by examining how
+well the contradiction and paraphrase pairs can be sepa-
+rated in the embedding space. Unlike the traditional NLI
+task where a model needs to be fine-tuned, here the models
+are probed in a zero-shot setting and the BERT output token
+embeddings are utilised. To do so, we encode the sentences
+from RadNLI and MS-CXR-T sentence similarity datasets
+19
+
+Category
+Description
+Examples
+Progression
+Pertaining to change or progression
+bigger, cleared, new
+Support devices
+Tubes, lines and implants
+nasogastric, pacemaker, cannula
+‘Other’
+No clear category
+can, relevant, overall
+Stop word
+‘Insignificant’ words
+the, no, of
+Positional
+Localisation (not anatomical)
+right, lower, bilateral
+Meta
+Pertaining to the report itself or practice of radiology
+evidence, radiograph, study
+Anatomy
+Anatomical locations
+pulmonary, chest, mediastinal
+Descriptive
+Qualitative appearance of a finding
+layering, focal, patchy
+Size or degree
+Quantifying extent or severity
+extensive, moderate, severe
+Finding
+Radiographic finding or pathology
+edema, penumonia, pneumothorax
+Uncertain
+Expression of certainty or doubt
+may, possible, concerning
+Table F.4. Semantic categories used in Figure 4.
+Prior image
+Current image
+(a) Improving consolidation
+Prior image
+Current image
+(b) Stable consolidation
+Prior image
+Current image
+(c) Worsening consolidation
+Prior image
+Current image
+(d) Improving pulmonary edema
+Prior image
+Current image
+(e) Stable pulmonary edema
+Prior image
+Current image
+(f) Worsening pulmonary edema
+Prior image
+Current image
+(g) Improving pleural effusion
+Prior image
+Current image
+(h) Stable pleural effusion
+Prior image
+Current image
+(i) Worsening pleural effusion
+Figure F.1. Examples of image pairs in our MS-CXR-T benchmark.
+with the [CLS] token from CXR-BERT-Specialised [9]
+and BioViL-T. For PubMedBERT [28] and CXR-BERT-
+General [9] which did not directly optimise the [CLS] to-
+ken during pretraining, we follow [55] to average the token
+output embeddings to represent each sentence.
+Cosine similarity is computed between the representa-
+tions of each sentence pair in the dataset [55] and is used as
+logits for the binary classification between paraphrase and
+contradiction. Note that for RadNLI, we use the subset of
+‘entailment’ and ‘contradiction’ pairs and discard the ’neu-
+tral’ pairs to unify the task across the two datasets. Given
+the similarities for each sentence pair, we report ROC-AUC
+and binary-accuracy. For the latter, a threshold value for
+each method is derived by setting aside a validation set.
+For this, we perform ten-fold cross validation and tune the
+threshold with step size of 0.005 on the validation set.
+20
+
+ERECT
+TABLEORTABLF.7. Image registration algorithm
+In Section 4.2, image registration is applied to pairs of
+images as a preprocessing step to enable a fair compari-
+son for the baseline approaches (e.g., BioViL [9]). We per-
+formed bidirectional multi-scale registration between image
+pairs optimising an affine transformation (4 degrees of free-
+dom), using mutual information (MI) [64] with 128 bins as
+the similarity criterion. In more detail, the spatial transfor-
+mation is characterised by four parameters: two for transla-
+tion, one for isotropic scaling, and one for rotation. The op-
+timisation is repeated five times with different random seeds
+for initialisation, and the run with the highest MI is selected
+to determine the final spatial alignment. To better identify
+the correspondences between the scans, bilateral filtering is
+applied to each image before registration to remove detailed
+texture whilst preserving edge information [37]. Our imple-
+mentation is based on the SimpleITK library [43].
+21
+
diff --git a/hNE3T4oBgHgl3EQfgArH/content/tmp_files/load_file.txt b/hNE3T4oBgHgl3EQfgArH/content/tmp_files/load_file.txt
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@@ -0,0 +1,1383 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf,len=1382
+page_content='Learning to Exploit Temporal Structure for  Biomedical Vision–Language Processing  Shruthi Bannur∗, Stephanie Hyland∗†, Qianchu Liu, Fernando P´erez-Garc´ıa, Maximilian Ilse,  Daniel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Castro, Benedikt Boecking, Harshita Sharma, Kenza Bouzid, Anja Thieme,  Anton Schwaighofer, Maria Wetscherek, Matthew P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Lungren, Aditya Nori  Javier Alvarez-Valle, and Ozan Oktay  Microsoft Health Futures  Abstract  Self-supervised learning in vision–language processing  (VLP) exploits semantic alignment between imaging and  text modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Prior work in biomedical VLP has mostly  relied on the alignment of single image and report pairs  even though clinical notes commonly refer to prior im-  ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  This does not only introduce poor alignment be-  tween the modalities but also a missed opportunity to ex-  ploit rich self-supervision through existing temporal con-  tent in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  In this work, we explicitly account for  prior images and reports when available during both train-  ing and fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Our approach, named BioViL-T, uses  a CNN–Transformer hybrid multi-image encoder trained  jointly with a text model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  It is designed to be versatile  to arising challenges such as pose variations and miss-  ing input images across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The resulting model excels  on downstream tasks both in single- and multi-image se-  tups, achieving state-of-the-art (SOTA) performance on (I)  progression classification, (II) phrase grounding, and (III)  report generation, whilst offering consistent improvements  on disease classification and sentence-similarity tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We  release a novel multi-modal temporal benchmark dataset,  MS-CXR-T, to quantify the quality of vision–language rep-  resentations in terms of temporal semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Our experi-  mental results show the advantages of incorporating prior  images and reports to make most use of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Introduction  Self-supervision from image–text pairs has enabled the  development of flexible general-purpose vision–language  models both in the general domain [39, 52, 76] and for  specialised domains such as biomedicine and radiology  ∗These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  †Corresponding author: stephanie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='hyland@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='com  "pleural fluid in the right base"  "lung nodule remains unchanged"  "pleural effusion is worsening"  Image  encoder  Text  encoder  Image  encoder  Text  encoder  ✓  ×  ✓  ×  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  ✓  ×  ×  ✓  ×  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  ×  ×  ×  ×  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  ×  Prior image  Current image  Spatiotemporal modelling  Spatial modelling  Current image  InfoNCE  affinity matrix  Current image  Prior image  (if available)  Clinical report  Existing methods  Proposed method  InfoNCE  affinity matrix  Clinical report  (b)  (a)  (c)  (d)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' (a) Existing visual–language pre-training approaches  [9, 31, 80] often use only a single image for contrastive learning  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=', InfoNCE [48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' (b) In such settings, discarding the temporal  connectivity of images limits the alignment of image–text pairs as  shown with the affinity matrix, leading to suboptimal pre-training  and missed opportunity to create additional model supervision for  free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' (c, d) Our approach exploits this domain knowledge by learn-  ing to incorporate a series of images and correlate them to reports,  leading to pre-trained models that can generalise to a wider range  of downstream tasks whilst achieving SOTA performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  [9, 31, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Vision–language processing (VLP) has shown  that cross-modal supervision can provide a richer signal for  training both image [18] and text [9] models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' However, the  success of VLP relies on paired samples sharing semantics,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=', given an image and text pair, the text should describe  the image with minimal extraneous detail [15,16,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  In this regard, VLP in biomedicine and radiology poses  a distinctive challenge, as reports routinely include compar-  isons to prior imaging studies [3, 46, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Without knowl-  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='04558v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='CV]  11 Jan 2023  edge of this prior image1, temporal information in the text  modality, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' “Pneumonia is improving”, could pertain to  any image containing “Pneumonia”, producing ambiguity  during contrastive training (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Despite this, the ex-  isting VLP work to date considers alignment between only  single images and reports [9,31,45,80], going so far as to re-  move temporal content from reports in training data to pre-  vent ‘hallucinations’ in downstream report generation [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  However, temporal information can provide complementary  self-supervision, solely by exploiting existing structure, and  without requiring any additional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  In this work, we neither ignore nor remove temporal in-  formation in the text modality, but explicitly account for it  during pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Rather than treating all image–report  pairs in the dataset as independent, we exploit temporal cor-  relations by making prior images available for comparison  to a given report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To learn from this structure, we develop  a temporal VLP pre-training framework named BioViL-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  A core component is its new multi-image encoder that can  handle the absence of prior images and potential spatial  misalignment between images across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' BioViL-T takes  into account prior images where available, removing cross-  modal ambiguity as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Linking multi-  ple images during pre-training proves beneficial to both im-  age and text models: we report state-of-the-art (SOTA) per-  formance on both temporal image classification and report  generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In the latter case, we show that prefixing the  prior report substantially increases performance, again re-  flecting the value of prior information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We emphasise that  the benefit is not restricted to temporal downstream tasks:  our approach also achieves SOTA on non-temporal tasks of  pneumonia detection [59] and phrase grounding [10], un-  derscoring the value of a cleaner learning signal during VLP  without needing to modify or add to the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Our contributions can be summarised as follows:  We introduce a novel pre-training framework, called  BioViL-T, which leverages the temporal relationship of  samples to self-supervise VLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' BioViL-T ex-  tends the applicability of biomedical pre-trained models  to a new set of downstream tasks without compromising  performance on existing benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  We develop a generic multi-image encoder that handles  missing image inputs and incorporates longitudinal in-  formation without requiring explicit image registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  We achieve SOTA results in biomedical report genera-  tion, temporal image classification, and phrase ground-  ing downstream benchmarks by accounting for prior  context in self-supervised training and fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  We release a new multimodal benchmark dataset,  MS-CXR-T, curated by an expert radiologist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' It enables  1In the MIMIC-CXR v2 dataset [35], around 40% of reports explicitly  reference a previous image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' See Appendix B for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  benchmarking of biomedical VLP models in terms of  temporal semantics extracted from image and text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Related work  Vision–language processing  Self-supervised VLP can  significantly reduce the need for manual labels required for  the training of image encoders [18, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The availability of  large-scale paired image–text datasets has thus led to rapid  development of general-purpose VLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Objectives  include contrastive and discriminative image–text matching  [39,52,68] including local variants [31,75], auto-regressive  (AR) captioning [4, 38, 76] and multi-modal masked mod-  elling objectives [13,39,60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Biomedical vision–language processing  Paired medical  image–report datasets were originally used for supervised  learning via (typically) automated label extraction from  clinical reports [32, 62, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Using such datasets, advances  in general-domain self-supervised VLP have been demon-  strated to benefit biomedical imaging applications [9, 31,  80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Work has incorporated ideas from general-domain  VLP such as the original CLIP-style cross-modal con-  trastive objective [80], multi-modal masking with merged  co-attention on image–text representations [45], and adap-  tations to the data of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For example, a radiology  report may have sparse image-specific details, prompting  a local modification to the contrastive loss enabling align-  ment between text tokens and image patches [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Domain-  specific pre-training of the text model is shown to benefit  biomedical VLP [9], and preferential masking of medical  terms during masked language modelling (MLM) was ex-  plored [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Here we use a local loss and domain-specific  pre-training of the text model, but did not find a benefit to  preferential masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Similarly, cross-attention [21] is used  rather than merged co-attention for image-guided MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Longitudinal modelling of medical images  While prior  images are used in unimodal supervised longitudinal analy-  sis of medical images [36,57,67,73], temporal information  has not directly been employed for self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The  closest work exploits patient metadata to select positive or  negative examples in unimodal contrastive learning [66,78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Existing models typically employ either late fusion of  global image representations [57,63,67,73], which can miss  fine-grained localised changes [31], or explicit spatial cor-  respondence of features, using fixed spatial grids [47] or  object detection [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Registering image pairs is commonly  used for change detection in other contexts [17,51,58], and  has been applied to medical imaging [5, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For chest  X-rays (CXRs) however, registration entails the ill-posed  problem of aligning 2D projections of 3D geometry, which  inevitably results in residual misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Our approach  does not rely on bounding boxes or explicit graph construc-  2  tion as it uses self-attention of visual tokens across time to  handle any spatial misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Self-supervision across time  Self-supervision has found  applications on densely-sampled time series data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=',  video) to capture temporal information [29,54,77,79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Our  problem setting involves sparsely and sporadically sampled  data where temporal pretext tasks are less applicable [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Similarly, it requires text supervision to enable both static  and temporal learning, when temporal structure is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' BioViL-T training framework  Our approach comprises a multi-image encoder designed  to extract spatio-temporal features from sequences of im-  ages (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1) and a text encoder incorporating optional  cross-attention on image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The models are trained  jointly with image-guided MLM and cross-modal global  and local contrastive objectives (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The resulting  image and text models are later adapted for uni- or multi-  modal downstream tasks as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Im-  plementation details are presented in Appendices E and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For a given image and report pair (xcurr  img ,xcurr  txt ), the re-  port xcurr  txt describes the current image content and changes  in reference to prior images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Our proposed formulation fo-  cuses on a single prior image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' however, it can be gener-  alised to multiple prior images depending on the applica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Hence, we construct datasets by including the prior  image whenever it exists2: (xcurr  img ,xprior  img ,xcurr  txt ) ∈ Dm or  (xcurr  img ,∅,xcurr  txt ) ∈ Ds with the resulting dataset being a  union of single and multi-image examples: D = Dm ∪ Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Extracting spatio-temporal image features  Clinical findings are often observed across different im-  age regions and co-occur simultaneously, which requires  dense level visual reasoning across time to capture both  static and temporal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In contrast to late global fusion  [63] and bounding-box based approaches [36], BioViL-T  leverages local correspondences between image regions  across time using transformer self-attention blocks [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Thus our method does not require an explicit image reg-  istration step between time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  We propose a hybrid CNN–Transformer encoder model  due to its data efficiency and spatial flexibility of  cross-attention across time points:  Eimg  ∶ RW ×H  →  RW ′×H′×Dimg and Aimg ∶ RT ×L×Dimg → RL×Dimg, where  W, H, and T correspond to spatiotemporal dimensions,  L = W ′H′ is the number of visual tokens per image,  and Dimg is the embedding dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Here Eimg (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=',  ResNet-50 [30]) serves as a stem network [50] to provide  visual token features of individual images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The CNN’s in-  ductive biases [23, 50] ensure data efficiency of our hy-  2The prior report is not included during pre-training as it may further  reference an earlier study, reintroducing temporal ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  brid model, making it ideal for smaller scale biomedical  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Eimg is initialised with BioViL weights [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The  main purpose of Aimg is to capture patch embedding inter-  actions across time when a prior image xprior  img is available  and to aggregate them into a fixed-length token represen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Input visual tokens, Hcurr  0  = Pcurr ∶= Eimg(xcurr  img ),  Hprior  0  ∶= Eimg(xprior  img ) are augmented with spatial and tem-  poral positional encodings and flattened across the spatial  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' They are then processed by K transformer en-  coder [65] layers A as follows:  [Hcurr  k  Hprior  k  ] = Ak([ Hcurr  k−1 + S + 1L ⊗ tcurr  Hprior  k−1 + S + 1L ⊗ tprior]),  (1)  for k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=',K, where S ∈ RL×Dimg denotes 2D sinusoidal  positional encodings [12] and T = [tcurr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='tprior] ∈ R2×Dimg  is its temporal counterpart, which is learnt (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' 2) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The  layer-normalised (LN) [6] output of the final transformer  encoder block Pdiff ∶= LN(Hcurr  K  ) is an ‘aggregated’ repre-  sentation of patch-level progression information anchored  on the current image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Figure 3 shows attention roll-out [1]  applied to Pdiff after pre-training, showing how the prior  image contributes to the fused representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3  further highlights the robustness to variations in pose un-  derlining that registration is not necessary for this encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Static-temporal feature decomposition  When a prior  image is available the final image representation V ∶=  Pcurr ⊕ Pdiff ∈ RW ′×H′×2Dimg is formed by concatenating  two sets of features (similar to [7]): those from the current  image alone (Pcurr) and the temporal features from cur-  rent and prior images (Pdiff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In this way, self-attention  is mainly required to cope with pose variations and patch  comparisons across time in extracting temporal content, re-  moving the need for registration or explicit spatial feature  alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  When no prior scan is available (x ∈ Ds),  Aimg is not used and Pdiff is replaced by a learnable to-  ken pmiss ∈ RDimg, replicated across the spatial dimen-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5 later demonstrates that Aimg highlights  the value of feature decomposition for tasks such as phrase  grounding which require well-localised features [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Hereafter, downstream tasks that require solely single  image features, Pcurr, are referred to as static tasks, and the  ones that benefit from additional progression information,  Pdiff, as temporal tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=', report decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Text-supervision for spatio-temporal learning  Let w = (w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=',wM) denote a vector of M tokens of  a report xtxt after tokenisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We first obtain contextu-  alised token features Etxt(w) ∈ RM×Dtxt by passing a se-  quence of text tokens w = (w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=',wM) through a BERT  encoder Etxt [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The input sequence is prepended with  either a [CLS] or [MLM] token associated with a down-  stream training objective, conditioning the output features  3  (If available)  Transformer  encoder  blocks  CXR-BERT  MLM loss  CNN  CNN  Global + local  contrastive loss  [CLS] increased right pleural  effusion [SEP] left lower lobe  pneumonia is improving [SEP]  [MLM] [MASK] right pleural  effusion [SEP] [MASK] lower lobe  pneumonia is [MASK] [SEP]  CXR-BERT  Increased right pleural  effusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Left lower lobe  pneumonia is improving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  (self-attention  + feed-forward network)  Temporal  encoding  Spatial  encoding  Flatten and  concatenate  If prior available  Otherwise  "Missing image"  embedding  Cross-attention  For masked language modelling  For contrastive loss  [CLS]  [MLM]  Shared  weights  Shared  weights  "Difference"  embedding  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The proposed self-supervised VLP training framework BioViL-T: Image representations V are extracted from single and  multiple input scans (whenever available) using a hybrid CNN and transformer encoder [23,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' This design choice is to increase the data-  efficiency and enable the fusion of temporal content without requiring image registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' They are later matched with their corresponding  text representations obtained with CXR-BERT [9] using local [31] and global InfoNCE [48] training objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' As an additional model  supervision, multi-modal fused representations, obtained with cross-attention, are used for image-guided masked language modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  similar to [38, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  During training, we do two forward  passes through Etxt: once with masking at 45% probabil-  ity (for the MLM objective) and once without masking for  contrastive learning, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The text encoder  is initialised with the weights of CXR-BERT3 [9], a model  pre-trained on domain-specific vocabulary and corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Both text and image features are later projected into a  joint latent space with φtxt ∶ RDtxt → RD, and similarly  vproj  w,h ∶= φimg(vw,h) where φimg ∶ RDimg → RD, with φ  being a two-layer perceptron in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Contrastive objectives  Let r ∶= [Etxt(w)][CLS] denote  the global representation of w, with rproj ∶= φtxt(r) its pro-  jected version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Given projected patch embeddings vproj  w,h , we  can compute a global cosine similarity SC(¯vproj,rproj) and  a local similarity using weighted pairwise cosine similari-  ties across text tokens and projected patch embeddings [31,  75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' These similarities are used in both global and local  contrastive objectives with the InfoNCE loss [48, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The  local loss proves crucial both for static phrase-grounding  and temporal image classification (see Table 4), highlight-  ing the importance of localised self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Image-guided masked language modelling  Prior work  [9,45] has shown that biomedical visual-language learning  benefits from an auxiliary task such as MLM since captur-  ing the joint distribution of tokens can stabilise and improve  3https : / / huggingface .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' co / microsoft / BiomedVLP -  CXR-BERT-general  language understanding during joint learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Given a batch  B of token vectors w, it is often defined as the cross-entropy  for predicting the randomly sampled masked tokens, m ⊂  {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=',M}, LMLM = − 1  ∣B∣ ∑w∈B log pθ(wm ∣w/m), where  θ are the weights of the text encoder Etxt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  In the absence of image information, however, certain  masked findings and attributes are not readily predicted,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=', “[MASK] is worsening”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' As shown in the general do-  main [13], visual information can help disambiguate such  masked predictions and provide additional cross-modal su-  pervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Thus, we use cross-attention [21,65] to the image  features vproj  w,h during this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Specifically, for our image-  guided MLM objective we model pθ(wm ∣w/m,vproj  w,h ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Adaptations to downstream tasks  BioViL-T can be adapted to various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For phrase-grounding and zero-shot inference, we rely  on SC(rproj, vproj  w,h ) similar to [9, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For multiple-text  prompts, projected text embeddings are marginalised prior  to ℓ2-normalisation [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  To enable language decoding,  vproj  w,h inputs are cross-attended by text queries w, and  causal-attention is utilised between text tokens [38,65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Dif-  fering from [9,31,80], we show that report generation tasks  can greatly benefit from temporal joint latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Conditioning on prior reports  In contrast to existing  work, we incorporate the prior report as a prompt to contex-  tualise the report generation task: pΦ(wcurr  txt ∣wprior  txt , vproj  w,h ),  4  where Φ are the multi-modal encoder–decoder network’s  weights, and wcurr  txt , wprior  txt  denote text tokens for current  and prior reports respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' This is analogous to fine-  tuning GPT-3 [11] with prompts and instructions [70], but  conditioning on both images and the previous report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' A  dedicated separation token [SEP] is added into the input  sequence [wprior  txt ,[SEP],wcurr  txt ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Curation of imaging datasets  CXR datasets [35] often  contain multiple image acquisitions Z = {ximg  1  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=',ximg  Z }  in a single visit due to data quality issues such as a lim-  ited field-of-view or scanning the wrong body part (Fig-  ure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Unlike [9,31,80], we conduct curation to choose  higher quality images among the potential candidates in-  stead of performing a random selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For this step, a  separate BioViL-T is trained on ‘clean’ studies with sin-  gle acquisitions and later used in a zero-shot setting to de-  tect out-of-distribution samples [25,26] arising from the re-  imaging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The candidate ˆz is selected as follows:  ˆz = arg maxz∈Z SC(¯vproj  z  , rproj) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' ∣sˆz − sZ/ˆz∣ > δ for a  margin δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' This approach is applied to enhance the quality  of the temporal classification dataset given its limited size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Datasets & experiments  Here, we demonstrate BioViL-T’s data efficiency and  adaptability to a wide range of applications, and show how  the model achieves SOTA performance on various down-  stream tasks by learning from data instances linked across  time, making effective use of domain priors and the avail-  able training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Specifically, our model is evaluated on  a diverse set of downstream tasks including zero- and few-  shot static and temporal image classification, language de-  coding (report generation), phrase-grounding [10], and sen-  tence similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  MS-CXR-T benchmark  We release a new multi-modal  benchmark dataset4, MS-CXR-T, to evaluate biomedical  VLP models on two distinct temporal tasks: image clas-  sification and sentence similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The former comprises  multi-image and ground-truth label pairs (N = 1326) across  5 findings, with classes corresponding to 3 states of disease  progression for each finding:  {Improving, Stable,  Worsening}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The latter quantifies the temporal-semantic  similarity of text embeddings extracted from pairs of sen-  tences (N = 361).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The pairs can be either paraphrases or  contradictions in terms of disease progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The data for  both tasks was manually annotated and reviewed by a board  certified radiologist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Appendix C provides further details on  its data distribution and annotation protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Datasets  For pre-training, we use the MIMIC-CXR v2  [27, 35] chest X-ray dataset, which contains longitudinal  4MS-CXR-T benchmark dataset will soon be released at: https://  aka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='ms/ms-cxr-t  Prior image  Current image  Prior image  Current image  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Attention rollout maps [1] from the reference patch  (marked with ★) to the current and prior images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The bounding  boxes, annotated by a radiologist, show the extent of consolida-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Note that the reference patch attends to its anatomical neigh-  bourhood in the prior image despite the misalignment between  prior and current images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The grid (14 × 14) represents the patch  tokens processed in the transformer encoder blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  imaging studies with corresponding radiological reports,  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1 for the distribution of studies per patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We  only use frontal view (AP/PA) scans and discard samples  where reports do not contain an IMPRESSION section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' From  this data, we gather 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1k and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='9k text-image pairs for  model training and validation respectively, with a major-  ity of text-image pairs including a prior image: ∣Dtrain  m  ∣ =  118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='8k, ∣Dtrain  s  ∣ = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The text consists of the IMPRES-  SION section and, for MLM additionally the FINDINGS sec-  tion if available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Note that no manual labels are used during  pre-training and no additional data is used for the methods  that leverage the link between current and prior images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For  early stopping we track the validation loss - see Appendix E  for implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Downstream evaluations are performed on a disjoint  held-out test set shared across all tasks, ∣Dtest∣ = 2971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For report generation, we extend this test set with samples  from healthy subjects (N = 815) to match the prevalence  of pathological studies used in prior work [14, 24, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For  fine-tuning on temporal image classification, we use labels  from the Chest ImaGenome dataset [71] as in [36] (statis-  tics in Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In detail, we use the following bench-  mark datasets: (I) MS-CXR [10] for phrase grounding, (II)  the RSNA Pneumonia dataset [59,69] to test zero-shot and  fine-tuned classification, (III) MS-CXR-T for temporal sen-  tence similarity and temporal image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Comparison approaches  We compare our approach to  other domain-specific SOTA pre-training frameworks [9,  31] specifically on phrase-grounding and zero-shot predic-  tive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The non-temporal BioViL framework [9]  is most similar to our approach and provides insight into  non-temporal pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Our ResNet model is initialised  with BioViL weights and architecturally have only added  the transformer encoder block to support multiple images,  and cross-attention is utilised for image-guided MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We  additionally compare to internal ablations such as remov-  ing the past report during report generation and masking  prior images during phrase grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For SOTA per-  5  357Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Results for report generation task:  Predictions are  evaluated in terms of lexical (BLEU-4, ROUGE) and factual-  ity metrics (CHEXBERT, TEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Approaches are grouped into  two broad categories: nearest-neighbour (NN) and auto-regressive  (AR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' BioViL-T pre-training consistently yields improved decod-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Further, the consistent performance gains of using prior im-  age and report demonstrate the importance of such domain priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  ‘PI / PR’ indicate usage of prior image and report, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Method  Pre-training PI / PR BLEU-4  ROUGE  CHEXBERT  TEM  NN  CXR-RePaiR-2 [24] BioViL  \x17 / \x17  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5  Baseline (NN) [9]  BioViL  \x17 / \x17  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='7  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  Proposed (NN)  BioViL-T  \x13/ \x17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0  AR  Baseline (AR) [9]  BioViL  \x17 / \x17  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  Proposed  BioViL-T  \x13/ \x17  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='7  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3  Proposed  BioViL-T  \x13/ \x13  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  formance comparison, various AR and nearest-neighbour  (NN) based language decoding approaches are used as base-  lines: IFCC [44], R2Gen [14], CXR-RePaiR-2 [24], and  CXR-RePaiR-Select [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For the temporal classification task, we compare against  a baseline exploiting the BioViL image encoder [9], and an  approach that makes use of graph convolutions across re-  gions of interest extracted from bounding boxes [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For  BioViL, we perform affine image registration (with 4 DoF)  for each pair of scans to cope with pose variations, and the  encoded images are concatenated along the feature dimen-  sion and classified via a multilayer perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For [36],  we compare to the three-class setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Lastly, we bench-  mark our final text model in isolation against domain spe-  cific SOTA models in a temporal sentence similarity task:  CXR-BERT [9] and PubMedBert [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Metrics  Due to class imbalance,  we report macro-  accuracy for temporal image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For phrase  grounding, we use mean Intersection-Over-Union (mIoU)  and Contrast-to-Noise-Ratio (CNR) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The latter mea-  sures the discrepancies between cosine similarities inside  and out of the bounding box region without requiring hard  thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  To evaluate the quality of generated reports,  we use both the standard lexical metrics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=', BLEU [49],  ROUGE-L [40], and also domain-specific factuality metric:  CheXbert5 [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To directly probe the generation of change-  related information, we introduce a new metric called tem-  poral entity matching (TEM) to compute the match score of  a fixed set of temporal entities (see Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Temporal pre-training yields data efficiency  Downstream tasks are enabled with minimal labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  The sections ‘NN’ and ‘Z&F’ on Tables 1 and 2 report  zero- and few-shot performance on tasks benefitting from  temporal information: temporal image classification and re-  5The average of the weighted-F1 score across 14 pathological observa-  tions labelled by CheXbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Temporal image classification results (repeated for 4  random seeds) on the MS-CXR-T benchmark for fully-supervised  and zero-/few-shot (Z&F) learning settings, in terms of macro-  accuracy across the three classes for each finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Affine regis-  tration is performed for the baseline method (denoted with suffix  ‘w/reg’), to partially address the pose variations across scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Method (% of labels) Pre-train Consolidation Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' effusion Pneumonia Pneumothorax Edema  Z&F  Z&F  Z&F  Z&F  Z&F  Z&F  Z&F  Z&F  Z&F  Z&F  Z&F  Z&F  Z&F  Z&F  Z&F  Z&F  Z&F  BioViL-T prompt (0%) Temporal  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='9  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='9  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5  BioViL-T (10%)  Temporal  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='7  Supervised  Supervised  Supervised  Supervised  Supervised  Supervised  Supervised  Supervised  Supervised  Supervised  Supervised  Supervised  Supervised  Supervised  Supervised  Supervised  Supervised  CNN + Transformer  ImageNet  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  CheXRelNet [36]  ImageNet  47  47  47  36  49  BioViL [9]  Static  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='8  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='8  BioViL w/reg [9]  Static  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='9  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='7  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='7  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='9  BioViL-T  Temporal  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='8  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='9  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='8  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Report generation results obtained with SOTA baseline  methods using the same train/test splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For comparison, we re-  port the same lexical (BLEU-2) and factuality (CHEXBERT) met-  rics from [24], and the specific sections decoded by each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Method  Decoded sections  BLEU-2  CHEXBERT  R2gen [14]  Findings & Impression  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='10  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='30  IFCC [44]  Findings  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='10  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='40  CXR-RePaiR-Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' [24]  Impression  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='10  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='30  BioViL-T  Impression  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='14  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='73  BioViL-T  Findings & Impression  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='19  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='35  port generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Here we measure the quality of the learnt  joint latent space and the extent to which BioViL-T enables  efficient use of raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For zero-shot classification we  prompt the AR fine-tuned model with prefix: “[FINDING]  is” and compare the next-token probability of words mean-  ing ‘improving’, ‘stable’, and ‘worsening’ (Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Without using any labelled data, Table 2 shows that the  proposed AR-based approach already yields performance  superior to prior fully-supervised work [36] on temporal  image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  With only 10% of labels, classifi-  cation fine-tuning provides a further boost, indicating that  BioViL-T produces a multi-image encoder readily adapted  to temporal tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Similarly, in a zero-shot report-retrieval  setting, the findings show that compared to temporally-  agnostic pre-training, BioViL-T leveraging prior images  improves across all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Consistent with prior work  [24], the retrieved reports already preserve factuality with  high CheXbert scores, more-so than the other metrics which  measure fine-grained specifics of phrasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' This demon-  strates that the latent space captures the high-level seman-  tics of the clinical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Fine-grained phrasing however  will be substantially improved by AR fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Achieving SOTA performance with BioViL-T  A wide range of downstream tasks benefit substantially  from temporally-aware pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Through downstream adaptations and fine-tuning our  model, we report SOTA performance on report generation  and temporal image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For the former, us-  6  ing both prior images and reports during fine-tuning sub-  stantially improves across metrics (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In particular,  TEM metric results show that temporal context is key for  accurately describing change in the generated report while  avoiding hallucinations (see Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1 for examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Com-  paring to published results on a comparable test split and  metrics (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1), we conclude that BioViL-T with fine-  tuning achieves SOTA on report generation, producing re-  ports that are lexically on par with prior work but substan-  tially more factually accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Note that we do ‘vanilla’  AR fine-tuning to focus on the impact of the pre-trained  encoders, so application-specific supervision [44] could be  used in conjunction to further boost performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  In temporal image classification (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' 2), BioViL-T pre-  training outperforms the non-temporal baseline (BioViL)  and improves on previously-reported results [36] by up to  20 percentage points (pp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Furthermore, baseline meth-  ods that rely on image registration (BioViL w/reg), under-  perform compared to the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Further anal-  ysis reveals that errors tend to be in cases with disagreement  between radiologists (Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We also note that pre-  training is critical for a hybrid CNN-transformer model on  this task, likely due to the small labelled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Lastly, cu-  ration of temporal training data is observed to improve the  classification results by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='68 pp aggregated across the find-  ings, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Static tasks benefit from temporal learning  BioViL-T broadens the range of applicable downstream  tasks whilst contributing to performance on static tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  In this section, we demonstrate that performance im-  provements afforded by BioViL-T are not restricted to tem-  poral tasks – static tasks also benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Below, we re-  port results on zero- and few-shot pneumonia classification  from single images [59], where BioViL-T establishes a new  SOTA compared to prior work [9,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  RSNA Pneumonia Detection Benchmark [59]  Classification results for train & test split of 70% – 30% respectively  Method  % of Labels  Supervision  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  F1  AUROC  GLoRIA [31]  \x17  Zero-shot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='58 BioViL [9]  \x17  Zero-shot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='732  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='665  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='831  BioViL-T  \x17  Zero-shot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='805  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='706  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='871  BioViL [9]  1%  Few-shot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='805  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='723  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='881  BioViL-T  1%  Few-shot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='814  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='730  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='890  We see a similar trend on the MS-CXR phrase grounding  benchmark (below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' This task can be solved with single  images, however we show that the inclusion of the prior  image (where available) does not impair the performance  of BioViL-T: feature decomposition effectively preserves  localised information from the current image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  MS-CXR benchmark results [10] (5-runs with different seeds)  “Multi-image” column indicates the input images used at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Method  Multi-Image  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' CNR  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' mIoU  BioViL [9]  \x17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='229 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='005  + Local loss [9,31]  \x17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='202 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='010  BioViL-T  \x17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='243 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='005  BioViL-T  \x13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='240 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='005  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Towards better sentence embedding quality  Language models acquire increased temporal sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  We hypothesise that text encoders learn temporal seman-  tics through supervision from longitudinal image series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To  verify this, RadNLI [44] and MS-CXR-T datasets are used  in a zero-shot binary classification setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Cosine similarity  of sentence pair embeddings [55] are treated as class-logits  to label each pair either as paraphrase or contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' See  Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6 for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Our text model is benchmarked against SOTA domain-  specific BERT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  The results below (MS-CXR-T),  collected over 4 runs, show that the proposed framework  greatly increases the sensitivity of sentence embeddings to  temporal content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' At same time, it learns to better capture  the static content (RadNLI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Here, CXR-BERT-Specialised  [9] is a text-model learnt through single-image based pre-  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Note that our text model is initialised with CXR-  BERT-General, illustrating the substantial increase in tem-  poral and static sensitivity due to BioViL-T pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  MS-CXR-T (361 pairs)  RadNLI (145 pairs)  Text Model  Accuracy  ROC-AUC  Accuracy  ROC-AUC  PubMedBERT [28]  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='39  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='542  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='38  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='727  CXR-BERT-G [9]  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='60  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='601  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='59  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='902  CXR-BERT-S [9]  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='837  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='66  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='932  BioViL-T  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='933 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='003  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='52 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='947 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='003  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Ablation experiments  In Table 4 we report extensive ablations across the multi-  image encoder architecture, pre-training choices, and AR  fine-tuning for report generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Image encoder  Table 4 shows that decomposition of  static and progression features is essential to ensure good  performance on single-image tasks, such as phrase ground-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For temporal representations, on the other hand, posi-  tional encodings are essential to disambiguate the order of  scans, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=', permutation variance across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Model pre-training  The corresponding results are shown  in the middle section of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The local contrastive loss  proves crucial to ensure meaningful language supervision  during pre-training, followed by the image-guided MLM  objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Lastly, use of the FINDINGS section results in  only minor performance gains as the key findings are al-  ready captured in the IMPRESSION section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  7  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Ablation experiments on image encoder, pre-training set-  tings, and report generation (repeated for 4 random seeds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Note  that for temporal classification, linear probing is applied to frozen  image embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In report generation, the baseline method is  fine-tuned with both prior image and report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Ablation  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' CNR (mIoU)  Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Effusion Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Encoder  Encoder  Encoder  Encoder  Encoder  Encoder  Encoder  Encoder  Encoder  Encoder  Encoder  Encoder  Encoder  Encoder  Encoder  Encoder  Encoder  Baseline  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='02 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='248)  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6  No temporal encoding  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='02 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='242)  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0  No decomposition  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='08 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='203)  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Pre-training  Baseline  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='02 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='248)  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6  Findings section discarded  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='01 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='246)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='8  No MLM loss  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='02 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='238)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='7  No local contrastive loss  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='02 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='236)  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6  Ablation  ROUGE  TEM  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Report gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Baseline  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='08  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='11  No prior image  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='25  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='40  No prior report  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='12  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='30  No prior image & report  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='09  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='48  No separation token  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='40  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='50 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='06  Report generation  The importance of prior image and re-  port is demonstrated by the substantial drop in the “no prior  image & report” ablation especially on TEM metric, con-  firming our hypothesis that temporal context is crucial for  improving report quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' While both inputs are crucial for  optimal performance, the prior report proves to be more im-  portant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' This is expected as the prior report is a summary of  the image and thus provides a stronger and clearer signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  The prior image however cannot be dismissed entirely as it  provides granular details which may not always be docu-  mented in a report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Finally, we found the separation token  is crucial in differentiating between the predicted tokens for  the current report and tokens from the prior report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='50  Progression  Support devices  Other  Stop word  Positional  Meta  Anatomy  Descriptive  Size or degree  Finding  Uncertain  ∆prior  img (w)  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Mean token-level increase in image-guided MLM loss  when prior image is discarded, grouped by token category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The  prior image is excluded during inference to measure its impact  on masked token predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Progression tokens are significantly  better predicted when prior images are incorporated into image  embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The top five Progression tokens are ‘persist’, ‘im-  proving’, ‘remains’, ‘unchanged’, and ‘residual’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Which tokens require a prior image in MLM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  We leverage the MLM objective in an inference setting to  analyse the influence of prior images in predicting masked  tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Inspired by the ∆ image loss of [8], we define  ∆prior  img as the change in loss by conditioning the estimation  with a prior image for a given token w as follows:  ∆prior  img (w) = l(w,xcurr  img ,∅) − l(w,xcurr  img ,xprior  img )  (2)  where l(w,xcurr  img ,xprior  img ) is the cross-entropy of predicting  the masked token w given visual features (MLM loss for a  single token), averaged over sentences in which w appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  ∆prior  img is a measure of how much that token benefits from  access to the prior image, as well as an assessment of the  contribution of the prior image to the image representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  In Figure 4 we show the distribution of ∆prior  img  as a func-  tion of token category (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=', Anatomy, Positional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' see F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5  for annotation details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For Progression-type terms in particular, the model  heavily relies on the prior image for image-guided MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  We further observe that this effect is specific to temporal to-  kens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' as expected, those from other semantic categories do  not consistently rely on the prior image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Conclusion  In this paper, we introduced BioViL-T, a vision–  language pre-training framework enabling alignment be-  tween text and multiple images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' BioViL-T makes use of  a novel multi-image encoder and explicitly decomposes  static–temporal features to augment the current image rep-  resentation with information from prior images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' This en-  ables the grounding of temporal references in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To  our knowledge, this is the first method capable of lever-  aging the temporal content commonly present in biomed-  ical text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' It addresses an important limitation in existing  VLP approaches, which simply discard such context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Also,  incorporating such multi-modal temporal content provides  strong learning signals to the model, resulting in richer rep-  resentations and improved downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  We demonstrate the value of this paradigm through ex-  tensive experiments: BioViL-T excels on both static and  temporal tasks, establishing new SOTA on report genera-  tion, temporal image classification, few/zero-shot pneumo-  nia detection, and phrase grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Furthermore, we re-  lease a new multi-modal benchmark (MS-CXR-T) to mea-  sure the quality of image and text representations in terms  of temporal semantics, enabling more diverse evaluation of  biomedical VLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The corresponding model weights  and code will be made publicly available6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For applications requiring the encoding of more than  2 images, the proposed model can scale linearly with the  6Code will be released at: https://aka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='ms/biovil-t-code  8  number of time points by switching to cross-attention of vi-  sual tokens across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Further exploration and evaluation  are required on diverse datasets to characterise what kinds  of tasks would benefit from a temporal modelling approach,  and specifically from the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Acknowledgements:  We would like to thank Hoifung  Poon, Melanie Bernhardt, Melissa Bristow and Naoto  Usuyama for their valuable feedback and contributions, and  Hannah Richardson for helping with the compliance review  of the datasets used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
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+page_content=' PhysioNet, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
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+page_content=' Rethinking and improving relative posi-  tion encoding for vision transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
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+page_content=' 19  [73] Yiwen Xu, Ahmed Hosny, Roman Zeleznik, Chintan Parmar,  Thibaud P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Coroller, Idalid Ivy Franco, Raymond H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Mak,  and Hugo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
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+page_content=' Aerts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Deep learning predicts lung cancer  treatment response from serial medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
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+page_content=' Clinical-BERT: Vision-language  pre-training for radiograph diagnosis and reports generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Proceedings of the AAAI Conference on Artificial Intelli-  gence, 36(3):2982–2990, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
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+page_content=' Filip: Fine-grained interactive language-image  pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
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+page_content=' Coca: Contrastive  captioners are image-text foundation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' arXiv preprint  arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
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+page_content='  Time is matter:  Temporal self-supervision for video transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In Inter-  national Conference on Machine Learning, pages 25804–  25816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
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+page_content=' Con-  trastive learning with temporal correlated medical images: A  case study using lung segmentation in chest x-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
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+page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
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+page_content=' Advances in Neural Information Processing Systems,  34:7025–7040, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' 3  [80] Yuhao Zhang, Hang Jiang, Yasuhide Miura, Christopher D  Manning, and Curtis P Langlotz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Contrastive learning of  medical visual representations from paired images and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='00747, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' 1, 2, 4, 5  [81] Yuhao Zhang, Yuhui Zhang, Peng Qi, Christopher D Man-  ning, and Curtis P Langlotz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Biomedical and clinical english  model packages for the stanza python nlp library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Journal of  the American Medical Informatics Association, 28(9):1892–  1899, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' 16, 17, 18  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Additional Results and Analyses  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Qualitative analysis of generated reports  Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1 shows example reports generated with  BioViL-T and BioViL models, which are compared to the  reference radiologist’s reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In comparison with BioViL  which only models the current image, BioViL-T shows the  benefit from incorporating prior study information and is  able to provide factually more accurate reports especially  in terms of describing temporal progression of the findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  This is showcased in the first two examples in the table: In  the first row, BioViL-T is able to comment on not only the  presence of the pleural effusion but also its improvement  while BioViL fails to mention the change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In the second  example, BioViL-T is able to correctly identify that there is  no relevant change by comparing with the previous study,  while BioViL wrongly hallucinates the tube in the current  image as a new placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' BioViL-T can also avoid hallu-  cination of the temporal information when there is no prior  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For instance, in the third example, BioViL-T cor-  rectly acknowledges that there is no prior image and gen-  erates the report based on information from the single cur-  rent image, while BioViL hallucinates a non-exisistent prior  study and wrongly generates temporal descriptions in the  report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Further analysis on temporal classification  A subset of the MS-CXR-T benchmark dataset is re-  annotated by an expert radiologist by blinding them to the  existing ground-truth labels and displaying only pairs of im-  ages obtained from each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' With the new set of labels,  the analysis focuses on measuring the correlation between  inter-rater agreement and image model’s prediction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1 shows the dependency between the two where  the x-axis corresponds to the cross entropy loss between the  MS-CXR-T benchmark labels and model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We  observe lower model performance on cases with smaller  inter-rater reliability for the three classes in the dataset, in-  dicating that the model’s prediction errors occur more often  for the cases where experts may disagree with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Self-attention visualisation  In Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2, we show examples of self-attention roll-  out [1] maps for pleural effusion and consolidation, includ-  ing radiologist-annotated bounding boxes surrounding the  corresponding pathology in each prior and current image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  To model the attention flow through the transformer en-  coder block, we first average each attention weight matrix  across all heads, subsequently we multiply the matrices be-  tween every two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For every block we add the identity  matrix in order to model the residual connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Last, we  only keep the top 10 % of attention weights per block to re-  duce noise in the final rollout map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In contrast to [20], we  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='00  Classification loss  Improving  Stable  Worsening  Agreement  Disagreement  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Cross entropy between model predictions and  MS-CXR-T temporal classification labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' ‘Disagreement’ indi-  cates cases for which annotations differed amongst radiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Model performance is higher for cases with with low ambiguity  (‘Agreement’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  do not visualize the rollout map with respect to a [CLS]  token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Instead, we choose a reference image patch from the  center of the radiologist-annotated bounding boxes, marked  with ★ in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  We find that the rollout maps in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2 are in good  agreement with radiologist-annotated bounding boxes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=',  the reference patch attends to other patches within the  bounding boxes in the prior and current image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In addition,  we find that BioViL-T is robust to pose variations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=', in  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2 (a) we show that despite the vertical shift be-  tween prior and current image, the reference patch attends  to the correct image patches in the prior image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  To further assess the robustness of BioViL-T against  pose variations between prior and current images, we per-  formed multiple rotations to the prior image within a pair  and computed rollout maps from the same reference patch  in the current image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3 shows that BioViL-T con-  sistently attends to the corresponding anatomical region in-  dependently of the spatial transformation applied, demon-  strating that registration is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Data curation of imaging datasets  Large datasets often contain instances that are misla-  belled or out of distribution [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We used BioViL-T to per-  form pairwise ranking of instances in MIMIC-CXR (Sec-  tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2) and selected representative examples  found in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Our method is able to select the most  appropriate image for a range of different image-acquisition  or image-processing issues (Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  We found that many lateral acquisitions in the dataset  were unexpectedly labelled as frontal (Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Some  images contained only noise (Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4b), non-human  samples (Figures A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4d and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4e) or incorrect anatomy (Fig-  ure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Often, acquisitions with an incomplete field of  view (FOV) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=', the lungs are not completely visible) were  repeated (Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Lastly, post-processed images were  13  Reference (by radiologist)  BioViL  BioViL-T  No evidence of acute cardiopul-  monary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Decreased  right pleural effusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Small right pleural effusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Small  right  pleural  effusion,  decreased since .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  No relevant change  as  com-  pared to the previous image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' No  evidence of pneumonia or other  parenchymal pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Normal  size of the heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  No pleural  effusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  In comparison with the study of , there has  been placement of a right ij port - a - cath that  extends to the lower svc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' No evidence of post  procedure pneumothorax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The cardiac silhouette  is within normal limits and there is no vascular  congestion or pleural effusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' There is some  asymmetry of opacification at the bases, more  prominent on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In the appropriate clinical  setting, this could possibly represent a develop-  ing consolidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  As compared to the previous ra-  diograph, no relevant change is  seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The lung volumes are nor-  mal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Normal size of the car-  diac silhouette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Normal hilar and  mediastinal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' No pneu-  monia, no pulmonary edema, no  pleural effusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  No previous images .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  The car-  diac silhouette is within normal  limits and there is no evidence of  vascular congestion, pleural effu-  sion, or acute focal pneumonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  In comparison  with the study of  , there is  little change  and no evidence of acute car-  diopulmonary disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' No pneumonia, vascular  congestion, or pleural effusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  No previous images .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  The car-  diac silhouette is within normal  limits and there is no vascular  congestion, pleural effusion, or  acute focal pneumonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Comparison between reports generated by radiologists, BioViL using only a single current image and BioViL-T using both  the current and previous study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' BioViL-T with access to longitudinal information can generate more accurate reports with more precise  details on the progression of findings (as in the first and second example) while avoiding hallucination (in the third example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Blue box  highlights the correct temporal information and brown box highlights incorrect temporal information including hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Prior image  Current image  (a) Example of improving pleural effusion  Prior image  Current image  (b) Example of stable pleural effusion  Prior image  Current image  (c) Example of worsening pleural effusion  Prior image  Current image  (d) Example of improving consolidation  Prior image  Current image  (e) Example of stable consolidation  Prior image  Current image  (f) Example of worsening consolidation  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Self-attention rollout maps [1] from the reference patch (marked with ★) to the current and prior images, overlaid on example  cases of (a) improving, (b) stable and (c) worsening pleural effusion (top row) and consolidation (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The bounding boxes,  annotated by a radiologist, show the area corresponding to the pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The centre patch in the bounding box for the current image was  selected as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The grid (14 × 14) represents the visual tokens processed in the transformer encoder blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  14  O★PORTABLEPORTABLE357Prior image  Current image  (a) Previous image rotated -30°  Prior image  Current image  (b) Original pair  Prior image  Current image  (c) Previous image rotated 30°  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Comparison of roll-out maps computed after applying in-plane spatial rotations to the prior image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The reference visual token  (★) attends to the corresponding anatomical region annotated by an expert independent of the underlying spatial transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  detected by the algorithm such as contrast-enhanced scans  (Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4i) that are not often used for diagnostic purposes  in clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Temporal aspects of the MIMIC-CXR v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2  dataset  Subjects in the MIMIC-CXR dataset often have multi-  ple associated studies that happened at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' A  study, sometimes referred to as an ‘exam’ or ‘procedure’,  refers to “one or more images taken on a single visit to a  medical facility”7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To assess pathology progression, radi-  ologists compare images (also referred to as ‘scans’ or ‘se-  ries’) from different studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In the MIMIC-CXR dataset,  each study (with one or more images) is accompanied by the  report written by the radiologist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1 represents the  distribution of studies per subject within MIMIC-CXR and  the corresponding cumulative distribution function, show-  ing that 67 % of the subjects have at least two different as-  sociated studies (and therefore at least two images acquired  at different stages of the disease).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Another way to quantify temporal information in  MIMIC-CXR is through the progression labels provided  by the Chest ImaGenome dataset [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' These progression  labels are extracted from the reports and thus identify the  cases when the radiologist explicitly describes changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We  found that in MIMIC, around 40 % of the reports are as-  sociated with a progression label from any of the available  findings defined by ImaGenome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' MS-CXR-T benchmark  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Temporal image classification  The MS-CXR-T temporal image classification contains  progression labels for five findings (Consolidation, Edema,  Pleural Effusion, Pneumonia and Pneumothorax) across  three progression classes (Improving, Stable, and  Worsening).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  This benchmark builds on the publicly  available Chest ImaGenome gold and Chest ImaGenome  7Adapted from https://ncithesaurus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='nci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='gov/  silver datasets [71] which provide progression labels auto-  matically derived from radiology reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We collected a set  of studies that are part of the ImaGenome silver dataset, ex-  cluding any studies that had been previously verified as part  of the ImaGenome gold dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Additionally, we excluded  studies where there are multiple progression labels for a sin-  gle pathology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' left pleural effusion has increased, right  pleural effusion remains stable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We conducted a review  process of the selected candidates, asking a board certified  radiologist to either accept or reject the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To inform  their review of the labels, the radiologist was given access to  the radiology report for the current image, and the sentence  from which the auto generated label had been extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  After collecting our curated labels and labels from the  ImaGenome gold dataset, we matched the report-based la-  bels to specific image pairs, performing a second data cu-  ration step to create the image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To ensure the di-  agnostic quality of all images in the dataset, if a study had  multiple frontal scans we performed a quality control step  asking a radiologist to select the best image for each study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1 shows examples from the benchmark across differ-  ent pathologies and progression labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  The class distribution for the image classification task in  MS-CXR-T is shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' As seen in the table, the  class distribution of the dataset skews towards the stable  and worsening classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' This could be explained as pa-  tients are more likely to get a chest X-ray scan when their  condition is stable or deteriorating as opposed to when there  is an improvement in patient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Temporal sentence similarity  In this section, we describe the process of creating the  MS-CXR-T temporal sentence similarity benchmark, which  consists of pairs of paraphrase or contradiction sentences in  terms of disease progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We create this dataset using  two different methods, RadGraph where paraphrase and  contradiction sentence pairs are discovered by analysing  graph representations of sentences and Swaps where para-  phrases and contradictions are created by swapping out tem-  poral keywords in the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  To create this dataset, we first collected a set of sentences  15  (a) Incorrect view  (b) Invalid acquisition  (c) Incomplete field of view  (d) Non-human sample  (e) Non-human sample  (f) Inverted intensities  (g) Non-chest sample  (h) Image orientation  (i) Post-processed image  (j) Processing artefacts  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Pairwise ranking of images performed by the proposed data curation method (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3) on images from the MIMIC-  CXR v2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Images highlighted with dashed green rectangles are automatically selected by our method and used for training to  improve model’s downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The rejected image samples may not be appropriate for training due to image acquisition or  image processing issues as shown in each subfigure above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  0  5  10  15  20  25  Number of studies per subject  0  7000  14000  21000  28000  35000  Number of subjects  0%  20%  40%  60%  80%  100%  Cumulative percentage  Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Number of studies per subject in the MIMIC-CXR  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' A study, sometimes referred to as an ‘exam’ or ‘proce-  dure’, refers to “one or more images taken on a single visit to  a medical facility” (adapted from https://ncithesaurus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  nci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='gov/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Note that 67 % of subjects have at least two  studies that happened at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' MS-CXR-T temporal image classification benchmark:  Showing the distribution of multi-image studies across different  clinical findings, distribution of classes {Improving, Stable,  Worsening} per finding, and number of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Findings  # of annotation pairs  Class distribution  # of subjects  Consolidation  201  14% / 42% / 44%  187  Edema  266  31% / 26% / 43%  241  Pleural effusion  411  19% / 49% / 32%  370  Pneumonia  237  8% / 25% / 67%  218  Pneumothorax  211  15% / 55% / 30%  148  Total  1326  18% / 40% / 42%  800  from the MIMIC dataset, using the Stanza constituency  parser [81] to extract individual sentences from reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Us-  ing the CheXbert labeller [62], we filtered this set to sen-  tences that described one of seven pathologies - Atelecta-  Subset  # of paraphrase pairs  # of contradiction pairs  Total  Radgraph  42  75  117  Swaps  99  145  244  Total  141  220  361  Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' MS-CXR-T temporal sentence similarity benchmark:  Number of paraphrase and contradiction examples in the full  dataset and across the RadGraph and Swaps subsets (see Ap-  pendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  sis, Consolidation, Edema, Lung Opacity, Pleural Effusion,  Pneumonia or Pneumothorax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We then filtered to sentences  which contained at least one mention of a temporal key-  word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Using this sentence pool, paraphrase and contradic-  tion pairs were constructed in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' (I) We paired sen-  tences from the sentence pool by matching on RadGraph  [33] entities, relaxing the matching constraint only for tem-  poral keywords and possible mentions of pathologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' (II)  We swapped out temporal keywords in a sentence to cre-  ate sentence pairs, choosing swap candidates from the top 5  masked token predictions from CXR-BERT-Specialized [9]  provided they were temporal keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' After creating can-  didate sentence pairs, we manually filtered out sentence  pairs with ambiguous differences in terms of disease pro-  gression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' A board certified radiologist then annotated each  sentence pair as either paraphrase or contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Sen-  tences were filtered out in the annotation process if (I) they  were not clear paraphrases or contradictions (II) the sen-  tences differed in meaning and this difference was not re-  lated to any temporal information (III) they were not gram-  matically correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Examples from the benchmark are shown in Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  The distribution of sentence pairs across the paraphrase and  contradiction classes are described in Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  16  SEMI-EREC  3013  PORTABLE00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='00SUPINE  PORTABLPORTABLISEMI-ERECT  PORTABLELabel  Sentence 1  Sentence 2  Swaps  Paraphrase  “Unchanged small-to-moderate right pleural effusion.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  “Stable small-to-moderate right pleural effusion.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Contradiction  “Interval worsening of the right-sided pneumothorax.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  “Interval resolution of the right-sided pneumothorax.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  RadGraph  Paraphrase  “There has also been a slight increase in left basal consolidation.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  “There is slight interval progression of left basal consolidation.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Contradiction  “Right mid and lower lung consolidations are unchanged.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  “There has been worsening of the consolidation involving the right  mid and lower lung fields.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Examples of paraphrase and contradiction sentence pairs from the MS-CXR-T temporal sentence similarity benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The  examples are selected from the RadGraph and Swaps subsets (see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Temporal entity matching  To quantify how well the generated report describes  progression-related information, we propose a new metric,  namely temporal entity matching (TEM) score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Metric Formulation  We first extract entities (tagged as “observation” or “ob-  servation modifier”) from the text by running the named en-  tity recognition model in the Stanza library [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Within the  extracted entities, we manually curated a list of temporal  entities that indicate progression (listed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The list is  reviewed by an expert radiologist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Given extracted tempo-  ral entities E in N pairs of reference and generated reports,  we calculate global precision (pE) and global recall (rE) as  below:  pE = ∑N  i=1 ∣Ei  gen ∩ Ei  ref∣  ∑N  i=1 ∣Eigen∣  (3)  rE = ∑N  i=1 ∣Ei  gen ∩ Ei  ref∣  ∑N  i=1 ∣Ei  ref∣  (4)  The TEM score is the harmonic mean of precision and  recall (also known as the F1 score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' List of temporal keywords  The list of temporal keywords used to compute the TEM  score are as follows: {bigger,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' change,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' cleared,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  decrease,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' decreased,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' decreasing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' elevated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' elevation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' en-  larged,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' enlargement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' enlarging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' expanded,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' greater,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' growing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  improved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' improvement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' improving,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' increase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' increased,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  increasing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' larger,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' new,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' persistence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' persistent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' persisting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  progression,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' progressive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' reduced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' removal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' resolution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' re-  solved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' resolving,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' smaller,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' stability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' stable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' stably,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' un-  changed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' unfolded,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' worse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' worsen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' worsened,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' worsening,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  unaltered}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Architecture and implementation details  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Hyper-parameters  The models are trained in a distributed setting across 8  GPU cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For pre-training, we use a batch size of 240  (30 * 8 GPUs) and the AdamW optimizer [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We use a  linear learning rate scheduler with a warm-up proportion of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='03 and base learning rate of 2 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We train for a max-  imum of 50 epochs and use validation set loss for check-  point selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The overall loss is a sum of components  with weighting factors: global contrastive (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0), local con-  trastive (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5), and image-guided MLM (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='0) respectively,  see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1 for further details on their formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Following [9] we use sentence permutation as text-based  data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Similarly, spelling errors in the reports  are corrected prior to tokenisation of the text data8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For  image augmentations, note that we apply the same augmen-  tation to current and prior images to prevent severe mis-  alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We resize the shorter edge to 512 and centre-  crop to (448, 448).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We apply random affine transformations  (rotation up to 30○ and shear up to 15○) and colour jitter  (brightness and contrast).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Training infrastructure  We train with distributed data processing (DDP) on eight  NVIDIA Tesla V100s with 32GB of memory each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To han-  dle inconsistently-present prior images with DDP, we define  a custom batch sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' This sampler is a mixture of two  samplers, in proportion to their dataset coverage: a sam-  pler which produces batches with only multi-image exam-  ples – (xcurr  img ,xprior  img ,xcurr  txt ) ∈ Dm and one with only single-  image examples – (xcurr  img ,∅,xcurr  txt ) ∈ Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Each GPU then  processes a batch which is entirely single or multi-image,  avoiding branching logic within the forward pass and en-  abling an efficient single pass through the CNN to process  all input images (current or prior) by concatenating them  along the batch dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  We confirmed that although the custom sampler theoret-  ically impacts the order in which the dataset is traversed,  it has a negligible effect on training metrics relative to fully  random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Since we train on eight GPUs and collect  negatives across all GPUs during contrastive training, each  update involves on average a representative mixture of both  single-image and multi-image samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  8https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='com/farrell236/mimic-cxr/blob/  master/txt/section_parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='py  17  Finally, following [9] we use the DICOM images from  MIMIC-CXR to avoid JPEG compression artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Adaptation and experimentation details  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Fine-tuning BioViL-T for report generation  During fine-tuning of BioViL-T for report generation,  we minimise the cross entropy loss to maximise the log like-  lihood of the report in an autoregressive manner given the  input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The model is initialised from the pretrained  weights of the image encoder and the text encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Sim-  ilar to the cross-modal masked language modelling task,  we additionally train a linear projection layer to map the  projected patch embeddings to the same hidden dimension  of the text encoder, and we train cross-attention layers in  each transformer block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The difference from the masked  language modelling task is that we change the bidirectional  self-attention to unidirectional causal attention that can only  access the past tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' If trained with prior report, we pass  the prior report as prefix to condition the generation of the  current report (the current and prior report are separated by  [SEP]), and we only back-propagate the gradients from  the loss on the tokens in the current report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For all experiments, we train the model for 100 epochs  and we chose the best checkpoint according to metrics on  the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We performed grid search for learning  rate in [10−5,2 × 10−5,5 × 10−5] and found 2 × 10−5 to be  optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We ran each experiment with 3 random seeds and  report mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  In addition to the metrics we reported in the main text,  we also evaluate the generated reports by named entity met-  ric (NEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' This metric was defined in [44] to measure the  accuracy of reporting clinically relevant entities in the gen-  erated reports (Similar to how TEM is computed to measure  the match of temporal entities in our study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Following [44],  we extract entities (tagged as “observation” or “observa-  tion modifier”) from the text by running the named entity  recognition model in the Stanza library [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The results are  presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Nearest-neighbour-based report retrieval  The joint latent space learnt by BioViL-T can also be  used to directly perform report retrieval without requiring  task-specific model fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Given the test image, we  retrieve its semantically closest report from the training set  in the joint latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Specifically, we encode each test  image with the image model in BioViL-T and collect its  projected image embeddings, and similarly we encode all  the reports in the training data with their projected text em-  beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For each test study, we compute cosine similarity  between the test image embedding and all the text embed-  dings from the training set in the joint latent space, and we  retrieve the closest text embedding and use its correspond-  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Results for report generation task: Predictions are  evaluated on NEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The approaches are grouped into two broad  categories: NN (Nearest Neighbour) and AR (Auto-Regressive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  BioViL-T pre-training consistently yields superior decoding per-  formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Further, the use of prior image and report consistently  yield performance gains demonstrating the importance of such do-  main priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Method  Pre-training Prior Img/Report  NEM  NN  CXR-RePaiR-2 [24]  BioViL  \x17 / \x17  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='36  Baseline (NN) [9]  BioViL  \x17 / \x17  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='25  Proposed (NN)  BioViL-T  \x13/ \x17  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='55  AR  Baseline (AR) [9]  BioViL  \x17 / \x17  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='22  Proposed  BioViL-T  \x13/ \x17  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='04  Proposed  BioViL-T  \x13/ \x13  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='17  ing report as the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To evaluate the retrieval perfor-  mance, we use the same decoding metrics on the retrieved  reports and report results in the top section of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  In a separate set of experiments, we also tried performing  nearest neighbour search only within the image embedding  space by retrieving the report associated with the closet im-  age embedding, but this yielded sub-optimal performance  compared with using the joint latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Fine-tuning for temporal image classification  In this section, we describe the training dataset and fine-  tuning procedure for the fully supervised and few-shot set-  tings of the temporal image classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For this task,  we finetune BioViL-T on a subset of the Chest ImaGenome  silver dataset [71] to predict progression labels for 5 differ-  ent pathologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To create our training dataset, we filter out  image pairs from this dataset where there are multiple di-  rections of progression of a single pathology in the image-  pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We additionally perform an automatic data curation  step to choose higher quality image pairs when possible, as  described in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2 shows the number of training  samples and label distribution for the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Statistics of the training dataset used for downstream  fine-tuning on temporal image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Findings  # labelled pairs  Class distribution  # of subjects  Consolidation  7012  15% / 42% / 43%  3308  Edema  14170  28% / 33% / 39%  4813  Pleural effusion  26320  16% / 53% / 31%  6838  Pneumonia  8471  12% / 29% / 59%  4197  Pneumothorax  3795  21% / 57% / 22%  1161  For the fully supervised setting, we add a multilayer clas-  sification head to the BioViL-T image encoder and fine-  tune the model independently for each pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We use  weighted cross entropy loss with a batch size of 128 and  the AdamW optimizer [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' During parameter optimisation,  positional encodings and missing-image embeddings are  18  Target class  Tokens  Improving  better, cleared, decreased, decreasing, im-  proved, improving, reduced, resolved, re-  solving, smaller  Stable  constant, stable, unchanged  Worsening  bigger, developing, enlarged, enlarging,  greater, growing, increased, increasing,  larger,  new,  progressing,  progressive,  worse, worsened, worsening  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Prompting the AR language decoding model for zero-  shot image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The list above shows the mapping from  decoded tokens to progression classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  exempt from weight decay penalty as in [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We train for  30 epochs, with a linear learning rate schedule, a warmup  proportion of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='03 and a base learning rate of 1 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For  data augmentation, we first resize the shorter edge of the im-  age to 512 and centre crop to (448, 448).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We apply random  horizontal flips, random cropping, random affine transfor-  mations (rotation up to 30○, shear up to 15○), colour trans-  forms (brightness and contrast) and Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For the few-shot setting we tune only a single-layer lin-  ear head on the BioViL-T image encoder and freeze the rest  of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We initialise the weight matrix of the linear  head with values from encoded text prompts [9] for each  of the three progression classes, and the bias matrix is ini-  tialised with zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To train, we again use weighted cross  entropy loss, with a batch size of 32 and the AdamW opti-  mizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We use a learning rate of 1 × 10−3 and train for 40  epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For data augmentation, we resize the shorter edge  of the image to 448 and center crop to (448, 488).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We ap-  ply random horizontal flips, random affine transformations  (rotation up to 45○ and shear up to 25○), colour transforms  (brightness and contrast).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' As in the pre-training step, we  always synchronise image data augmentations to apply the  identical transforms to the current and prior images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Auto-regressive prompting for zero-shot tem-  poral image classification  Following the GPT-3 style language prompting [11], we  prompt the fine-tuned AR language decoding model with  the template: “[FINDING] is” and infer the next token  to perform temporal classification for each of the five find-  ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The mapping from the predicted next token to the  three progression classes is characterised by a short list of  tokens provided in Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' After computing the posterior  for each token in the list, the obtained values are normalised  across the three classes, and the class with the highest score  is selected as the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The corresponding results are  reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Further analysis of image-guided MLM  In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6 we used a simplified notation for the  computation of ∆prior  img (m) for ease of exposition – here  we provide further detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Recall that w = (w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=',wM)  is a sequence of tokens and w/m is that sequence  with token m masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Let pθ(wm ∣w/m,xcurr  img ,xprior  img )  be the text model’s predicted probability of token m  given xcurr  img ,xprior  img , and w/m (θ are the weights of the  model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Then, l(w,pθ(wm ∣w/m,xcurr  img ,xprior  img )) is the  cross-entropy loss of predicting token m given those inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  It is possible for different sentences in a report to refer  to the same image finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Since we mask single tokens  at a time, to prevent information leakage from other sen-  tences we consider each sentence in a report independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Suppose report xcurr  txt consists of S sentences, so we have  xcurr  txt = [w1,[SEP],.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=',[SEP],wS], where ws is the to-  kens of sentence s and [SEP] separates sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For a given sample (xcurr  img ,xprior  img ,xcurr  txt ) ∈ Dm in the test  set indexed by i, we define  δi(m) = ∑  s∈S  [l(m,pθ(ws  m ∣ws  /m,xcurr  img ,∅))  −l(m,pθ(ws  m ∣ws  /m,xcurr  img ,xprior  img ))]  This is the MLM loss for predicting m given each sentence  in the report with and without the prior image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Note that if  m does not appear in a given sentence, its contribution to  the sum is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The overall ∆prior  img (m) is computed across  all samples:  ∆prior  img =  1  Nm  ⎛  ⎝ ∑  i∈Dtest  m  δi(m)⎞  ⎠  (5)  where Nm is the number of sentences in reports in Dtest  m  in which token m appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' This estimate is subject to high  variance when Nm is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Hence, for Figure 4 we filter  to tokens m with Nm ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We collected 931 tokens with  Nm ≥ 10 from the validation set for manual annotation by  a board-certified radiologist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The categories, shown in Fig-  ure 4 and described in Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4 are specific to the radiology  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Sentence similarity experiment  The text models are evaluated in isolation to observe if  their encoding is sensitive to key clinical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To  achieve this, we assess the quality of sentence represen-  tations obtained from our text model by examining how  well the contradiction and paraphrase pairs can be sepa-  rated in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Unlike the traditional NLI  task where a model needs to be fine-tuned, here the models  are probed in a zero-shot setting and the BERT output token  embeddings are utilised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To do so,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' we encode the sentences  from RadNLI and MS-CXR-T sentence similarity datasets  19  Category  Description  Examples  Progression  Pertaining to change or progression  bigger,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' cleared,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' new  Support devices  Tubes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' lines and implants  nasogastric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' pacemaker,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' cannula  ‘Other’  No clear category  can,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' relevant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' overall  Stop word  ‘Insignificant’ words  the,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' no,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' of  Positional  Localisation (not anatomical)  right,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' lower,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' bilateral  Meta  Pertaining to the report itself or practice of radiology  evidence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' radiograph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' study  Anatomy  Anatomical locations  pulmonary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' chest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' mediastinal  Descriptive  Qualitative appearance of a finding  layering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' focal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' patchy  Size or degree  Quantifying extent or severity  extensive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' moderate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' severe  Finding  Radiographic finding or pathology  edema,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' penumonia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' pneumothorax  Uncertain  Expression of certainty or doubt  may,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' possible,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' concerning  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Semantic categories used in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Prior image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Current image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='(a) Improving consolidation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Prior image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Current image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='(b) Stable consolidation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Prior image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Current image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='(c) Worsening consolidation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Prior image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Current image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='(d) Improving pulmonary edema  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Prior image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Current image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='(e) Stable pulmonary edema  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Prior image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Current image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='(f) Worsening pulmonary edema  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Prior image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Current image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='(g) Improving pleural effusion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Prior image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Current image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='(h) Stable pleural effusion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Prior image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Current image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='(i) Worsening pleural effusion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='Figure F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Examples of image pairs in our MS-CXR-T benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  with the [CLS] token from CXR-BERT-Specialised [9]  and BioViL-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For PubMedBERT [28] and CXR-BERT-  General [9] which did not directly optimise the [CLS] to-  ken during pretraining, we follow [55] to average the token  output embeddings to represent each sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  Cosine similarity is computed between the representa-  tions of each sentence pair in the dataset [55] and is used as  logits for the binary classification between paraphrase and  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Note that for RadNLI, we use the subset of  ‘entailment’ and ‘contradiction’ pairs and discard the ’neu-  tral’ pairs to unify the task across the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Given  the similarities for each sentence pair, we report ROC-AUC  and binary-accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' For the latter, a threshold value for  each method is derived by setting aside a validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  For this, we perform ten-fold cross validation and tune the  threshold with step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='005 on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  20  ERECT  TABLEORTABLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Image registration algorithm  In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='2, image registration is applied to pairs of  images as a preprocessing step to enable a fair compari-  son for the baseline approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=', BioViL [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' We per-  formed bidirectional multi-scale registration between image  pairs optimising an affine transformation (4 degrees of free-  dom), using mutual information (MI) [64] with 128 bins as  the similarity criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' In more detail, the spatial transfor-  mation is characterised by four parameters: two for transla-  tion, one for isotropic scaling, and one for rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' The op-  timisation is repeated five times with different random seeds  for initialisation, and the run with the highest MI is selected  to determine the final spatial alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' To better identify  the correspondences between the scans, bilateral filtering is  applied to each image before registration to remove detailed  texture whilst preserving edge information [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content=' Our imple-  mentation is based on the SimpleITK library [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
+page_content='  21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQfgArH/content/2301.04558v1.pdf'}
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+Prepared for submission to JHEP
+Scale-free non-Hermitian skin effect in a
+boundary-dissipated spin chain
+He-Ran Wang, Bo Li, Fei Song, Zhong Wang
+Institute for Advanced Study, Tsinghua University, Beijing, 100084, China
+E-mail: whr21@mails.tsinghua.edu.cn, boliwalker@outlook.com,
+songf18@mails.tsinghua.edu.cn, wangzhongemail@tsinghua.edu.cn
+Abstract: We study the open XXZ spin chain with a PT-symmetric non-Hermitian
+boundary field. We find an interaction-induced scale-free non-Hermitian skin effect by using
+the coordinate Bethe ansatz. The steady state and the ground state in the PT broken phase
+are constructed, and the formulas of their eigen-energies in the thermodynamic limit are
+obtained. The differences between the many-body scale-free states and the boundary string
+states are explored, and the transition between the two at isotropic point is investigated.
+We also discuss an experimental scheme to verify our results.
+Keywords: Bethe Ansatz, Quantum Dissipative Systems, Spin Chain, Non-Hermitian
+Skin Effect
+arXiv:2301.11896v1  [quant-ph]  27 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Non-Hermitian XXZ model and the phase diagram
+3
+3
+Bethe ansatz solutions for scale-free skin modes
+6
+3.1
+Single-magnon state
+6
+3.2
+Two-magnon state
+6
+3.3
+Imaginary Bethe equation at 0 < ∆ < 1
+8
+3.4
+Boundary bound state at ∆ > 1 and phase transition
+12
+4
+Ground state phase diagram
+13
+4.1
+Scale-free solutions for −1 < ∆ < 0
+14
+4.2
+Imaginary Bethe equation at ∆ < −1
+14
+5
+Experimental Realizations
+16
+6
+Conclusion
+18
+1
+Introduction
+Exactly solvable models play important roles in condensed matter physics, statistical
+physics, and mathematical physics. Certain experimentally relevant one-dimensional sys-
+tems can be modeled by open spin chains with boundary fields, some of which belong to the
+category of Yang-Baxter integrability. Examples include spin chains with diagonal [1–8]
+or off-diagonal [9–12] magnetic field. The problem is also related to classical dynamics of
+molecules with drain and source [13–17], and spin transport within the framework of Lind-
+blad master equation [18–25]. Many mathematical tools, such as coordinate Bethe ansatz
+[1, 2, 6, 7, 14, 26], Sklyanin’s reflection algebra (an open boundary version of algebraic
+Bethe ansatz) [3–5, 8–12, 27], matrix product operator ansatz [13, 15–19, 21–24], etc, have
+been developed to treat those systems.
+In this article, we investigate a non-Hermitian open XXZ chain using coordinated Bethe
+ansatz. The chain is subjected to opposite imaginary magnetic field on two ends, pointing
+to a prescribed direction called z.
+Here, the non-Hermiticity naturally stems from the
+ubiquitous coupling with the environment. A special case has been studied thoroughly in
+previous literature, where the strength of the boundary field takes a specific value depending
+on the anisotropic interaction strength between adjacent spins.
+The Hamiltonian then
+respects the q−deformed SU(2) symmetry [28] with |q| = 1, and serves as a representation
+of the Temperley-Lieb algebra [29, 30]. The spectrum of the model is purely real, though
+the Hamiltonian is non-Hermitian. Furthermore, when q is a root of unity for some values
+– 1 –
+
+of the boundary field, the representation of the symmetry group enjoys richer structures,
+such that an exact duality between the spin model and free-end quantum Potts model. The
+duality leads to the same conformal field theory (CFT) structure of two models, thus the
+negative central charge obtained from the side of Potts model makes the non-Hermitian
+spin chain a typical example of recently introduced “non-unitary” CFT [31, 32].
+A significant consequence of q being a root of unity is that the spin Hamiltonian
+develops Jordan blocks, which feature exceptional points. The number of Jordan blocks for
+given q and size N has been counted [33], followed by the constructions of the corresponding
+generalized eigenstates [34]. Given the existence of many-body exceptional points, it is
+natural to identify different phases around them.
+Our article exhausts the parameter
+space of boundary imaginary field and anisotropic interaction. For a small boundary field,
+the spectrum remains real, and the Bethe roots of the ground state only shift slightly
+compared with the Hermitian case. The spectrum becomes complex, however, when the
+boundary field exceeds the q−deformed SU(2) symmetric value. We show that, despite the
+breaking of the quantum group invariance, the model possesses a novel behaviour of scale-
+free localization. We figure out the structure of the steady state (with the largest imaginary
+part of energy) and the ground state (with the lowest real part of energy) as the combination
+of a boundary Bethe root and a set of continuous Bethe roots in the thermodynamic limit.
+The continuous Bethe roots all have an imaginary part proportional inversely to the system
+size N, corresponding to a small imaginary wavevector (or momentum) κ ∼ α/N.
+In
+the single-particle context, when the localization length of wavefunction is proportional
+to the system size N, the density |ψN(x)|2 ∼ exp(2αx/N) is invariant under re-scaling
+transformation with factor s: |ψN(x)|2 = |ψsN(sx)|2, and therefore called scale-free non-
+Hermitian skin effect (NHSE) or critical NHSE [35–37]. Scale-free NHSE has also been
+found in Hermitian systems with non-Hermitian boundary field, though the mechanism
+is different [38]. In the present work, the imaginary part of wavevector is attributed to
+the scattering between the boundary mode and magnons traveling in the bulk, and these
+Bethe roots contribute a non-negligible imaginary part to the energy. Thus, unlike previous
+works, our scale-free behaviour has a many-body origin. More precisely, it originates from
+the interplay between boundary dissipation and many-body interactions. On one hand,
+the interaction among magnons is an indispensable ingredient for the scale-free NHSE. On
+the other hand, we also note that the Hermitian counterpart, namely the open XXZ model
+subjected to real boundary field, has only isolated boundary modes, and such continuous
+skin modes are lacking [6–8]. We derive an integral equation, dubbed imaginary Bethe
+equation, to solve the scale-free localization length in the thermodynamic limit. We then
+give an exact formula for the imaginary part of the steady state energy, which are then
+compared to finite-size numerical results. We also explain how to measure these physical
+quantities in cold-atom experiments.
+Before proceeding, we compare our results to earlier studies on boundary-driven spin
+chains as open quantum systems. The evolution of those open quantum systems is gen-
+erated by the Linbladian operator, composed of an integrable Hamiltonian and quantum
+– 2 –
+
+jump operators on the boundary. A typical example relevant to our work is [19]
+L(ρ) = −i[HXXZ, ρ] +
+�
+µ=1,N
+LµρL†
+µ − 1
+2{L†
+µLµ, ρ}
+= −iHeffρ + iρH†
+eff +
+�
+µ=1,N
+LµρL†
+µ,
+with L1 = √gS−
+1 , LN = √gS+
+N and Heff = HXXZ − i
+2
+�
+µ=1,N L†
+µLµ. Here, Heff is the non-
+Hermitian Hamiltonian we shall focus on below (see eq.(2.1)). Although the Lindbladian
+breaks integrability, the density matrix of non-equilibrium steady state (NESS) has been
+established by the matrix product operator (MPO) ansatz exactly. Furthermore, it has
+been found that the local matrix of MPO ansatz is indeed the infinite-dimensional solution
+of Yang-Baxter relations, and thus exterior integrability emerges in the NESS [39, 40].
+However, the dynamics towards NESS is unknown yet. Our work about the non-Hermitian
+effective Hamiltonian is complementary to the NESS solution because Heff governs the
+time evolution of the open quantum system under post-selection, which is relevant to
+numerous experiments [41]. Our solution is enabled by the Yang-Baxter integrability of
+the model. Another related system is the XXZ model with only one jump operator L1
+on the left boundary [26]. Since the dissipator is purely lossy, the Lindbladian becomes
+upper-triangular under an appropriate basis choice, so that the Liouvillian spectrum can
+be completely determined by the effective non-Hermitian Hamiltonian. The Hamiltonian
+has scale-free eigenstates even in the single-magnon sector, but PT symmetry is absent due
+to that the dissipator occurs only on one of the two ends. By contrast, our Hamiltonian
+preserves PT symmetry, and in single-magnon sector there are only Bloch-wave modes and
+exponentially localized states. Scale-free modes originate from many-body interactions in
+our model.
+The rest part is organized as follows. In the next section, we introduce the model
+Hamiltonian, its general Bethe equations, and the phase diagram. In Sections 3.1 and 3.2,
+we consider the single-magnon and two-magnon state as a warm-up. We then generalize the
+results to the many-body cases to obtain the steady state with scale-free NHSE in Section
+3.3. Section 3.4 is devoted to another type of steady state solution, the boundary string
+states, which emerges for the highly anisotropic case. In Section 4, we apply the ansatz of
+scale-free solutions to the ground state for different parameters. A possible experimental
+setup for the non-Hermitian model is discussed in section 5. We give some concluding
+remarks in Section 6 .
+2
+Non-Hermitian XXZ model and the phase diagram
+The Hamiltonian reads:
+H = −
+N−1
+�
+j=1
+(Sx
+j Sx
+j+1 + Sy
+j Sy
+j+1 + ∆Sz
+j Sz
+j+1) + ig
+2 (Sz
+N − Sz
+1).
+(2.1)
+where Sα = 1
+2σα(α = x, y, z) is the spin-1/2 operator; the anisotropic interaction strength
+∆ and boundary field strength g are purely real, with g > 0. The model respects the PT
+– 3 –
+
+symmetry with TiT = −i and PSα
+j P = Sα
+N−j, and therefore the eigenvalues are either real
+or form complex conjugate pairs. When the whole spectrum is purely real, the model is
+said to be in the PT exact (or PT-symmetric) phase, otherwise it is in the PT broken
+phase [42, 43]. The steady state, which has the largest imaginary part of eigen-energy in
+the PT broken phase, is of great importance because it captures the long-time behaviour
+of the system. A generic initial wavefunction evolving for a sufficiently long time under
+exp(−iHt) will converge to the steady state; we shall study the phase diagram of this steady
+state. The Hamiltonian also commutes with total z magnetization m = �N
+j=1 Sz
+j , so that
+it can be block diagonalized in each sector with definite total magnetization. Furthermore,
+there is another symmetry operator PX with X = �N
+j=1 σx
+j which sends m to −m, and
+therefore it suffices to study non-positive magnetization (m ≤ 0) states. For the odd length
+chain, PX symmetry leads to the two-fold degeneracy of the steady states. Thus, we only
+take even site number N throughout the paper.
+Ferromagnetic (∆ > 0) and anti-ferromagnetic (∆ < 0) models can be related by the
+transformation Z = �N/2
+j
+σz
+2j−1:
+ZTH(∆, g)ZT = ZH(∆, −g)Z = −H(−∆, g).
+(2.2)
+As such, an eigenstate of ferromagnetic Hamiltonian with H(∆, g)|ψ⟩ = E|ψ⟩ can be trans-
+formed to an eigenstate of the anti-ferromagnetic one: H(−∆, g)(ZT|ψ⟩) = −E∗(ZT|ψ⟩).
+If E has the largest imaginary part among the spectrum, so does −E∗. Thus, the steady
+state properties remain the same for ±∆, and it suffices to study the ferromagnetic case.
+Δ
+g
+PT exact
+Scale-free skin effect 
+Boundary 
+string
+Boundary 
+string
+1
+-1
+1
+Figure 1.
+Steady-state phase diagram of model Eq.
+(2.1) in the zero magnetization sector.
+The critical curve ∆2 + g2 = 1 separates PT exact and broken phases.
+When |∆| < 1 and
+g > gc =
+√
+1 − ∆2, the steady state is the many-body scale-free state; when |∆| > 1, the steady
+state is the boundary string state.
+Eigenstates of the model can be solved by coordinate Bethe ansatz [1]. Take all spin
+down state as the reference state with energy E0 = − 1
+4(N −1)∆, we can excite M magnons
+by flipping M spins up (M ≤ N/2).
+We can construct the ansatz state |ψ⟩ , whose
+– 4 –
+
+wavefunction in the onsite magnon number basis is given by:
+⟨n1 · · · nj · · · nM|ψ⟩ =
+�
+P
+�
+{ηj}
+(−1)PA({eikj}; P, {ηj})eiη1kP1n1 · · · eiηjkPjnj · · · eiηMkPMnM ,
+where n1 < · · · < nj < · · · < nM are the positions of up spin, P refers to all possible
+permutations, and chirality ηj = ±1 corresponds to right-moving and left-moving magnons.
+Relabeling the momentum of magnon by βj = eikj [44, 45], we have the equivalent form:
+⟨n1 · · · nj · · · nM|ψ⟩ =
+�
+P
+�
+{ηj}
+(−1)PA({βj}; P, {ηj})βη1n1
+P1
+· · · βηjnj
+Pj
+· · · βηMnM
+PM
+.
+(2.3)
+The coefficient A({βj}; P, {ηj}) is a function of magnon momenta {βj}, permutation P,
+and chirality {ηj}. Imposing the condition that |ψ⟩ is an eigenstate of H with energy
+EM({βj}) = E0 +
+M
+�
+j=1
+(∆ − (βj + β−1
+j )/2)
+(2.4)
+and the boundary condition, those coefficients can be found as [1]:
+A({βj}; P, {ηj}) =
+M
+�
+j
+(1 − ∆+βηj
+Pj)β−(N+1)ηj
+Pj
+�
+j<l
+(1 − 2∆βηl
+Pl + βηl
+Plβ−ηj
+Pj )(1 − 2∆βηj
+Pj + βηj
+Pjβηl
+Pl)β−ηl
+Pl ,
+(2.5)
+where ∆± = ∆ ± ig, and the constraints of M momenta, so-called Bethe equations, can be
+derived:
+β2(N−1)
+j
+(βj − ∆+)(βj − ∆−)
+(β−1
+j
+− ∆+)(β−1
+j
+− ∆−) =
+M
+�
+l̸=j
+1 − 2∆βj + βjβl
+1 − 2∆βl + βjβl
+1 − 2∆βj + βjβ−1
+l
+1 − 2∆β−1
+l
++ βjβ−1
+l
+.
+(2.6)
+Heuristically, the left hand side is the phase accumulated by a magnon when it travels
+freely from one end of the chain to the other and then gets reflected back; each term of
+the right hand side represents the scattering phase between magnon j and l. Note that for
+the periodic XXZ model there is only the βj, βl term in the scattering phase, while for the
+open boundary condition βj also scatter with β−1
+l
+. The solutions are inversion-symmetric
+in the sense that βj and β−1
+j
+always appear in pairs in the solutions. After solving a set of
+Bethe roots {βj}, the energy of M magnon state is
+EM({βj}) = E0 +
+M
+�
+j=1
+(∆ − (βj + β−1
+j )/2).
+(2.7)
+Although it remains unclear whether all eigenstates of the non-Hermitian XXZ model have
+the form of Bethe ansatz, it turns out that the ground state and the steady state can be
+solved exactly.
+Our results on the steady states for different parameters are summarized in Fig. 1. We
+focus on the zero magnetization sector (m = 0) since for the Hermitian XXZ model (with
+– 5 –
+
+∆ < 1) the ground state lies in this sector. Our numerical results support that the steady
+state belongs to the zero magnetization sector. The phase boundary between PT-exact and
+broken phase is ∆2 + g2 = 1. Given the transformation between H(∆, g) and H(−∆, g),
+the steady-state phase diagram is symmetric about the g axis. Apart from the many-body
+scale-free modes which will be investigated in detail in Section 3, we identify the “boundary
+string” state as the steady state when |∆| > 1. A boundary string corresponds to a multi-
+magnon bound state localized at the boundary, which will be explained in Section 3.4.
+3
+Bethe ansatz solutions for scale-free skin modes
+3.1
+Single-magnon state
+In the single-magnon sector (M = 1), the Bethe equation (2.6) is simplified to
+β2(N−1)
+(β − ∆+)(β − ∆−)
+(β−1 − ∆+)(β−1 − ∆−) = 1.
+(3.1)
+When |∆±| < 1, or equivalently g < gc, solutions of the equation have been found to
+be on the unit circle [46]. We review the proof briefly: define complex variable function
+h(β) : C → C as
+h(β) = βN−1(β − ∆+)/(β−1 − ∆−).
+(3.2)
+Eq.(3.1) is then transformed to h(β) = h(β−1). For g < gc, the image of the disk |β| < 1
+under h is still inside the disk, that is, |h(β)| < 1, and vice versa. The statement can be
+verified by writing β = ρ exp(iφ), ∆+ = ρ0 exp(iφ0) with ρ0 < 1, then
+|β − ∆+|2 − |β−1 − ∆−|2 = (ρ − ρ−1)(ρ + ρ−1 − 2ρ0 cos(φ − φ0)).
+Since ρ + ρ−1 ≥ 1 > 2ρ0 cos(φ − φ0), we have |β − ∆+| < |β−1 − ∆−| when |β| < 1
+(ρ < 1 < ρ−1), therefore |h(β)| < 1.
+A similar argument works for |β| > 1.
+Thus,
+possible solutions of h(β) = h(β−1) must be on the unit circle, corresponding to purely
+real momentum.
+When g > gc, no theorem prohibits the existence of non-unitary solutions, and one can
+notice that a pair of isolated boundary modes with β± ≈ ∆± is possible. For such a solution,
+|β±| > 1 leads to the divergence of the term β2(N−1) in Eq. (3.1) in the thermodynamic
+limit, but this can be compensated by the factor (β − ∆+)(β − ∆−), which is close to zero.
+Most of the energy levels remain real, and the scale-free localization behavior is absent in
+this sector. We define the boundary imaginary energy contributed by one boundary mode
+as
+Eb = −1
+2Im(∆− + ∆−1
+− ) = 1
+2(g −
+g
+∆2 + g2 ).
+(3.3)
+3.2
+Two-magnon state
+In the M = 2 sector, scale-free modes appear and contribute to the 1/N scaling behaviour
+of energy. Figure 2(a1) shows a typical finite-size two-magnon spectrum. We color the
+– 6 –
+
+eigenvalues by many-body participation entropy [47–49], a measure of localization gener-
+alized from single-particle inverse participation ratio (IPR):
+S2(|ψ⟩) = − log
+�
+i |ψi|4
+(�
+i |ψi|2)2 ,
+(3.4)
+where the index i sums over all the basis functions in the relevant Hilbert space, |ψ⟩ is
+the many-body eigenstate concerned.
+The participation entropy gets smaller when the
+eigenstate is more localized in the Hilbert space, as we observed in figure 2(a1): Two dark
+points correspond to isolated states bounded to the boundaries, and both two magnons are
+localized to the same side; around Im(E) = ±Eb there is a continuum of states, which are
+the combination of a boundary mode with β1 ≈ ∆± and a scale-free mode with β2 ≈ eik; the
+continuum on the real axis is brighter, though there is one localized mode, corresponding
+to the state with two magnons localized on different ends.
+We apply Bethe equation to the state with one magnon bounded to the boundary
+while the other has momentum β2:
+|β2|2N ≈ | 1 − 2∆β2 + ∆−β2
+1 − 2∆∆− + ∆−β2
+1 − 2∆β2 + ∆−1
+− β2
+1 − 2∆∆−1
+− + ∆−β2
+|.
+(3.5)
+The right hand side, which will be denoted by exp(2κ), is of order O(1). Taking logarithm
+of both sides, we have 2Nln|β2| = 2κ+O(1/N). Thus, lnβ2 acquires a first order correction
+to the real part:
+Re(lnβ2) = ln|β2| = κ/N + O(1/N 2).
+Traveling along the chain, the corresponding magnon accumulates an amplitude change
+A = |β2|N ∼ exp (κ).
+The localized mode can be distinguished from disorder-induced
+localization or non-Hermitian skin effect, both of which has the exponential decay behavior
+ψ(x) ∼ exp(κ′x). In contrast, the leading order of A shows no dependence on the system
+size N. Here, the scale-free localization has an intrinsic many-body origin because in the
+non-interacting limit ∆ = 0, the right hand side of Eq. (3.5) equals 1, and the imaginary
+momentum vanishes. Therefore, the phenomenon in our model is dramatically different
+from that of free-particle models [35–37].
+The energy of such two-magnon state is
+Im(E) = Eb − 1
+2Im(β2 + β−1
+2 ) = Eb − 1
+N κ sin(k).
+(3.6)
+The statement is verified by finite-size scaling of the average of those complex energy. As
+shown in Fig. 2(b1), the imaginary part scales linearly with 1/N. In the thermodynamic
+limit, it will converge to ±Eb. Moreover, we add a next-nearest-neighbor zz interaction
+Hnn = −
+N−2
+�
+j=1
+∆′Sz
+j Sz
+j+2
+(3.7)
+to break integrability, yet the numerical results [Fig. 2(b1)(b2)] show no qualitative differ-
+ences, which supports the universality of scale-free behaviour.
+– 7 –
+
+8
+6
+4
+Re(E)
+0.2
+0.1
+0.0
+0.1
+0.2
+Im(E)
+(a1)
+3.5
+4.5
+5.5
+6.5
+10
+8
+6
+Re(E)
+0.30
+0.15
+0.00
+0.15
+0.30
+Im(E)
+(b1)
+3
+4
+5
+6
+0.02
+0.022
+0.024
+0.026
+1/N
+0.0852
+0.0855
+0.0858
+AVG[Im(E)]
+(a2)
+= 0.76 g = 0.83
+= 0.80 g = 0.80
+= 0.85 g = 0.76
+0.02
+0.022
+0.024
+0.026
+1/N
+0.1295
+0.1315
+0.1335
+AVG[Im(E)]
+(b2)
+= 0.77 g = 0.84
+= 0.80 g = 0.80
+= 0.85 g = 0.73
+Figure 2. (a1) and (b1) The spectrum of M = 2 sector for N = 40, ∆ = 0.8, g = 0.8. Eigenvalues
+are colored by the participation entropy of the corresponding eigenstate. (a2) and (b2) Finite-size
+scaling for the average of the imaginary part of the energy. Only those states with Im(E) > 0 in the
+continuous spectrum are taken into consideration. (a1) and (a2) are based on the pristine model
+Eq. (2.1); for (b1) and (b2), additional next-nearest-neighbor coupling ∆′ = 0.3 is added [see Eq.
+(3.7)].
+3.3
+Imaginary Bethe equation at 0 < ∆ < 1
+After the warm-up on two-body scale-free modes, we now generalize it to the many-body
+cases. We will study the parameter space 0 < ∆ < 1, where it is the Luttinger liquid
+phase in the Hermitian limit and the ground state lies in M = N/2 sector. We assume
+that the steady state is composed of a boundary mode and a set of continuous scale-free
+Bethe roots, and then derive the Bethe equations in the thermodynamic limit.
+We adopt a conventional parametrization of magnon momentum [50, 51]:
+βj = −sinh[γ(xj − i)/2]
+sinh[γ(xj + i)/2],
+(3.8)
+where γ = arccos(−∆) such that 0 < γ < π. The kinetic energy of the magnon is
+Exj = −1
+2(βj + β−1
+j ) = 1 − cos(γ) cosh(γxj)
+cos(γ) − cosh(γxj) .
+(3.9)
+– 8 –
+
+Taking the logarithm of Bethe equations (2.6), we have
+2Nθ1(xj) + φb(xj) = 2πIj +
+�
+l̸=j
+[θ2(xj − xl) + θ2(xj + xl)],
+(3.10)
+where the function θn is defined as
+θn(x) = 2 arctan[cot(nγ/2) tanh(γx/2)].
+(3.11)
+The second term φb(xj) on the left of Eq. (3.10) is the scattering phase between the magnon
+j and the boundary, which has the form:
+φb(x) = θm+(x) + θm−(x),
+where θm± is obtained by taking n in Eq. (3.11) as complex numbers
+m± = 1
+γ θ1(ln(cos(γ) ± ig)
+γ
+) + iπ
+2γ .
+This involved boundary term will not have significance in the rest part of solution. For
+the right hand side, Ij is an integer and the set of {Ij} determines the set of Bethe roots.
+We take the occupation of one boundary mode and Ij = j on the steady state, and the
+corresponding boundary Bethe root β0 = ∆− has the parametrization
+x0 = 1
+γ ln g − sin(γ)
+g + sin(γ) − i = λ0 − i.
+(3.12)
+The steady-state Bethe equations becomes
+2Nθ1(xj)+φb(xj) = 2πj +
+�
+l̸=j
+[θ2(xj −xl)+θ2(xj +xl)]+θ2(xj −x0)+θ2(xj +x0). (3.13)
+We rewrite xj = λj + iσj/N with purely real λj, σj. We note that |σj/N| ≪ λj, and
+therefore any real function of xj can be expanded as
+f(xj) = f(λj) + if′(λj)σj/N + O(1/N 2).
+(3.14)
+The real and imaginary part of Bethe equations are:
+2Nθ1(λj) + φb(λj) + O(1) = 2πj + [
+�
+l̸=j
+θ2(λj − λl) + θ2(λj + λl)]
++Re[θ2(λj − x0) + θ2(λj + x0)],
+(2θ′
+1(λj) + 1
+N φ′
+b(λj))σj + O( 1
+N ) = 1
+N
+�
+l̸=j
+θ′
+2(λj − λl)(σj − σl) + θ2(λj + λl)(σj + σl)]
++Im[θ2(λj − x0) + θ2(λj + x0)].
+(3.15)
+For the real (imaginary) part only O(N) (O(1)) terms are included, and the leading terms
+of the real and imaginary part of Bethe equations are:
+2Nθ1(λj) = 2πj + [
+�
+l̸=j
+θ2(λj − λl) + θ2(λj + λl)],
+(3.16)
+2θ′
+1(λj)σj = 1
+N
+�
+l̸=j
+θ′
+2(λj − λl)(σj − σl) + θ2(λj + λl)(σj + σl)]
++Im[θ2(λj − x0) + θ2(λj + x0)].
+(3.17)
+– 9 –
+
+Eq. (3.16) is the same as the ground state Bethe equations of the Hermitian open XXZ
+model [1]. It is standard to calculate the difference between (j + 1) − th and the j − th
+equation, taking f(λj+1) − f(λj) = f′(λj)(λj+1 − λj):
+2Nθ′
+1(λj)(λj+1 − λj) = 2π + [
+�
+l̸=j
+θ′
+2(λj − λl) + θ′
+2(λj + λl)](λj+1 − λj).
+(3.18)
+The thermodynamic limit is taken by sending
+lim
+N→∞
+1
+N(λj − λj+1) = ρ(λj),
+lim
+N→∞
+1
+N
+�
+l
+f(λl) =
+� ∞
+0
+dλρ(λ)f(λ),
+(3.19)
+then the integral equation of ρ(λ) is
+1
+2ρ(λ) +
+� +∞
+−∞
+dλ′ K2(λ − λ′)
+2π
+1
+2ρ(λ′) = K1(λ)
+2π
+,
+(3.20)
+where Kn(λ) = θ′
+n(λ). Note that only when the steady state belongs to the zero magne-
+tization sector, the integral interval of λ can be taken as (−∞, ∞). This is the case here,
+and it corresponds to filling the Fermi sea k ∈ (−π +γ, π −γ). The integral equation (3.20)
+is commonly solved by Fourier transformation:
+˜ρ(ω) =
+� +∞
+−∞
+dλ
+2πρ(λ)eiωλ,
+˜Kn(ω) =
+� +∞
+−∞
+dλ
+2πKn(λ)eiωλ =
+sinh( π
+γ − n)ω
+sinh π
+γ ω
+.
+(3.21)
+Applying the convolution formula on Eq. (3.20), we have a linear equation
+1
+2(1 + ˜K2(ω))˜ρ(ω) =
+˜K1(ω)
+2π
+,
+(3.22)
+then the distribution function is solved: ˜ρ(ω) = 1/(2π cosh ω), ρ(λ) = 1/(2 cosh(πλ/2)).
+Eq. (3.17) counts two kinds of mechanisms of scale-free localization. On the right
+hand side, the first term sums interactions between magnons in the bulk, while the second
+term is the scattering with the boundary mode. The last one is much smaller than the first
+in a few-body state, but becomes comparable when the magnon number is the same order
+of system size N, e.g. in the zero magnetization sector. Define σ(λ) as a function of λ, the
+continuous version of this equation is
+ρ(λ)σ(λ) +
+� +∞
+−∞
+dλ′ K2(λ − λ′)
+2π
+ρ(λ′)σ(λ′) = 1
+2πIm[θ2(λ − x0) + θ2(λ + x0)].
+(3.23)
+Dubbed “imaginary Bethe equation”, it is a central result of this article. To derive the
+linear equation of ˜σρ(ω) =
+� +∞
+−∞
+dλ
+2πeiωλρ(λ)σ(λ), we need to find the Fourier transformation
+– 10 –
+
+of the right hand side:
+� +∞
+−∞
+dλ
+2πeiωλIm[θ2(λ − x0) + θ2(λ + x0)] =
+� +∞
+−∞
+dλ
+2πeiωλIm[θ2(λ − λ0 + i) + θ2(λ + λ0 − i)]
+= 2i sin(ωλ0)
+� +∞
+−∞
+dλ
+2πeiωλIm[θ2(λ + i)]
+= −2 sin(ωλ0)
+ω
+� +∞
+−∞
+dλ
+2πeiωλ d
+dλIm[θ2(λ + i)]
+= 2 sin(ωλ0)
+ω
+� +∞
+−∞
+dλ
+2πeiωλ
+2γ cos(γ) sin2(γ) sinh(γλ)
+(cosh(γλ) − cos(γ))(cosh(γλ) − cos(3γ))
+= 2i sin(ωλ0) sinh(ω)
+sinh[( π
+γ − 2)ω]
+ω sinh( π
+γ ω)
+.
+(3.24)
+Denoting the result above by Θ(ω), we can solve σ(λ) formally:
+˜σρ(ω) =
+1
+1 + ˜K2(ω)
+Θ(ω)
+2π .
+(3.25)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+g
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Im(Es)
+Ferromagnetic:
+= 0.8
+BA
+Eb
+ED
+Figure 3.
+Imaginary part of the steady state energy Es as a function of the non-Hermitian
+parameter g. We take ∆ = 0.8, so that gc = 0.6. Red curve is from the Bethe ansatz (BA). As a
+comparsion, blue line is imaginary energy of a boundary mode (Eb). The black dots are computed
+from exact diagonalization (ED) in the zero-magnetization subspace, with system size N = 16.
+The summation of energy of all those scale-free modes becomes an integral over λ in
+the thermodynamic limit. The imaginary part of energy formula of a single magnon is
+Im(E(x)) = 1
+N E′(λ)σ(λ) = −sin(γ)
+Nγ K′
+1(λ)σ(λ).
+(3.26)
+While each one contributes O(1/N) to the total imaginary part, the sum of contributions
+– 11 –
+
+from all scale-free magnons is comparable to the boundary mode contribution Eb:
+�
+j
+Im(E(xj)) = −sin(γ)
+γ
+� +∞
+0
+dλρ(λ)K′
+1(λ)σ(λ)
+= −sin(γ)
+2γ
+� +∞
+−∞
+dω 2πiω ˜K1(ω)˜σρ(ω)
+= −sin(γ)
+2γ
+� +∞
+−∞
+dω iω
+˜K1(ω)
+1 + ˜K2(ω)
+Θ(ω)
+= −sin(γ)
+γ
+� +∞
+0
+dω sin(−ωλ0) tanh(ω)
+sinh[(π
+γ − 2)ω]
+sinh( π
+γ ω)
+.
+(3.27)
+The imaginary part of the steady-state eigen-energy is then given by adding Eb:
+Im(Es) =
+�
+j
+Im(E(xj)) + Eb.
+(3.28)
+In Fig. 3, we compare our formula with the exact diagonalization (ED) results in the
+M = N/2 sector, which agrees excellently. The boundary field g controls the imaginary
+part of the energy totally via λ0 =
+1
+γ ln g−gc
+g+gc . As g crosses gc, the steady state energy
+becomes complex.
+3.4
+Boundary bound state at ∆ > 1 and phase transition
+Anisotropic interaction ∆ > 1 prefers bounding all magnons together, and therefore the
+magnons in the steady state tend to localize to the boundary. Specifically, the first magnon
+is bound to the boundary, and the next one is bounded to the previous one recursively.
+In the context of integrable spin models, the bound state is named an “string”; we shall
+follow this terminology and call our bound state near the boundary a “boundary string”.
+We note that similar states have been identified in the spin model subjected to a non-
+Hermitian magnetic field at only one end [26]. In the thermodynamic limit, Bethe roots
+{βj} satisfies a recursive relation:
+βj+1 + β−1
+j
+= 2∆, β1 = ∆ − ig.
+(3.29)
+The imaginary part of energy is given by
+Im(Es) = −1
+2
+M
+�
+j=1
+Im(βj + β−1
+j )
+= −1
+2
+M−1
+�
+j=1
+Im(β−1
+j
++ βj+1) − 1
+2Im(β1 + β−1
+M )
+= −1
+2Im(β1 + β−1
+M ).
+For large N, βN/2 approaches the fixed point of recursive relations Eq.
+(3.29): β∞ =
+∆ ±
+√
+∆2 − 1, which is purely real. It follows that Im(Es) = − 1
+2Im(β1) = g
+2.
+– 12 –
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+0.0
+0.2
+0.4
+Im(Es)
+(a)
+g=0.8
+g=0.6
+0.06
+0.08
+0.10
+0.12
+1/N
+0.02
+0.04
+0.06
+0.08
+0-Im(Es)
+(b)
+g = 0.2
+g = 1.0
+g = 1.6
+Figure 4. (a) The imaginary part of the steady state energy in the zero magnetization sector.
+The size N = 16. (b) Finite-size scaling of the steady-state energy at the isotropic point ∆ = 1.
+γ0 = g/2 [see the paragraph above Eq. (3.31)].
+It is clear that the structure of Bethe roots of the steady state is different for 0 < ∆ < 1
+and ∆ > 1. We may take the isotropic limit ∆ = 1 from the two sides to understand the
+phase transition point.
+In the scale-free phase, we have to deal with the γ → π limit of Eq. (3.27) carefully.
+Note that the Fermi sea ranging from −π+γ to π−γ shrinks to a Fermi point when γ → π.
+We take the limit by substituting ω by ωγ/ sin γ with sin γ ≪ 1, and the integral can be
+simplified to
+� +∞
+0
+dω sin(2ω/g) exp(−2ω) = g/2(1 + g2),
+(3.30)
+so that Im(Es) = Eb + g/2(1 + g2) = g/2.
+In the boundary-string phase, the imaginary part is a constant γ0 = g/2. Bethe roots
+can be determined by the recursive equation Eqs. (3.29) at ∆ = 1, which results in an
+explicit solution:
+βn = 1 +
+1
+n − 1 + i
+g
+.
+(3.31)
+For small n, the magnon localizes exponentially at the boundary. However, for n sufficiently
+large (n/N ∼ O(1)), it behaves in a scale-free fashion.
+This result is also confirmed
+by comparing the imaginary part with γ0 for different system sizes, and the differences
+scale linearly with 1/N (see Fig.4). We emphasize that the solution Eq. (3.31), though
+qualitatively valid, is not exact because the scattering between the large n scale-free modes
+have been neglected. This approximation has been implied in Eq. (3.29).
+4
+Ground state phase diagram
+For the ferromagnetic case ∆ > 0, the steady state and the ground state coincide, and the
+phase boundary ∆ = 1 separates the scale-free phase and the boundary string. However,
+the transformation ZT relating the ferromagnetic and the anti-ferromagnetic steady state
+– 13 –
+
+Δ
+g
+Luttinger liquid
+Luttinger liquid 
+(Scale-free) 
+Boundary 
+string
+Gapped spinon
+(Scale-free)
+1
+-1
+1
+Figure 5. Ground state phase diagram in the zero magnetization sector. The right half of the
+diagram, ∆ > 0, coincides with the counterpart of steady state (see figure 1). In the PT exact
+phase (g < gc ≡
+√
+1 − ∆2), the ground state is Luttinger liquid (LL); for g > gc, gapless excitations
+become scale-free. When ∆ < −1, the gap between the ground state and excited states opens, yet
+our ansatz of scale-free skin modes remains valid.
+is not applicable to the ground state, because it changes to the highest-energy (real part)
+state when reversing the sign of ∆. Therefore, one cannot borrow the ground-state phase
+diagram from that of the steady state. Moreover, the comparison between ground states
+at zero and finite non-Hermiticity provides another perspective on the effect of boundary
+dissipation.
+In Fig.
+5, we summarize the results of the ground state.
+The ansatz of
+many-body scale-free state in the region ∆ < 0 is studied in the following subsections.
+4.1
+Scale-free solutions for −1 < ∆ < 0
+For −1 < ∆ < 0, we find that Eq. (3.27) can still be applied to the ground state, though
+it does not coincide with the steady state anymore. The formula is compared with exact
+diagonalization results in Fig. 6. It seems that the data does not agree as well as in the
+ferromagnetic case (see Fig. 3). This is due to the larger finite-size error. In fact, we
+observe that as size N increases, the ED results become closer to the analytical ones.
+4.2
+Imaginary Bethe equation at ∆ < −1
+The imaginary Bethe equation also works for ∆ < −1, with some technical modifications.
+Retaining the parametrization ∆ = − cos(γ), |∆| > 1 now leads to a purely imaginary γ,
+and it is convenient to write γ → iφ:
+βj = −sin[φ(xj − i)/2]
+sin[φ(xj + i)/2],
+(4.1)
+where φ = arccos h(−∆). The boundary Bethe root is
+x0 = − 2
+φ arctan(sinh(φ)
+g
+) − i = λ0 − i.
+(4.2)
+– 14 –
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+g
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Im(Eg)
+Antiferromagnetic:
+=
+0.8
+BA
+Eb
+ED,N=10
+ED,N=16
+Figure 6.
+Imaginary part of the ground state energy Eg as a function of the non-Hermitian
+parameter g. We take ∆ = −0.8, so that gc = 0.6. Red curve is from the Bethe ansatz (BA). As a
+comparison, blue curve is imaginary part of the energy of a boundary mode (Eb). The hollow dots
+are obtained from exact diagonalization (ED) with N = 10, and solid dots with N = 16.
+On the ground state the whole Brillouin zone k ∈ (−π, π) is filled, so that Re(x) ∈
+(−π/φ, π/φ). The single-magnon kinetic energy is
+Exj = 1 − cosh(φ) cos(φxj)
+cosh(φ) − cos(φxj) .
+(4.3)
+We also adopted here a new definition of the function θn:
+θn(x) = 2 arctan[coth(nφ/2) tan(φx/2)].
+The imaginary Bethe equation is then given by
+ρ(λ)σ(λ) +
+� +π/φ
+−π/φ
+dλ′ K2(λ − λ′)
+2π
+ρ(λ′)σ(λ′) = 1
+2πIm[θ2(λ − x0) + θ2(λ + x0)].
+(4.4)
+Since functions of λ are periodic functions with periodicity 2π/φ, we can expand them
+as Fourier series to solve the integral equation:
+˜f(m) =
+� +π/φ
+−π/φ
+dλ
+2πf(λ)eimφλ, m ∈ Z
+(4.5)
+Notably, ˜
+Kn(m) = exp(−n|m|φ), and the right hand side of the Eq. (4.4) transforms as
+� +π/φ
+−π/φ
+dλ
+2πeimφλIm[θ2(λ − x0) + θ2(λ + x0)] = 2i sin(mφλ0)
+� +π/φ
+−π/φ
+dλ
+2πeimφλIm[θ2(λ + i)]
+= 2 sin(mφλ0)
+mφ
+� +π/φ
+−π/φ
+dλ
+2πeimφλ
+2φ cosh(φ) sinh2(φ) sin(φλ)
+(cos(φλ) − cosh(φ))(cos(φλ) − cosh(3φ))
+= 2isin(mφλ0)
+mφ
+sinh(mφ)e−2mφ = Θ(m).
+(4.6)
+– 15 –
+
+The imaginary part of energy is:
+�
+j
+Im(E(xj)) = −sinh(φ)
+φ
+� +π/φ
+0
+dλρ(λ)K′
+1(λ)σ(λ)
+= −sinh(φ)
+2
+�
+m∈Z
+2πimφ ˜K1(m)˜σρ(m)
+= −sinh(φ)
+2
+�
+m∈Z
+imφ
+˜K1(m)
+1 + ˜K2(m)
+Θ(m)
+= sinh(φ)
+�
+m∈Z+
+sin(mφλ0) tanh(mφ)e−2mφ.
+(4.7)
+As illustrated in Fig. 7, there are finite-size errors between the numerical results and
+our Bethe ansatz formula, which is similar to the gapless antiferromagnetic case.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+g
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Im(Eg)
+=
+1.2
+BA
+ED,N=10
+ED,N=16
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+g
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+Im(Eg)
+=
+2.0
+BA
+ED,N=10
+ED,N=16
+Figure 7. Imaginary part of the ground state energy as a function of the non-Hermitian parameter
+g. Left panel: ∆ = −1.2; Right panel: ∆ = −2.0. Red curve is obtained from the Bethe ansatz
+(BA). Green triangles and blue squares are numerical results from exact diagonalization of N = 10
+and N = 16, respectively.
+5
+Experimental Realizations
+The onsite non-Hermiticity can be realized in cold atom systems by coupling the spin down
+(up) degrees of freedom of the first (last) site to an auxiliary state by optical pumping
+[52, 53]. Each atom in the bulk has two effective energy levels to mimic a 1/2 spin. XXZ
+interaction can be induced by the mixing of even and odd parity states, e.g. in Rydberg
+atoms [54]. We may introduce the third energy levels on the two ends so that the effective
+spin down (up) state on the left (right) end can decay to it spontaneously. The effective
+loss is described by a non-Hermitian term
+Hloss = −ig
+2 | ↑⟩⟨↑ |1 − ig
+2 | ↓⟩⟨↓ |N = −ig
+2 + ig
+2 (Sz
+N − Sz
+1).
+To evolve the open system under the non-Hermitian Hamiltonian without quantum jump,
+i.e., by post-selection, the population of auxiliary energy levels should be monitored by
+– 16 –
+
+0
+20
+40
+60
+80
+100
+time[1/g]
+0.15
+0.15
+0.45
+Sz
+j  
+(a)
+j=14
+j=10
+j=8
+0
+20
+40
+60
+80
+time[1/g]
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Sz
+j
+(b)
+j=14
+j=10
+j=8
+0
+20
+40
+60
+80
+100
+time[1/g]
+0.20
+0.15
+0.50
+Sz
+j  
+(c)
+j=14
+j=10
+j=8
+0
+20
+40
+60
+80
+time[1/g]
+0.3
+0.4
+0.5
+Sz
+j  
+(d)
+j=14
+j=10
+j=8
+2
+4
+6
+8
+10
+12
+14
+position
+0.5
+0.0
+0.5
+Sz
+j  
+(e)
+2
+4
+6
+8
+10
+12
+14
+position
+0.5
+0.0
+0.5
+Sz
+j  
+(f)
+Figure 8. (a)-(d) Time evolution of spin polarization under post-selection dynamics. The dashed
+lines indicate the values of Im(Es)
+g
+obtained from the Bethe ansatz. For (a)(b), the time evolution
+starts from a local quench; in (c)(d), it starts from the “domain-wall” initial state (see text). The
+time is measured in unit of 1/g. (e)(f) Steady-state spin polarization profile obtained from exact
+diagonalization. Parameter values: N = 14 and g = 0.8 are fixed. For (a)(c)(e), ∆ = 0.8; for
+(b)(d)(f), ∆ = 1.2.
+exciting the states with laser.
+The absence of fluorescence signals the absence of the
+quantum jump from the magnetization-conserved quantum trajectory. The evolution of
+the many-body state is then governed by the effective Hamiltonian Heff = HXXZ + Hloss,
+– 17 –
+
+which differs from our initial Hamiltonian Eq. (2.1) only by an imaginary constant ig
+2 .
+During the time interval [t, t + δt], the state |ψ⟩ evolves as
+|ψ(t + δt)⟩ = exp(−iHeffδt)|ψ(t)⟩
+| exp(−iHeffδt)|ψ(t)⟩|,
+Starting from an initial state in the zero magnetization sector, the system will relax to
+the steady state after sufficiently long time.
+The imaginary part of the corresponding
+eigenvalue can be obtained by measuring the expectation of boundary spin polarization:
+Im(Es) = Im⟨ψs|H|ψs⟩ = Im⟨ψs|Heff|ψs⟩ + g
+2 = g
+2⟨ψs|(Sz
+N − Sz
+1)|ψs⟩ = g⟨ψs|Sz
+N|ψs⟩, (5.1)
+where |ψs⟩ is the steady state. The measured boundary spin polarization can be compared
+to our analytical result of Im(Es).
+Numerical simulations for post-selection evolution under Heff are conducted on N = 14
+chain to back up the above proposal. We consider two kinds of initial states. The first one
+is a “local quench”, in which the spin chain is prepared in the ground state of HXXZ, and
+boundary coupling to the auxiliary energy levels is turned on at certain moment. The other
+initial state is a domain-wall configuration, in which the spins of the left half chain point
+down while those of the right half point up. We discretize the continuous time evolution by
+fourth order Runge-Kutta method, and obtain the spin polarizations in Fig. 8(a-d). It is
+clear that for both local quench and domain wall initial states, the edge spin polarization
+converges to the predictions of Bethe ansatz solution. For certain parameter choices, e.g.,
+Fig. 8(b), the relaxation time towards the steady state is comparable to 1/g, which is most
+convenient from experimental perspective. The results from Bethe ansatz and numerical
+simulation are also confirmed by exact diagonalization Fig. 8(e)(f). Fig. 8(e)(f) also shows
+clearly that the steady-state expectation of Sz
+N is well below 1/2 for |∆| < 1 , while it
+saturates to 1/2 for |∆| > 1.
+6
+Conclusion
+In this work, we applied coordinate Bethe ansatz to solve the steady state and the ground
+state of a PT symmetric one-dimensional boundary-dissipated spin chain, focusing on the
+PT broken phase. We found the many-body scale-free state, which is composed of one
+boundary mode and a continuum of scale-free modes in our particular model. We then
+derived the Bethe equations of the scale-free Bethe roots, and obtained a compact formula
+for the eigen-energy in the thermodynamic limit.
+We then proposed an experimental
+scheme to measure the dissipative part of the energy, and discussed how to compare it
+with our analytical results.
+Our findings shed a light on exceptional points and PT transition in many-body
+physics. Particularly, our solution is a generalization of the concept of scale-free NHSE from
+free-particle to many-body systems. Although we focused on the scale-free behaviour in the
+XXZ spin chain, it is expected that this feature is universal in a family of non-Hermitian
+models with interactions in the bulk and dissipation-induced defect mode at the boundary.
+For example, non-integrable models also exhibit scale-free properties, as demonstrated in
+– 18 –
+
+Fig. 2. Moreover, in integrable models solvable by nested Bethe ansatz (Fermi-Hubbard
+model, higher spin XXX chain, etc.), boundary-operator induced PT symmetry transition
+and the corresponding steady states may have richer structures to uncover.
+We have demonstrated the scale-free skin effect by the difference between the imaginary
+part of the steady state energy and that of a single boundary mode. Intuitively, the scale-
+free skin effect may also manifest itself in the excitation spectra near the steady state.
+A thorough analysis of those eigenstates requires preserving the O(1/N 2)-order terms in
+the imaginary Bethe equations, which is left for future studies. Algebraic Bethe ansatz is
+another possible approach to the solutions of excitations, though it remains a question how
+scale-free Bethe roots emerge in the monodromy and transfer matrices.
+Acknowledgments
+This work is supported by NSFC under Grant No. 12125405.
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new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf,len=887
+page_content='Prepared for submission to JHEP  Scale-free non-Hermitian skin effect in a  boundary-dissipated spin chain  He-Ran Wang, Bo Li, Fei Song, Zhong Wang  Institute for Advanced Study, Tsinghua University, Beijing, 100084, China  E-mail: whr21@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='cn, boliwalker@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='com,  songf18@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='cn, wangzhongemail@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='cn  Abstract: We study the open XXZ spin chain with a PT-symmetric non-Hermitian  boundary field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We find an interaction-induced scale-free non-Hermitian skin effect by using  the coordinate Bethe ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The steady state and the ground state in the PT broken phase  are constructed, and the formulas of their eigen-energies in the thermodynamic limit are  obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The differences between the many-body scale-free states and the boundary string  states are explored, and the transition between the two at isotropic point is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  We also discuss an experimental scheme to verify our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Keywords: Bethe Ansatz, Quantum Dissipative Systems, Spin Chain, Non-Hermitian  Skin Effect  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='11896v1  [quant-ph]  27 Jan 2023  Contents  1  Introduction  1  2  Non-Hermitian XXZ model and the phase diagram  3  3  Bethe ansatz solutions for scale-free skin modes  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1  Single-magnon state  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  Two-magnon state  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='3  Imaginary Bethe equation at 0 < ∆ < 1  8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4  Boundary bound state at ∆ > 1 and phase transition  12  4  Ground state phase diagram  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1  Scale-free solutions for −1 < ∆ < 0  14  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  Imaginary Bethe equation at ∆ < −1  14  5  Experimental Realizations  16  6  Conclusion  18  1  Introduction  Exactly solvable models play important roles in condensed matter physics, statistical  physics, and mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Certain experimentally relevant one-dimensional sys-  tems can be modeled by open spin chains with boundary fields, some of which belong to the  category of Yang-Baxter integrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Examples include spin chains with diagonal [1–8]  or off-diagonal [9–12] magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The problem is also related to classical dynamics of  molecules with drain and source [13–17], and spin transport within the framework of Lind-  blad master equation [18–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Many mathematical tools, such as coordinate Bethe ansatz  [1, 2, 6, 7, 14, 26], Sklyanin’s reflection algebra (an open boundary version of algebraic  Bethe ansatz) [3–5, 8–12, 27], matrix product operator ansatz [13, 15–19, 21–24], etc, have  been developed to treat those systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  In this article, we investigate a non-Hermitian open XXZ chain using coordinated Bethe  ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The chain is subjected to opposite imaginary magnetic field on two ends, pointing  to a prescribed direction called z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Here, the non-Hermiticity naturally stems from the  ubiquitous coupling with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' A special case has been studied thoroughly in  previous literature, where the strength of the boundary field takes a specific value depending  on the anisotropic interaction strength between adjacent spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The Hamiltonian then  respects the q−deformed SU(2) symmetry [28] with |q| = 1, and serves as a representation  of the Temperley-Lieb algebra [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The spectrum of the model is purely real, though  the Hamiltonian is non-Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Furthermore, when q is a root of unity for some values  – 1 –  of the boundary field, the representation of the symmetry group enjoys richer structures,  such that an exact duality between the spin model and free-end quantum Potts model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The  duality leads to the same conformal field theory (CFT) structure of two models, thus the  negative central charge obtained from the side of Potts model makes the non-Hermitian  spin chain a typical example of recently introduced “non-unitary” CFT [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  A significant consequence of q being a root of unity is that the spin Hamiltonian  develops Jordan blocks, which feature exceptional points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The number of Jordan blocks for  given q and size N has been counted [33], followed by the constructions of the corresponding  generalized eigenstates [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Given the existence of many-body exceptional points, it is  natural to identify different phases around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Our article exhausts the parameter  space of boundary imaginary field and anisotropic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' For a small boundary field,  the spectrum remains real, and the Bethe roots of the ground state only shift slightly  compared with the Hermitian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The spectrum becomes complex, however, when the  boundary field exceeds the q−deformed SU(2) symmetric value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We show that, despite the  breaking of the quantum group invariance, the model possesses a novel behaviour of scale-  free localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We figure out the structure of the steady state (with the largest imaginary  part of energy) and the ground state (with the lowest real part of energy) as the combination  of a boundary Bethe root and a set of continuous Bethe roots in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The continuous Bethe roots all have an imaginary part proportional inversely to the system  size N, corresponding to a small imaginary wavevector (or momentum) κ ∼ α/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  In  the single-particle context, when the localization length of wavefunction is proportional  to the system size N, the density |ψN(x)|2 ∼ exp(2αx/N) is invariant under re-scaling  transformation with factor s: |ψN(x)|2 = |ψsN(sx)|2, and therefore called scale-free non-  Hermitian skin effect (NHSE) or critical NHSE [35–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Scale-free NHSE has also been  found in Hermitian systems with non-Hermitian boundary field, though the mechanism  is different [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' In the present work, the imaginary part of wavevector is attributed to  the scattering between the boundary mode and magnons traveling in the bulk, and these  Bethe roots contribute a non-negligible imaginary part to the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Thus, unlike previous  works, our scale-free behaviour has a many-body origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' More precisely, it originates from  the interplay between boundary dissipation and many-body interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' On one hand,  the interaction among magnons is an indispensable ingredient for the scale-free NHSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' On  the other hand, we also note that the Hermitian counterpart, namely the open XXZ model  subjected to real boundary field, has only isolated boundary modes, and such continuous  skin modes are lacking [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We derive an integral equation, dubbed imaginary Bethe  equation, to solve the scale-free localization length in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We then  give an exact formula for the imaginary part of the steady state energy, which are then  compared to finite-size numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We also explain how to measure these physical  quantities in cold-atom experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Before proceeding, we compare our results to earlier studies on boundary-driven spin  chains as open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The evolution of those open quantum systems is gen-  erated by the Linbladian operator, composed of an integrable Hamiltonian and quantum  – 2 –  jump operators on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' A typical example relevant to our work is [19]  L(ρ) = −i[HXXZ, ρ] +  �  µ=1,N  LµρL†  µ − 1  2{L†  µLµ, ρ}  = −iHeffρ + iρH†  eff +  �  µ=1,N  LµρL†  µ,  with L1 = √gS−  1 , LN = √gS+  N and Heff = HXXZ − i  2  �  µ=1,N L†  µLµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Here, Heff is the non-  Hermitian Hamiltonian we shall focus on below (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Although the Lindbladian  breaks integrability, the density matrix of non-equilibrium steady state (NESS) has been  established by the matrix product operator (MPO) ansatz exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Furthermore, it has  been found that the local matrix of MPO ansatz is indeed the infinite-dimensional solution  of Yang-Baxter relations, and thus exterior integrability emerges in the NESS [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  However, the dynamics towards NESS is unknown yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Our work about the non-Hermitian  effective Hamiltonian is complementary to the NESS solution because Heff governs the  time evolution of the open quantum system under post-selection, which is relevant to  numerous experiments [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Our solution is enabled by the Yang-Baxter integrability of  the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Another related system is the XXZ model with only one jump operator L1  on the left boundary [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Since the dissipator is purely lossy, the Lindbladian becomes  upper-triangular under an appropriate basis choice, so that the Liouvillian spectrum can  be completely determined by the effective non-Hermitian Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The Hamiltonian  has scale-free eigenstates even in the single-magnon sector, but PT symmetry is absent due  to that the dissipator occurs only on one of the two ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' By contrast, our Hamiltonian  preserves PT symmetry, and in single-magnon sector there are only Bloch-wave modes and  exponentially localized states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Scale-free modes originate from many-body interactions in  our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The rest part is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' In the next section, we introduce the model  Hamiltonian, its general Bethe equations, and the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' In Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2,  we consider the single-magnon and two-magnon state as a warm-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We then generalize the  results to the many-body cases to obtain the steady state with scale-free NHSE in Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4 is devoted to another type of steady state solution, the boundary string  states, which emerges for the highly anisotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' In Section 4, we apply the ansatz of  scale-free solutions to the ground state for different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' A possible experimental  setup for the non-Hermitian model is discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We give some concluding  remarks in Section 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  2  Non-Hermitian XXZ model and the phase diagram  The Hamiltonian reads:  H = −  N−1  �  j=1  (Sx  j Sx  j+1 + Sy  j Sy  j+1 + ∆Sz  j Sz  j+1) + ig  2 (Sz  N − Sz  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1)  where Sα = 1  2σα(α = x, y, z) is the spin-1/2 operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' the anisotropic interaction strength  ∆ and boundary field strength g are purely real, with g > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The model respects the PT  – 3 –  symmetry with TiT = −i and PSα  j P = Sα  N−j, and therefore the eigenvalues are either real  or form complex conjugate pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' When the whole spectrum is purely real, the model is  said to be in the PT exact (or PT-symmetric) phase, otherwise it is in the PT broken  phase [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The steady state, which has the largest imaginary part of eigen-energy in  the PT broken phase, is of great importance because it captures the long-time behaviour  of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' A generic initial wavefunction evolving for a sufficiently long time under  exp(−iHt) will converge to the steady state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' we shall study the phase diagram of this steady  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The Hamiltonian also commutes with total z magnetization m = �N  j=1 Sz  j , so that  it can be block diagonalized in each sector with definite total magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Furthermore,  there is another symmetry operator PX with X = �N  j=1 σx  j which sends m to −m, and  therefore it suffices to study non-positive magnetization (m ≤ 0) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' For the odd length  chain, PX symmetry leads to the two-fold degeneracy of the steady states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Thus, we only  take even site number N throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Ferromagnetic (∆ > 0) and anti-ferromagnetic (∆ < 0) models can be related by the  transformation Z = �N/2  j  σz  2j−1:  ZTH(∆, g)ZT = ZH(∆, −g)Z = −H(−∆, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2)  As such, an eigenstate of ferromagnetic Hamiltonian with H(∆, g)|ψ⟩ = E|ψ⟩ can be trans-  formed to an eigenstate of the anti-ferromagnetic one: H(−∆, g)(ZT|ψ⟩) = −E∗(ZT|ψ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  If E has the largest imaginary part among the spectrum, so does −E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Thus, the steady  state properties remain the same for ±∆, and it suffices to study the ferromagnetic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Δ  g  PT exact  Scale-free skin effect  Boundary  string  Boundary  string  1  1  1  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Steady-state phase diagram of model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1) in the zero magnetization sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The critical curve ∆2 + g2 = 1 separates PT exact and broken phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  When |∆| < 1 and  g > gc =  √  1 − ∆2, the steady state is the many-body scale-free state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' when |∆| > 1, the steady  state is the boundary string state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Eigenstates of the model can be solved by coordinate Bethe ansatz [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Take all spin  down state as the reference state with energy E0 = − 1  4(N −1)∆, we can excite M magnons  by flipping M spins up (M ≤ N/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  We can construct the ansatz state |ψ⟩ , whose  – 4 –  wavefunction in the onsite magnon number basis is given by:  ⟨n1 · · · nj · · · nM|ψ⟩ =  �  P  �  {ηj}  (−1)PA({eikj};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' P, {ηj})eiη1kP1n1 · · · eiηjkPjnj · · · eiηMkPMnM ,  where n1 < · · · < nj < · · · < nM are the positions of up spin, P refers to all possible  permutations, and chirality ηj = ±1 corresponds to right-moving and left-moving magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Relabeling the momentum of magnon by βj = eikj [44, 45], we have the equivalent form:  ⟨n1 · · · nj · · · nM|ψ⟩ =  �  P  �  {ηj}  (−1)PA({βj};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' P, {ηj})βη1n1  P1  · · βηjnj  Pj  · · βηMnM  PM  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='3)  The coefficient A({βj};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' P, {ηj}) is a function of magnon momenta {βj}, permutation P,  and chirality {ηj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Imposing the condition that |ψ⟩ is an eigenstate of H with energy  EM({βj}) = E0 +  M  �  j=1  (∆ − (βj + β−1  j )/2)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4)  and the boundary condition, those coefficients can be found as [1]:  A({βj};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' P, {ηj}) =  M  �  j  (1 − ∆+βηj  Pj)β−(N+1)ηj  Pj  �  j<l  (1 − 2∆βηl  Pl + βηl  Plβ−ηj  Pj )(1 − 2∆βηj  Pj + βηj  Pjβηl  Pl)β−ηl  Pl ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5)  where ∆± = ∆ ± ig, and the constraints of M momenta, so-called Bethe equations, can be  derived:  β2(N−1)  j  (βj − ∆+)(βj − ∆−)  (β−1  j  − ∆+)(β−1  j  − ∆−) =  M  �  l̸=j  1 − 2∆βj + βjβl  1 − 2∆βl + βjβl  1 − 2∆βj + βjβ−1  l  1 − 2∆β−1  l  + βjβ−1  l  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6)  Heuristically, the left hand side is the phase accumulated by a magnon when it travels  freely from one end of the chain to the other and then gets reflected back;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' each term of  the right hand side represents the scattering phase between magnon j and l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Note that for  the periodic XXZ model there is only the βj, βl term in the scattering phase, while for the  open boundary condition βj also scatter with β−1  l  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The solutions are inversion-symmetric  in the sense that βj and β−1  j  always appear in pairs in the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' After solving a set of  Bethe roots {βj}, the energy of M magnon state is  EM({βj}) = E0 +  M  �  j=1  (∆ − (βj + β−1  j )/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='7)  Although it remains unclear whether all eigenstates of the non-Hermitian XXZ model have  the form of Bethe ansatz, it turns out that the ground state and the steady state can be  solved exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Our results on the steady states for different parameters are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We  focus on the zero magnetization sector (m = 0) since for the Hermitian XXZ model (with  – 5 –  ∆ < 1) the ground state lies in this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Our numerical results support that the steady  state belongs to the zero magnetization sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The phase boundary between PT-exact and  broken phase is ∆2 + g2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Given the transformation between H(∆, g) and H(−∆, g),  the steady-state phase diagram is symmetric about the g axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Apart from the many-body  scale-free modes which will be investigated in detail in Section 3, we identify the “boundary  string” state as the steady state when |∆| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' A boundary string corresponds to a multi-  magnon bound state localized at the boundary, which will be explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  3  Bethe ansatz solutions for scale-free skin modes  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1  Single-magnon state  In the single-magnon sector (M = 1), the Bethe equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6) is simplified to  β2(N−1)  (β − ∆+)(β − ∆−)  (β−1 − ∆+)(β−1 − ∆−) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1)  When |∆±| < 1, or equivalently g < gc, solutions of the equation have been found to  be on the unit circle [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We review the proof briefly: define complex variable function  h(β) : C → C as  h(β) = βN−1(β − ∆+)/(β−1 − ∆−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1) is then transformed to h(β) = h(β−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' For g < gc, the image of the disk |β| < 1  under h is still inside the disk, that is, |h(β)| < 1, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The statement can be  verified by writing β = ρ exp(iφ), ∆+ = ρ0 exp(iφ0) with ρ0 < 1, then  |β − ∆+|2 − |β−1 − ∆−|2 = (ρ − ρ−1)(ρ + ρ−1 − 2ρ0 cos(φ − φ0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Since ρ + ρ−1 ≥ 1 > 2ρ0 cos(φ − φ0), we have |β − ∆+| < |β−1 − ∆−| when |β| < 1  (ρ < 1 < ρ−1), therefore |h(β)| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  A similar argument works for |β| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Thus,  possible solutions of h(β) = h(β−1) must be on the unit circle, corresponding to purely  real momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  When g > gc, no theorem prohibits the existence of non-unitary solutions, and one can  notice that a pair of isolated boundary modes with β± ≈ ∆± is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' For such a solution,  |β±| > 1 leads to the divergence of the term β2(N−1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1) in the thermodynamic  limit, but this can be compensated by the factor (β − ∆+)(β − ∆−), which is close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Most of the energy levels remain real, and the scale-free localization behavior is absent in  this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We define the boundary imaginary energy contributed by one boundary mode  as  Eb = −1  2Im(∆− + ∆−1  − ) = 1  2(g −  g  ∆2 + g2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='3)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  Two-magnon state  In the M = 2 sector, scale-free modes appear and contribute to the 1/N scaling behaviour  of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Figure 2(a1) shows a typical finite-size two-magnon spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We color the  – 6 –  eigenvalues by many-body participation entropy [47–49], a measure of localization gener-  alized from single-particle inverse participation ratio (IPR):  S2(|ψ⟩) = − log  �  i |ψi|4  (�  i |ψi|2)2 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4)  where the index i sums over all the basis functions in the relevant Hilbert space, |ψ⟩ is  the many-body eigenstate concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The participation entropy gets smaller when the  eigenstate is more localized in the Hilbert space, as we observed in figure 2(a1): Two dark  points correspond to isolated states bounded to the boundaries, and both two magnons are  localized to the same side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' around Im(E) = ±Eb there is a continuum of states, which are  the combination of a boundary mode with β1 ≈ ∆± and a scale-free mode with β2 ≈ eik;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' the  continuum on the real axis is brighter, though there is one localized mode, corresponding  to the state with two magnons localized on different ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  We apply Bethe equation to the state with one magnon bounded to the boundary  while the other has momentum β2:  |β2|2N ≈ | 1 − 2∆β2 + ∆−β2  1 − 2∆∆− + ∆−β2  1 − 2∆β2 + ∆−1  − β2  1 − 2∆∆−1  − + ∆−β2  |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5)  The right hand side, which will be denoted by exp(2κ), is of order O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Taking logarithm  of both sides, we have 2Nln|β2| = 2κ+O(1/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Thus, lnβ2 acquires a first order correction  to the real part:  Re(lnβ2) = ln|β2| = κ/N + O(1/N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Traveling along the chain, the corresponding magnon accumulates an amplitude change  A = |β2|N ∼ exp (κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The localized mode can be distinguished from disorder-induced  localization or non-Hermitian skin effect, both of which has the exponential decay behavior  ψ(x) ∼ exp(κ′x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' In contrast, the leading order of A shows no dependence on the system  size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Here, the scale-free localization has an intrinsic many-body origin because in the  non-interacting limit ∆ = 0, the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5) equals 1, and the imaginary  momentum vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Therefore, the phenomenon in our model is dramatically different  from that of free-particle models [35–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The energy of such two-magnon state is  Im(E) = Eb − 1  2Im(β2 + β−1  2 ) = Eb − 1  N κ sin(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6)  The statement is verified by finite-size scaling of the average of those complex energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' As  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 2(b1), the imaginary part scales linearly with 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' In the thermodynamic  limit, it will converge to ±Eb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Moreover, we add a next-nearest-neighbor zz interaction  Hnn = −  N−2  �  j=1  ∆′Sz  j Sz  j+2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='7)  to break integrability, yet the numerical results [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 2(b1)(b2)] show no qualitative differ-  ences, which supports the universality of scale-free behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  – 7 –  8  6  4  Re(E)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  Im(E)  (a1)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5  10  8  6  Re(E)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='30  Im(E)  (b1)  3  4  5  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='022  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='024  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='026  1/N  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0852  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0855  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0858  AVG[Im(E)]  (a2)  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='76 g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='83  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='80 g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='80  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='85 g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='022  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='024  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='026  1/N  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1295  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1315  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1335  AVG[Im(E)]  (b2)  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='77 g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='84  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='80 g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='80  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='85 g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='73  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (a1) and (b1) The spectrum of M = 2 sector for N = 40, ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Eigenvalues  are colored by the participation entropy of the corresponding eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (a2) and (b2) Finite-size  scaling for the average of the imaginary part of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Only those states with Im(E) > 0 in the  continuous spectrum are taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (a1) and (a2) are based on the pristine model  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' for (b1) and (b2), additional next-nearest-neighbor coupling ∆′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='3 is added [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='7)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='3  Imaginary Bethe equation at 0 < ∆ < 1  After the warm-up on two-body scale-free modes, we now generalize it to the many-body  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We will study the parameter space 0 < ∆ < 1, where it is the Luttinger liquid  phase in the Hermitian limit and the ground state lies in M = N/2 sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We assume  that the steady state is composed of a boundary mode and a set of continuous scale-free  Bethe roots, and then derive the Bethe equations in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  We adopt a conventional parametrization of magnon momentum [50, 51]:  βj = −sinh[γ(xj − i)/2]  sinh[γ(xj + i)/2],  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8)  where γ = arccos(−∆) such that 0 < γ < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The kinetic energy of the magnon is  Exj = −1  2(βj + β−1  j ) = 1 − cos(γ) cosh(γxj)  cos(γ) − cosh(γxj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='9)  – 8 –  Taking the logarithm of Bethe equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6), we have  2Nθ1(xj) + φb(xj) = 2πIj +  �  l̸=j  [θ2(xj − xl) + θ2(xj + xl)],  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='10)  where the function θn is defined as  θn(x) = 2 arctan[cot(nγ/2) tanh(γx/2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='11)  The second term φb(xj) on the left of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='10) is the scattering phase between the magnon  j and the boundary, which has the form:  φb(x) = θm+(x) + θm−(x),  where θm± is obtained by taking n in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='11) as complex numbers  m± = 1  γ θ1(ln(cos(γ) ± ig)  γ  ) + iπ  2γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  This involved boundary term will not have significance in the rest part of solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' For  the right hand side, Ij is an integer and the set of {Ij} determines the set of Bethe roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  We take the occupation of one boundary mode and Ij = j on the steady state, and the  corresponding boundary Bethe root β0 = ∆− has the parametrization  x0 = 1  γ ln g − sin(γ)  g + sin(γ) − i = λ0 − i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='12)  The steady-state Bethe equations becomes  2Nθ1(xj)+φb(xj) = 2πj +  �  l̸=j  [θ2(xj −xl)+θ2(xj +xl)]+θ2(xj −x0)+θ2(xj +x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='13)  We rewrite xj = λj + iσj/N with purely real λj, σj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We note that |σj/N| ≪ λj, and  therefore any real function of xj can be expanded as  f(xj) = f(λj) + if′(λj)σj/N + O(1/N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='14)  The real and imaginary part of Bethe equations are:  2Nθ1(λj) + φb(λj) + O(1) = 2πj + [  �  l̸=j  θ2(λj − λl) + θ2(λj + λl)]  +Re[θ2(λj − x0) + θ2(λj + x0)],  (2θ′  1(λj) + 1  N φ′  b(λj))σj + O( 1  N ) = 1  N  �  l̸=j  θ′  2(λj − λl)(σj − σl) + θ2(λj + λl)(σj + σl)]  +Im[θ2(λj − x0) + θ2(λj + x0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='15)  For the real (imaginary) part only O(N) (O(1)) terms are included, and the leading terms  of the real and imaginary part of Bethe equations are:  2Nθ1(λj) = 2πj + [  �  l̸=j  θ2(λj − λl) + θ2(λj + λl)],  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='16)  2θ′  1(λj)σj = 1  N  �  l̸=j  θ′  2(λj − λl)(σj − σl) + θ2(λj + λl)(σj + σl)]  +Im[θ2(λj − x0) + θ2(λj + x0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='17)  – 9 –  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='16) is the same as the ground state Bethe equations of the Hermitian open XXZ  model [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' It is standard to calculate the difference between (j + 1) − th and the j − th  equation, taking f(λj+1) − f(λj) = f′(λj)(λj+1 − λj):  2Nθ′  1(λj)(λj+1 − λj) = 2π + [  �  l̸=j  θ′  2(λj − λl) + θ′  2(λj + λl)](λj+1 − λj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='18)  The thermodynamic limit is taken by sending  lim  N→∞  1  N(λj − λj+1) = ρ(λj),  lim  N→∞  1  N  �  l  f(λl) =  � ∞  0  dλρ(λ)f(λ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='19)  then the integral equation of ρ(λ) is  1  2ρ(λ) +  � +∞  −∞  dλ′ K2(λ − λ′)  2π  1  2ρ(λ′) = K1(λ)  2π  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='20)  where Kn(λ) = θ′  n(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Note that only when the steady state belongs to the zero magne-  tization sector, the integral interval of λ can be taken as (−∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' This is the case here,  and it corresponds to filling the Fermi sea k ∈ (−π +γ, π −γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The integral equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='20)  is commonly solved by Fourier transformation:  ˜ρ(ω) =  � +∞  −∞  dλ  2πρ(λ)eiωλ,  ˜Kn(ω) =  � +∞  −∞  dλ  2πKn(λ)eiωλ =  sinh( π  γ − n)ω  sinh π  γ ω  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='21)  Applying the convolution formula on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='20), we have a linear equation  1  2(1 + ˜K2(ω))˜ρ(ω) =  ˜K1(ω)  2π  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='22)  then the distribution function is solved: ˜ρ(ω) = 1/(2π cosh ω), ρ(λ) = 1/(2 cosh(πλ/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='17) counts two kinds of mechanisms of scale-free localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' On the right  hand side, the first term sums interactions between magnons in the bulk, while the second  term is the scattering with the boundary mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The last one is much smaller than the first  in a few-body state, but becomes comparable when the magnon number is the same order  of system size N, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' in the zero magnetization sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Define σ(λ) as a function of λ, the  continuous version of this equation is  ρ(λ)σ(λ) +  � +∞  −∞  dλ′ K2(λ − λ′)  2π  ρ(λ′)σ(λ′) = 1  2πIm[θ2(λ − x0) + θ2(λ + x0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='23)  Dubbed “imaginary Bethe equation”, it is a central result of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' To derive the  linear equation of ˜σρ(ω) =  � +∞  −∞  dλ  2πeiωλρ(λ)σ(λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' we need to find the Fourier transformation  – 10 –  of the right hand side:  � +∞  −∞  dλ  2πeiωλIm[θ2(λ − x0) + θ2(λ + x0)] =  � +∞  −∞  dλ  2πeiωλIm[θ2(λ − λ0 + i) + θ2(λ + λ0 − i)]  = 2i sin(ωλ0)  � +∞  −∞  dλ  2πeiωλIm[θ2(λ + i)]  = −2 sin(ωλ0)  ω  � +∞  −∞  dλ  2πeiωλ d  dλIm[θ2(λ + i)]  = 2 sin(ωλ0)  ω  � +∞  −∞  dλ  2πeiωλ  2γ cos(γ) sin2(γ) sinh(γλ)  (cosh(γλ) − cos(γ))(cosh(γλ) − cos(3γ))  = 2i sin(ωλ0) sinh(ω)  sinh[( π  γ − 2)ω]  ω sinh( π  γ ω)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='24)  Denoting the result above by Θ(ω), we can solve σ(λ) formally:  ˜σρ(ω) =  1  1 + ˜K2(ω)  Θ(ω)  2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='25)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  g  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5  Im(Es)  Ferromagnetic:  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8  BA  Eb  ED  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Imaginary part of the steady state energy Es as a function of the non-Hermitian  parameter g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We take ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8, so that gc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Red curve is from the Bethe ansatz (BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' As a  comparsion, blue line is imaginary energy of a boundary mode (Eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The black dots are computed  from exact diagonalization (ED) in the zero-magnetization subspace, with system size N = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The summation of energy of all those scale-free modes becomes an integral over λ in  the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The imaginary part of energy formula of a single magnon is  Im(E(x)) = 1  N E′(λ)σ(λ) = −sin(γ)  Nγ K′  1(λ)σ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='26)  While each one contributes O(1/N) to the total imaginary part, the sum of contributions  – 11 –  from all scale-free magnons is comparable to the boundary mode contribution Eb:  �  j  Im(E(xj)) = −sin(γ)  γ  � +∞  0  dλρ(λ)K′  1(λ)σ(λ)  = −sin(γ)  2γ  � +∞  −∞  dω 2πiω ˜K1(ω)˜σρ(ω)  = −sin(γ)  2γ  � +∞  −∞  dω iω  ˜K1(ω)  1 + ˜K2(ω)  Θ(ω)  = −sin(γ)  γ  � +∞  0  dω sin(−ωλ0) tanh(ω)  sinh[(π  γ − 2)ω]  sinh( π  γ ω)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='27)  The imaginary part of the steady-state eigen-energy is then given by adding Eb:  Im(Es) =  �  j  Im(E(xj)) + Eb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='28)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 3, we compare our formula with the exact diagonalization (ED) results in the  M = N/2 sector, which agrees excellently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The boundary field g controls the imaginary  part of the energy totally via λ0 =  1  γ ln g−gc  g+gc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' As g crosses gc, the steady state energy  becomes complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4  Boundary bound state at ∆ > 1 and phase transition  Anisotropic interaction ∆ > 1 prefers bounding all magnons together, and therefore the  magnons in the steady state tend to localize to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Specifically, the first magnon  is bound to the boundary, and the next one is bounded to the previous one recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  In the context of integrable spin models, the bound state is named an “string”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' we shall  follow this terminology and call our bound state near the boundary a “boundary string”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  We note that similar states have been identified in the spin model subjected to a non-  Hermitian magnetic field at only one end [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' In the thermodynamic limit, Bethe roots  {βj} satisfies a recursive relation:  βj+1 + β−1  j  = 2∆, β1 = ∆ − ig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='29)  The imaginary part of energy is given by  Im(Es) = −1  2  M  �  j=1  Im(βj + β−1  j )  = −1  2  M−1  �  j=1  Im(β−1  j  + βj+1) − 1  2Im(β1 + β−1  M )  = −1  2Im(β1 + β−1  M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  For large N, βN/2 approaches the fixed point of recursive relations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='29): β∞ =  ∆ ±  √  ∆2 − 1, which is purely real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' It follows that Im(Es) = − 1  2Im(β1) = g  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  – 12 –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4  Im(Es)  (a)  g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8  g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='12  1/N  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='08  0-Im(Es)  (b)  g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (a) The imaginary part of the steady state energy in the zero magnetization sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The size N = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (b) Finite-size scaling of the steady-state energy at the isotropic point ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  γ0 = g/2 [see the paragraph above Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='31)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  It is clear that the structure of Bethe roots of the steady state is different for 0 < ∆ < 1  and ∆ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We may take the isotropic limit ∆ = 1 from the two sides to understand the  phase transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  In the scale-free phase, we have to deal with the γ → π limit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='27) carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Note that the Fermi sea ranging from −π+γ to π−γ shrinks to a Fermi point when γ → π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  We take the limit by substituting ω by ωγ/ sin γ with sin γ ≪ 1, and the integral can be  simplified to  � +∞  0  dω sin(2ω/g) exp(−2ω) = g/2(1 + g2),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='30)  so that Im(Es) = Eb + g/2(1 + g2) = g/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  In the boundary-string phase, the imaginary part is a constant γ0 = g/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Bethe roots  can be determined by the recursive equation Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='29) at ∆ = 1, which results in an  explicit solution:  βn = 1 +  1  n − 1 + i  g  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='31)  For small n, the magnon localizes exponentially at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' However, for n sufficiently  large (n/N ∼ O(1)), it behaves in a scale-free fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  This result is also confirmed  by comparing the imaginary part with γ0 for different system sizes, and the differences  scale linearly with 1/N (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We emphasize that the solution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='31), though  qualitatively valid, is not exact because the scattering between the large n scale-free modes  have been neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' This approximation has been implied in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  4  Ground state phase diagram  For the ferromagnetic case ∆ > 0, the steady state and the ground state coincide, and the  phase boundary ∆ = 1 separates the scale-free phase and the boundary string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' However,  the transformation ZT relating the ferromagnetic and the anti-ferromagnetic steady state  – 13 –  Δ  g  Luttinger liquid  Luttinger liquid  (Scale-free)  Boundary  string  Gapped spinon  (Scale-free)  1  1  1  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Ground state phase diagram in the zero magnetization sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The right half of the  diagram, ∆ > 0, coincides with the counterpart of steady state (see figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' In the PT exact  phase (g < gc ≡  √  1 − ∆2), the ground state is Luttinger liquid (LL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' for g > gc, gapless excitations  become scale-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' When ∆ < −1, the gap between the ground state and excited states opens, yet  our ansatz of scale-free skin modes remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  is not applicable to the ground state, because it changes to the highest-energy (real part)  state when reversing the sign of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Therefore, one cannot borrow the ground-state phase  diagram from that of the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Moreover, the comparison between ground states  at zero and finite non-Hermiticity provides another perspective on the effect of boundary  dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  5, we summarize the results of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The ansatz of  many-body scale-free state in the region ∆ < 0 is studied in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1  Scale-free solutions for −1 < ∆ < 0  For −1 < ∆ < 0, we find that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='27) can still be applied to the ground state, though  it does not coincide with the steady state anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The formula is compared with exact  diagonalization results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' It seems that the data does not agree as well as in the  ferromagnetic case (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' This is due to the larger finite-size error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' In fact, we  observe that as size N increases, the ED results become closer to the analytical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  Imaginary Bethe equation at ∆ < −1  The imaginary Bethe equation also works for ∆ < −1, with some technical modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Retaining the parametrization ∆ = − cos(γ), |∆| > 1 now leads to a purely imaginary γ,  and it is convenient to write γ → iφ:  βj = −sin[φ(xj − i)/2]  sin[φ(xj + i)/2],  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1)  where φ = arccos h(−∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The boundary Bethe root is  x0 = − 2  φ arctan(sinh(φ)  g  ) − i = λ0 − i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2)  – 14 –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  g  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='30  Im(Eg)  Antiferromagnetic:  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8  BA  Eb  ED,N=10  ED,N=16  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Imaginary part of the ground state energy Eg as a function of the non-Hermitian  parameter g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We take ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8, so that gc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Red curve is from the Bethe ansatz (BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' As a  comparison, blue curve is imaginary part of the energy of a boundary mode (Eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The hollow dots  are obtained from exact diagonalization (ED) with N = 10, and solid dots with N = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  On the ground state the whole Brillouin zone k ∈ (−π, π) is filled, so that Re(x) ∈  (−π/φ, π/φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The single-magnon kinetic energy is  Exj = 1 − cosh(φ) cos(φxj)  cosh(φ) − cos(φxj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='3)  We also adopted here a new definition of the function θn:  θn(x) = 2 arctan[coth(nφ/2) tan(φx/2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The imaginary Bethe equation is then given by  ρ(λ)σ(λ) +  � +π/φ  −π/φ  dλ′ K2(λ − λ′)  2π  ρ(λ′)σ(λ′) = 1  2πIm[θ2(λ − x0) + θ2(λ + x0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4)  Since functions of λ are periodic functions with periodicity 2π/φ, we can expand them  as Fourier series to solve the integral equation:  ˜f(m) =  � +π/φ  −π/φ  dλ  2πf(λ)eimφλ, m ∈ Z  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5)  Notably, ˜  Kn(m) = exp(−n|m|φ), and the right hand side of the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4) transforms as  � +π/φ  −π/φ  dλ  2πeimφλIm[θ2(λ − x0) + θ2(λ + x0)] = 2i sin(mφλ0)  � +π/φ  −π/φ  dλ  2πeimφλIm[θ2(λ + i)]  = 2 sin(mφλ0)  mφ  � +π/φ  −π/φ  dλ  2πeimφλ  2φ cosh(φ) sinh2(φ) sin(φλ)  (cos(φλ) − cosh(φ))(cos(φλ) − cosh(3φ))  = 2isin(mφλ0)  mφ  sinh(mφ)e−2mφ = Θ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6)  – 15 –  The imaginary part of energy is:  �  j  Im(E(xj)) = −sinh(φ)  φ  � +π/φ  0  dλρ(λ)K′  1(λ)σ(λ)  = −sinh(φ)  2  �  m∈Z  2πimφ ˜K1(m)˜σρ(m)  = −sinh(φ)  2  �  m∈Z  imφ  ˜K1(m)  1 + ˜K2(m)  Θ(m)  = sinh(φ)  �  m∈Z+  sin(mφλ0) tanh(mφ)e−2mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='7)  As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 7, there are finite-size errors between the numerical results and  our Bethe ansatz formula, which is similar to the gapless antiferromagnetic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  g  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='30  Im(Eg)  =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  BA  ED,N=10  ED,N=16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  g  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='40  Im(Eg)  =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  BA  ED,N=10  ED,N=16  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Imaginary part of the ground state energy as a function of the non-Hermitian parameter  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Left panel: ∆ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Right panel: ∆ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Red curve is obtained from the Bethe ansatz  (BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Green triangles and blue squares are numerical results from exact diagonalization of N = 10  and N = 16, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  5  Experimental Realizations  The onsite non-Hermiticity can be realized in cold atom systems by coupling the spin down  (up) degrees of freedom of the first (last) site to an auxiliary state by optical pumping  [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Each atom in the bulk has two effective energy levels to mimic a 1/2 spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' XXZ  interaction can be induced by the mixing of even and odd parity states, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' in Rydberg  atoms [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We may introduce the third energy levels on the two ends so that the effective  spin down (up) state on the left (right) end can decay to it spontaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The effective  loss is described by a non-Hermitian term  Hloss = −ig  2 | ↑⟩⟨↑ |1 − ig  2 | ↓⟩⟨↓ |N = −ig  2 + ig  2 (Sz  N − Sz  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  To evolve the open system under the non-Hermitian Hamiltonian without quantum jump,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=', by post-selection, the population of auxiliary energy levels should be monitored by  – 16 –  0  20  40  60  80  100  time[1/g]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='45  Sz  j  (a)  j=14  j=10  j=8  0  20  40  60  80  time[1/g]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5  Sz  j  (b)  j=14  j=10  j=8  0  20  40  60  80  100  time[1/g]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='50  Sz  j  (c)  j=14  j=10  j=8  0  20  40  60  80  time[1/g]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5  Sz  j  (d)  j=14  j=10  j=8  2  4  6  8  10  12  14  position  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5  Sz  j  (e)  2  4  6  8  10  12  14  position  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='5  Sz  j  (f)  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (a)-(d) Time evolution of spin polarization under post-selection dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The dashed  lines indicate the values of Im(Es)  g  obtained from the Bethe ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' For (a)(b), the time evolution  starts from a local quench;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' in (c)(d), it starts from the “domain-wall” initial state (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The  time is measured in unit of 1/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (e)(f) Steady-state spin polarization profile obtained from exact  diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Parameter values: N = 14 and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8 are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' For (a)(c)(e), ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' for  (b)(d)(f), ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  exciting the states with laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The absence of fluorescence signals the absence of the  quantum jump from the magnetization-conserved quantum trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The evolution of  the many-body state is then governed by the effective Hamiltonian Heff = HXXZ + Hloss,  – 17 –  which differs from our initial Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1) only by an imaginary constant ig  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  During the time interval [t, t + δt], the state |ψ⟩ evolves as  |ψ(t + δt)⟩ = exp(−iHeffδt)|ψ(t)⟩  | exp(−iHeffδt)|ψ(t)⟩|,  Starting from an initial state in the zero magnetization sector, the system will relax to  the steady state after sufficiently long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  The imaginary part of the corresponding  eigenvalue can be obtained by measuring the expectation of boundary spin polarization:  Im(Es) = Im⟨ψs|H|ψs⟩ = Im⟨ψs|Heff|ψs⟩ + g  2 = g  2⟨ψs|(Sz  N − Sz  1)|ψs⟩ = g⟨ψs|Sz  N|ψs⟩, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='1)  where |ψs⟩ is the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The measured boundary spin polarization can be compared  to our analytical result of Im(Es).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Numerical simulations for post-selection evolution under Heff are conducted on N = 14  chain to back up the above proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We consider two kinds of initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The first one  is a “local quench”, in which the spin chain is prepared in the ground state of HXXZ, and  boundary coupling to the auxiliary energy levels is turned on at certain moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The other  initial state is a domain-wall configuration, in which the spins of the left half chain point  down while those of the right half point up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We discretize the continuous time evolution by  fourth order Runge-Kutta method, and obtain the spin polarizations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 8(a-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' It is  clear that for both local quench and domain wall initial states, the edge spin polarization  converges to the predictions of Bethe ansatz solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' For certain parameter choices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=',  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 8(b), the relaxation time towards the steady state is comparable to 1/g, which is most  convenient from experimental perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' The results from Bethe ansatz and numerical  simulation are also confirmed by exact diagonalization Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 8(e)(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 8(e)(f) also shows  clearly that the steady-state expectation of Sz  N is well below 1/2 for |∆| < 1 , while it  saturates to 1/2 for |∆| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  6  Conclusion  In this work, we applied coordinate Bethe ansatz to solve the steady state and the ground  state of a PT symmetric one-dimensional boundary-dissipated spin chain, focusing on the  PT broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We found the many-body scale-free state, which is composed of one  boundary mode and a continuum of scale-free modes in our particular model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' We then  derived the Bethe equations of the scale-free Bethe roots, and obtained a compact formula  for the eigen-energy in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  We then proposed an experimental  scheme to measure the dissipative part of the energy, and discussed how to compare it  with our analytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Our findings shed a light on exceptional points and PT transition in many-body  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Particularly, our solution is a generalization of the concept of scale-free NHSE from  free-particle to many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Although we focused on the scale-free behaviour in the  XXZ spin chain, it is expected that this feature is universal in a family of non-Hermitian  models with interactions in the bulk and dissipation-induced defect mode at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  For example, non-integrable models also exhibit scale-free properties, as demonstrated in  – 18 –  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Moreover, in integrable models solvable by nested Bethe ansatz (Fermi-Hubbard  model, higher spin XXX chain, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' ), boundary-operator induced PT symmetry transition  and the corresponding steady states may have richer structures to uncover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  We have demonstrated the scale-free skin effect by the difference between the imaginary  part of the steady state energy and that of a single boundary mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Intuitively, the scale-  free skin effect may also manifest itself in the excitation spectra near the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  A thorough analysis of those eigenstates requires preserving the O(1/N 2)-order terms in  the imaginary Bethe equations, which is left for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Algebraic Bethe ansatz is  another possible approach to the solutions of excitations, though it remains a question how  scale-free Bethe roots emerge in the monodromy and transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  Acknowledgments  This work is supported by NSFC under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' 12125405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
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+page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Ramette, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Metlitski, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Vuletic, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Ho and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content=' Choi, Landau-forbidden  quantum criticality in rydberg quantum simulators, arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='08829 (2022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
+page_content='  – 22 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFKT4oBgHgl3EQfvy5I/content/2301.11896v1.pdf'}
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+arXiv:2301.00679v1  [math.CA]  28 Dec 2022
+On a Cyclic Inequality Related to Chebyshev
+Polynomials
+Mohammad Javaheri
+Harry Shen
+mjavaheri@siena.edu
+hx21shen@siena.edu
+515 Loudon Road
+Siena College, School of Science
+Loudonville, NY 12211
+Abstract
+We show that any weighted geometric mean of Chebyshev polyno-
+mials is bounded from above by another Chebyshev polynomial. We
+also study a related homogeneous cyclic inequality
+� n
+�
+i=1
+x(a+b+1)/2
+i
+�2
+≥
+n
+�
+i=1
+xi
+n
+�
+i=1
+xa
+i xb
+i+1,
+where a, b, x1, . . . , xn (with xn+1 = x1) are nonnegative. In particular,
+we prove that the inequality holds when a = b = 1 and n ≤ 8 for all
+nonnegative numbers x1, . . . , xn. 1
+1
+Introduction
+Chebyshev polynomials of the first kind are defined by the recurrence rela-
+tion:
+Tn+1(x) = 2xTn(x) − Tn−1(x),
+(1)
+where T0(x) = 1 and T1(x) = x. The most well-known property of Chebyshev
+polynomials is that they express cos(nθ) in terms of cos(θ) via the equation
+cos(nθ) = Tn(cos θ). Chebyshev polynomials are a special kind of Jacobi
+polynomials (also known as hypergeometric polynomials), a class of classical
+orthogonal polynomials.
+Chebyshev was the first mathematician to have
+1Mathematics Subject Classification (2010): 26D07.
+Keywords: Chebyshev polynomials, cyclic homogeneous inequality.
+1
+
+noticed them in 1854, but their importance was not noticed until Hans Hahn
+rediscovered them and named them after Chebyshev. The polynomials Tn(x)
+are orthogonal with respect to the inner product
+⟨f, g⟩ = 2
+π
+� 1
+−1
+f(x)g(x)
+dx
+√
+1 − x2.
+In other words, ⟨Tm, Tn⟩ = 0 for all positive integers m ̸= n, and ⟨Tn, Tn⟩ = 1
+for all integers n ≥ 0.
+Chebyshev polynomials have the explicit expression
+Tn(x) = 1
+2
+�
+x −
+√
+x2 − 1
+�n
++ 1
+2
+�
+x +
+√
+x2 − 1
+�n
+,
+(2)
+for x ≥ 1. By replacing n with α in Equation (2), one can generalize Cheby-
+shev polynomials to functions Tα : [1, ∞) → [1, ∞) for all values of α ∈ R.
+Equivalently, Tα is defined by
+Tα
+�x + x−1
+2
+�
+= xα + x−α
+2
+,
+(3)
+for all x > 0. From Equation (1), we can see by induction that Tn+1(x) ≥
+Tn(x) for all x ≥ 1. One sees that if |α| ≥ |β|, then Tα(x) > Tβ(x) for
+all x > 1 (see Lemma 3). More generally, any weighted geometric mean of
+functions of the form Tα is bounded by another function of the same form:
+Theorem 1. Let ti, 1 ≤ i ≤ n, be nonnegative real numbers such that
+�n
+i=1 ti = 1. If α2 ≥ �n
+i=1 tiα2
+i , then we have
+n
+�
+i=1
+(Tαi(x))ti ≤ Tα(x),
+(4)
+for all x ≥ 1, where the equality occurs if and only if x = 1 or |α| = |αi|
+for all 1 ≤ i ≤ n with ti ̸= 0. Conversely, if the inequality (4) holds for all
+x ≥ 1, then α2 ≥ �n
+i=1 tiα2
+i .
+When n = 2, Theorem 1 can be used to show that if (c − 1)2 ≤ 2ab, then
+(xc + yc)2 ≥ (x + y)(xayb + yaxb),
+for all x, y ≥ 0, where c = (a + b + 1)/2. This inequality is a homogeneous
+cyclic inequality on two variables. We are interested in a general form of this
+inequality on n arbitrary nonnegative variables:
+� n
+�
+i=1
+xc
+i
+�2
+≥
+n
+�
+i=1
+xi
+n
+�
+i=1
+xa
+i xb
+i+1.
+(5)
+2
+
+It turns out that, given fixed values of a, b ≥ 0, the validity of the cyclic
+homogeneous inequality (5) depends on n. One can make the following ob-
+servations regarding the inequality (5):
+i) If a + b = 1, then the inequality holds for all n ≥ 1 (by the Rearrange-
+ment inequality [3]).
+ii) Given a, b ≥ 0 with a + b ̸= 1, there exists an integer N(a, b) such that
+the inequality (5) holds for all x1, . . . , xn ≥ 0 if and only if n ≤ N(a, b);
+[4].
+iii) Given a positive integer n, the set On of (a, b) ∈ [0, ∞) × [0, ∞) for
+which the inequality (5) holds for all x1, . . . , xn ≥ 0 is a topologically
+closed and convex subset of R2.
+Theorem 5 shows that O2 = {(a, b) : a, b ≥ 0 and (a+b−1)2 ≤ 8ab}. The
+problem of completely characterizing On for n > 2 seems difficult, however,
+a necessary condition is that
+(a + b − 1)2 ≤ 8ab sin2(π/n).
+For n = 3, in Theorem 6, we will show that
+{(a, b) : a, b ≥ 0 and 2a + 1 ≥ b ≥ (a − 1)/2 ≥ −b/2} ⊆ O3.
+As an example, in section 4, we will consider the case of a = b = 1 and prove
+the following theorem.
+Theorem 2. Let x1, . . . , xn be nonnegative real numbers. If n ≤ 8, then
+� n
+�
+i=1
+x3
+i
+�2
+≥
+� n
+�
+i=1
+x2
+i
+� � n
+�
+i=1
+x2
+i x2
+i+1
+�
+,
+(6)
+where the equality occurs if and only if x1 = x2 = · · · = xn. Moreover, the
+inequality (6) does not hold in general if n > 8.
+2
+Inequalities on Chebyshev polynomials
+In this section, we prove Theorem 1 which gives an upper bound for geometric
+means of Chebyshev polynomials. We first show that Tα(x) is an increasing
+function of |α| for a fixed x > 1.
+Lemma 3. If |α| ≥ |β|, then Tα(x) ≥ Tβ(x) for all x ≥ 1. The equality
+occurs if and only if x = 1 or |α| = |β|.
+3
+
+Proof. Without loss of generality, suppose that α ≥ β > 0. By virtue of
+Equation 3, we need to show that the function α �→ xα + x−α is a strictly
+increasing function of α > 0 for a fixed positive x ̸= 1. The claim then
+follows from
+d
+dα(xα + x−α) = (xα − x−α) ln x > 0,
+which holds for all positive x ̸= 1 and α > 0.
+Next, we prove Theorem 1 in the case of n = 2.
+Lemma 4. If 2α2 ≥ β2 + γ2, then
+(Tα(x))2 ≥ Tβ(x)Tγ(x),
+(7)
+for all x ≥ 1. The equality occurs if and only if x = 1 or |α| = |β| = |γ|.
+Conversely, if the inequality (7) holds for all x ≥ 1, then 2α2 ≥ β2 + γ2.
+Proof. We let
+G(x) = (xα + x−α)2 − (xβ + x−β)(xγ + x−γ).
+To prove the inequality (7), it is sufficient to show that G(x) ≥ 0 for all
+x > 0. For H(x) = xG′(x), one has H(1) = 0 and
+xH′(x) = 4α2(x2α + x−2α) − (β + γ)2(xβ+γ + x−β−γ) − (β − γ)2(xβ−γ + x−β+γ)
+≥ 4α2(x2α + x−2α) − (β + γ)2(x2α + x−2α) − (β − γ)2(x2α + x−2α)
+≥ 2(2α2 − β2 − γ2)(x2α + x−2α) ≥ 0,
+since |2α| ≥ |β + γ| and |2α| ≥ |β − γ|. It follows that H′(x) ≥ 0 for all
+x > 0. Since H(1) = 0, we must have H(x) ≥ 0 for all x ≥ 1 and H(x) ≤ 0
+for all 0 < x ≤ 1. Therefore, G′(x) ≥ 0 for all x ≥ 1 and G′(x) ≤ 0 for all
+0 < x ≤ 1. Since G(1) = 0, it follows that G(x) ≥ 0 for all x > 0. The
+equality occurs if and only if x = 1 or |2α| = |β + γ| or |2α| = |β − γ|.
+Therefore, the equality occurs if and only if x = 1 or |α| = |β| = |γ|.
+For the converse, suppose that G(x) ≥ 0 for all x > 0. From the calcu-
+lations above, we have G′(1) = 0 and G′′(1) = 4(2α2 − β2 − γ2). Since G
+attains a minimum at x = 1, we must have G′′(1) ≥ 0, which implies that
+2α2 ≥ β2 + γ2.
+A function f(x) is said to be concave on an interval [a, b], if
+f(tx1 + (1 − t)x2) ≥ tf(x1) + (1 − t)f(x2),
+4
+
+for all x1, x2 ∈ [a, b]. A function f(x) is said to be midpoint-concave on an
+interval [a, b], if
+f
+�x1 + x2
+2
+�
+≥ f(x1) + f(x2)
+2
+,
+for all x1, x2 ∈ [a, b]. We are now ready to prove Theorem 1.
+Proof of Theorem 1. Let fx : (0, ∞) → R be defined by
+fx(α) = ln T√α(x).
+We show that fx is a midpoint-concave function of α > 0 for any fixed value
+of x ≥ 1. It follows from Lemma 4 that
+fx(α) + fx(β) = ln T√α(x) + ln T√β(x) = ln T√α(x)T√β(x) ≤ ln T√γ(x)2,
+where γ = (α+β)/2. Therefore, fx(α)+fx(β) ≤ 2fx((α+β)/2), which means
+that fx is a midpoint-concave function on (0, ∞). A theorem of Jensen states
+that if a function is continuous and midpoint-concave, then it is concave [5, 6].
+It follows that fx is concave. By Jensen’s inequality [3], we conclude that
+n
+�
+i=1
+tifx(α2
+i ) ≤ fx
+� n
+�
+i=1
+tiα2
+i
+�
+,
+which implies the inequality (4).
+For the converse, by replacing x with (x + x−1)/2 in (4) and taking the
+natural logarithm of both sides, suppose that
+G(x) = ln(xα + x−α) −
+n
+�
+i=1
+ti ln(xαi + x−αi) ≥ 0,
+for all x > 0. It is straightforward to show that G(1) = G′(1) = 0 and
+G′′(1) = 1
+2
+�
+α2 −
+�
+i=1
+tiα2
+i
+�
+.
+Since G attains a minimum at x = 1, we conclude that G′′(1) ≥ 0, and the
+claim follows.
+□
+3
+A related cyclic homogeneous inequality
+In this section, we study the cyclic homogeneous inequality (5). We first
+consider the case of n = 2.
+5
+
+Theorem 5. Let a, b ≥ 0 and c = (a + b + 1)/2. If (c − 1)2 ≤ 2ab, then
+(xc + yc)2 ≥ (x + y)(xayb + yaxb),
+(8)
+for all x, y ≥ 0, and the equality occurs if and only if x = y or {a, b} = {0, 1}.
+Proof. With α = c/2, β = 1/2, and γ = (a − b)/2, Lemma 4 implies that
+((x/y)c/2 + (x/y)−c/2)2 ≥ ((x/y)1/2 + (x/y)−1/2)((x/y)(a−b)/2 + (x/y)(b−a)/2),
+for all x ≥ 1, if 2α2 ≥ β2 + γ2 or equivalently (c − 1)2 ≤ 2ab. The equality
+occurs if and only if x/y = 1 or |α| = |β| = |γ|, or equivalently, if and only
+if x = y or {a, b} = {0, 1}.
+Next, we consider the following general homogeneous cyclic inequality
+� n
+�
+i=1
+xc
+i
+�2
+≥
+n
+�
+i=1
+xi
+n
+�
+i=1
+xa
+i xb
+i+1,
+(9)
+where c = (a + b + 1)/2 and a, b, x1, . . . , xn ≥ 0.
+One asks that under
+what conditions on a, b, c, the inequality (9) holds for all x1, . . . , xn ≥ 0.
+It is straightforward to see that if a + b = 1, then the inequality (9) for
+all x1, . . . , xn ≥ 0, follows from the Rearrangement inequality [2, Ch. 6].
+However, if a+b ̸= 1, then the inequality (9) fails to hold if n is large enough
+[4]. In other words, the validity of the inequality (9) for all x1, . . . , xn ≥ 0
+for fixed values of a, b ≥ 0 with a + b ̸= 1 depends on n. Similarly, given a
+fixed value of n, the inequality (9) holds for a specific subset On of values
+(a, b) ∈ [0, ∞)×[0, ∞). Theorem 5 describes O2 completely. For n > 2, such
+a complete description seems to be difficult to find. However, in the following
+theorem, we derive a sufficient condition for the inequality (9) in the case of
+n = 3.
+Theorem 6. If 2a + 1 ≥ b ≥ (a − 1)/2 ≥ −b/2, then
+(xc + yc + zc)2 ≥ (x + y + z)(xayb + yazb + zaxb),
+for all x, y, z ≥ 0. The equality occurs if and only if x = y = z.
+Proof. Without loss of generality, we assume that a ≥ b. If b ≥ a − 1, then
+the claim follows from [4, Prop. 2.1]. Thus, suppose that a − b − 1 ≥ 0. Let
+x, y, z ≥ 0. By Jensen’s inequality [2, Ch. 7]:
+a + 1 − b
+2c
+x2c + b
+cxcyc ≥ xa+1yb,
+b
+2cx2c + 2b − a + 1
+2c
+y2c + a − b
+c
+xcyc ≥ xayb+1,
+a − b − 1
+2c
+x2c + 1
+cxczc + b
+cxcyc ≥ xaybz.
+6
+
+Adding these inequalities yields
+2a − b
+2c
+x2c + 2b − a + 1
+2c
+y2c + a + b
+c
+xcyc + 1
+cxczc ≥ (x + y + z)xayb.
+(10)
+Similarly
+2a − b
+2c
+y2c + 2b − a + 1
+2c
+z2c + a + b
+c
+yczc + 1
+cxcyc ≥ (x + y + z)yazb,
+(11)
+2a − b
+2c
+z2c + 2b − a + 1
+2c
+x2c + a + b
+c
+xczc + 1
+cyczc ≥ (x + y + z)zaxb.
+(12)
+The claim follows from adding inequalities (10)-(12).
+4
+The case of a = b = 1
+In this section, we prove Theorem 2 which states that the inequality (5) holds
+in the case of a = b = 1 if and only if n ≤ 8. To show that the inequality
+(5) does not hold for n ≥ 9, it is sufficient to find a counterexample for the
+inequality with 9 variables, since if the inequality (5) holds for n variables,
+then it holds for n − 1 variables. Here is one such counterexample [4]:
+x1 = x9 = 8.5, x2 = x8 = 9, x3 = x7 = 10, x4 = x6 = 11.5, x5 = 12.
+It is then left to prove that the inequality (5) holds with a = b = 1 for all
+x1, . . . , x8 ≥ 0. We first need a lemma.
+Lemma 7. Let x1, . . . , x8 be nonnegative real numbers. Then
+8
+�
+i=1
+x3
+i ≥ 1
+8
+�
+8
+�
+i=1
+xi
+� �
+8
+�
+i=1
+x2
+i
+�
+.
+Proof. By the Power Mean Inequality [1, Ch. III], one has
+�
+1
+8
+8
+�
+i=1
+x3
+i
+�1/3
+≥ 1
+8
+8
+�
+i=1
+xi and
+�
+1
+8
+8
+�
+i=1
+x3
+i
+�1/3
+≥
+�
+1
+8
+8
+�
+i=1
+x2
+i
+�1/2
+.
+The claim follows from these inequalities.
+Now, we are ready to prove Theorem 2.
+7
+
+Proof of Theorem 2.
+Equivalently, we show that the maximum value of
+the function f : U → R defined by
+f(x1, . . . , x8) =
+�8
+i=1 x4
+i x4
+i+1
+��8
+i=1 x6
+i
+�2 ,
+is 1, where
+U =
+�
+(x1, . . . , x8) :
+8
+�
+i=1
+x4
+i = 1
+�
+.
+one has (�8
+i=1 x6
+i /8)1/6 ≥ (�8
+i=1 x4
+i /8)1/4 by the Power Mean Inequality [1,
+Ch. III]. Therefore, �8
+i=1 x6
+i ≥
+�
+1/8 and so the function f is bounded from
+above on U, hence it attains a positive absolute maximum on the compact set
+U, say at (x1, . . . , x8). Without loss of generality, we can assume x1, . . . , x8 ≥
+0. By the method of Lagrange multipliers, there exists a real number λ such
+that
+1
+A4
+�
+4x3
+i (x4
+i−1 + x4
+i+1)A2 − 2AB(6x5
+i )
+�
+= λ(4x3
+i ), ∀i = 1, . . . , 8,
+(13)
+where A = x6
+1 + · · · + x6
+8 and B = x4
+1x4
+2 + · · · + x4
+8x4
+1. Therefore,
+x4
+i (x4
+i−1 + x4
+i+1)A − 3Bx6
+i = λA3x4
+i , ∀1 ≤ i ≤ 8.
+(14)
+By summing the equations (14), we have λ = −B/A2. We need to show
+that A2 ≥ B.
+On the contrary, suppose B > A2, and we will derive a
+contradiction.
+Equations (13) imply that, if xi ̸= 0, then
+3Bx2
+i = A(x4
+i−1 + x4
+i+1) + AB.
+(15)
+First, we show that xi ̸= 0 for all i ∈ {1, . . . , 8}. On the contrary, and
+without loss of generality, suppose x8 = 0 and x7 > 0. Given ǫ ∈ (0, x7), let
+δ = δ(ǫ) = (x4
+7 − (x7 − ǫ)4)1/4,
+such that (x7 − ǫ)4 + δ4 = x4
+7, and so δ3δ′ = (x7 − ǫ)3. We define
+F(ǫ) = f(x1, . . . , x6, x7 − ǫ, δ) = Bǫ
+A2ǫ
+,
+and compute
+F ′(ǫ) = 4(x7 − ǫ)3
+A4ǫ
+�
+A2
+ǫ(−x4
+6 − δ4 + (x7 − ǫ)4 + x4
+1) + 3AǫBǫ((x7 − ǫ)2 − δ2)
+�
+.
+8
+
+It follows that
+lim
+ǫ→0+ F ′(ǫ) = x3
+7
+A4(A2(+x4
+7 + x4
+1) − A2x2
+6 + 3ABx2
+7)
+= x3
+7
+A4(A2(x4
+1 + x4
+7) + A2B) > 0,
+(16)
+where we have used equation (15) with i = 7 to obtain −A2x2
+6 + 3ABx2
+7 =
+A2B. The inequality (16) is a contradiction with the assumption that F(ǫ)
+attains a maximum as ǫ → 0+. We conclude that xi > 0 for all i = 1, . . . , 8.
+In particular, equations (15) hold for all i = 1, . . . , 8. In the rest of the proof,
+we let yi = x2
+i . Hence, with C = B/A, the equations (15) turn into
+3yiC = y2
+i−1 + y2
+i+1 + B, ∀1 ≤ i ≤ 8.
+(17)
+It follows that
+3(yi + yi+4)C = 2B + y2
+i−1 + y2
+i+1 + y2
+i+3 + y2
+i+5,
+3(yi+2 + yi+6)C = 2B + y2
+i+1 + y2
+i+3 + y2
+i+5 + y2
+i+7,
+which imply that yj + yj+4 = yj+2 + yj+6 for all j, since yi−1 = yi+7 as
+the indices are computed modulo 8. Therefore, there exist nonnegative real
+numbers r, s such that
+y1 + y5 = y3 + y7 = r,
+(18)
+y2 + y6 = y4 + y8 = s.
+(19)
+Equations (17) imply that
+3(y1 − y3)C = y2
+8 − y2
+4
+3(y2 − y4)C = y2
+1 − y2
+5
+3(y3 − y5)C = y2
+2 − y2
+6
+3(y4 − y6)C = y2
+3 − y2
+7.
+Let ¯yi = yi − r/2 if i is odd, and ¯yi = yi − s/2 if i is even. It follows that
+¯yi + ¯yi+4 = 0 for all i. Moreover, for i odd, we have
+3C(¯yi − ¯yi+4) = 3C(yi − yi+4) = y2
+i−1 − y2
+i+1 + y2
+i+3 − y2
+i+5
+= (yi−1 − yi+3)(yi−1 + yi+3) + (yi+1 − yi+5)(yi+1 + yi+5)
+= 2¯yi−1s + 2¯yi+1s,
+which implies that ¯yi = (¯yi−1 + ¯yi+1)s/(3C) for odd i. Similarly, ¯yi = (¯yi−1 +
+¯yi+1)r/(3C) for even i. It then follows that
+¯yi = s
+3C (¯yi−1 + ¯yi+1) = rs
+9C2(¯yi−2 + ¯yi + ¯yi + ¯yi+2) = 2rs
+9C2 ¯yi,
+9
+
+for odd i, and similarly for even i. We claim that 9C2 ̸= 2rs. On the contrary,
+suppose 9C2 = 2rs, and so, since C = B/A > A, we must have
+6
+√
+2A < 6
+√
+2C ≤ 4√rs ≤ 2(r + s) ≤
+8
+�
+i=1
+yi.
+(20)
+However, by Lemma 7, we have 8A ≥ �8
+i=1 yi which contradicts (20), since
+6
+√
+2 > 8. Thus 9C2 ̸= 2rs, and so ¯yi = 0 for all i. Therefore, y1 = y3 =
+y5 = y7 = r/2, and y2 = y4 = y6 = y8 = s/2.
+So we have r2 + s2 =
+4((r/2)2 + (s/2)2) = �8
+i=1 y2
+i = 1 and
+f(x1, . . . , x8) =
+8(r/2)2(s/2)2
+(4(r/2)3 + 4(s/2)3)2 =
+2r2s2
+(r3 + s3)2 ≤ 1,
+(21)
+since it follows from r2 + s2 = 1 that
+r3 + s3 ≥ 2
+�r2 + s2
+2
+�3/2
+≥ 2
+�1
+2
+�3/2
+≥ r2 + s2
+√
+2
+≥
+√
+2rs.
+The equality occurs in (21) if and only if r = s (and so y1 = y2 = · · · = y8);
+hence, the equality in (6) occurs if and only if x1 = x2 = · · · = x8.
+□
+References
+[1] P. S. Bullen, Handbook of Means and Their Inequalities, Springer (2003).
+[2] Z. Cvetkovski, Inequalities, Theorems, Techniques, and Selected Prob-
+lems, Springer, Berlin (2012).
+[3] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities , Cambridge Univ.
+Press (1934).
+[4] M. Javaheri, A new arrangement inequality, J. Ineq. Pure Appl. Math.
+7(5) (2006), Article 162.
+[5] J. L. W. V. Jensen, Sur les fonctions convexes et les inégalités entre
+les valeurs moyennes, Acta Mathematica (Institut Mittag-Leffler) 1906,
+Vol. 30, No. 1, pp 175–193.
+[6] C.P. Niculescu and L-E. Persson, Convex Functions and Their Applica-
+tions, Springer International Publishing, (2018).
+10
+
diff --git a/j9AyT4oBgHgl3EQfyPkN/content/tmp_files/load_file.txt b/j9AyT4oBgHgl3EQfyPkN/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ee65344b8f10de93a3c418b06ffcd7e36c205db2
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@@ -0,0 +1,261 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf,len=260
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='00679v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='CA]  28 Dec 2022  On a Cyclic Inequality Related to Chebyshev  Polynomials  Mohammad Javaheri  Harry Shen  mjavaheri@siena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='edu  hx21shen@siena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='edu  515 Loudon Road  Siena College, School of Science  Loudonville, NY 12211  Abstract  We show that any weighted geometric mean of Chebyshev polyno-  mials is bounded from above by another Chebyshev polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' We  also study a related homogeneous cyclic inequality  � n  �  i=1  x(a+b+1)/2  i  �2  ≥  n  �  i=1  xi  n  �  i=1  xa  i xb  i+1,  where a, b, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , xn (with xn+1 = x1) are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' In particular,  we prove that the inequality holds when a = b = 1 and n ≤ 8 for all  nonnegative numbers x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' 1  1  Introduction  Chebyshev polynomials of the first kind are defined by the recurrence rela-  tion:  Tn+1(x) = 2xTn(x) − Tn−1(x),  (1)  where T0(x) = 1 and T1(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' The most well-known property of Chebyshev  polynomials is that they express cos(nθ) in terms of cos(θ) via the equation  cos(nθ) = Tn(cos θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Chebyshev polynomials are a special kind of Jacobi  polynomials (also known as hypergeometric polynomials), a class of classical  orthogonal polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Chebyshev was the first mathematician to have  1Mathematics Subject Classification (2010): 26D07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Keywords: Chebyshev polynomials, cyclic homogeneous inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  1  noticed them in 1854, but their importance was not noticed until Hans Hahn  rediscovered them and named them after Chebyshev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' The polynomials Tn(x)  are orthogonal with respect to the inner product  ⟨f, g⟩ = 2  π  � 1  −1  f(x)g(x)  dx  √  1 − x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  In other words, ⟨Tm, Tn⟩ = 0 for all positive integers m ̸= n, and ⟨Tn, Tn⟩ = 1  for all integers n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Chebyshev polynomials have the explicit expression  Tn(x) = 1  2  �  x −  √  x2 − 1  �n  + 1  2  �  x +  √  x2 − 1  �n  ,  (2)  for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' By replacing n with α in Equation (2), one can generalize Cheby-  shev polynomials to functions Tα : [1, ∞) → [1, ∞) for all values of α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Equivalently, Tα is defined by  Tα  �x + x−1  2  �  = xα + x−α  2  ,  (3)  for all x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' From Equation (1), we can see by induction that Tn+1(x) ≥  Tn(x) for all x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' One sees that if |α| ≥ |β|, then Tα(x) > Tβ(x) for  all x > 1 (see Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' More generally, any weighted geometric mean of  functions of the form Tα is bounded by another function of the same form:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Let ti, 1 ≤ i ≤ n, be nonnegative real numbers such that  �n  i=1 ti = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' If α2 ≥ �n  i=1 tiα2  i , then we have  n  �  i=1  (Tαi(x))ti ≤ Tα(x),  (4)  for all x ≥ 1, where the equality occurs if and only if x = 1 or |α| = |αi|  for all 1 ≤ i ≤ n with ti ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Conversely, if the inequality (4) holds for all  x ≥ 1, then α2 ≥ �n  i=1 tiα2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  When n = 2, Theorem 1 can be used to show that if (c − 1)2 ≤ 2ab, then  (xc + yc)2 ≥ (x + y)(xayb + yaxb),  for all x, y ≥ 0, where c = (a + b + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' This inequality is a homogeneous  cyclic inequality on two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' We are interested in a general form of this  inequality on n arbitrary nonnegative variables:  � n  �  i=1  xc  i  �2  ≥  n  �  i=1  xi  n  �  i=1  xa  i xb  i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  (5)  2  It turns out that, given fixed values of a, b ≥ 0, the validity of the cyclic  homogeneous inequality (5) depends on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' One can make the following ob-  servations regarding the inequality (5):  i) If a + b = 1, then the inequality holds for all n ≥ 1 (by the Rearrange-  ment inequality [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  ii) Given a, b ≥ 0 with a + b ̸= 1, there exists an integer N(a, b) such that  the inequality (5) holds for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , xn ≥ 0 if and only if n ≤ N(a, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  iii) Given a positive integer n, the set On of (a, b) ∈ [0, ∞) × [0, ∞) for  which the inequality (5) holds for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , xn ≥ 0 is a topologically  closed and convex subset of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Theorem 5 shows that O2 = {(a, b) : a, b ≥ 0 and (a+b−1)2 ≤ 8ab}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' The  problem of completely characterizing On for n > 2 seems difficult, however,  a necessary condition is that  (a + b − 1)2 ≤ 8ab sin2(π/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  For n = 3, in Theorem 6, we will show that  {(a, b) : a, b ≥ 0 and 2a + 1 ≥ b ≥ (a − 1)/2 ≥ −b/2} ⊆ O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  As an example, in section 4, we will consider the case of a = b = 1 and prove  the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , xn be nonnegative real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' If n ≤ 8, then  � n  �  i=1  x3  i  �2  ≥  � n  �  i=1  x2  i  � � n  �  i=1  x2  i x2  i+1  �  ,  (6)  where the equality occurs if and only if x1 = x2 = · · · = xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Moreover, the  inequality (6) does not hold in general if n > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  2  Inequalities on Chebyshev polynomials  In this section, we prove Theorem 1 which gives an upper bound for geometric  means of Chebyshev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' We first show that Tα(x) is an increasing  function of |α| for a fixed x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' If |α| ≥ |β|, then Tα(x) ≥ Tβ(x) for all x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' The equality  occurs if and only if x = 1 or |α| = |β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Without loss of generality, suppose that α ≥ β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' By virtue of  Equation 3, we need to show that the function α �→ xα + x−α is a strictly  increasing function of α > 0 for a fixed positive x ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' The claim then  follows from  d  dα(xα + x−α) = (xα − x−α) ln x > 0,  which holds for all positive x ̸= 1 and α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Next, we prove Theorem 1 in the case of n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' If 2α2 ≥ β2 + γ2, then  (Tα(x))2 ≥ Tβ(x)Tγ(x),  (7)  for all x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' The equality occurs if and only if x = 1 or |α| = |β| = |γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Conversely, if the inequality (7) holds for all x ≥ 1, then 2α2 ≥ β2 + γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' We let  G(x) = (xα + x−α)2 − (xβ + x−β)(xγ + x−γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  To prove the inequality (7), it is sufficient to show that G(x) ≥ 0 for all  x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' For H(x) = xG′(x), one has H(1) = 0 and  xH′(x) = 4α2(x2α + x−2α) − (β + γ)2(xβ+γ + x−β−γ) − (β − γ)2(xβ−γ + x−β+γ)  ≥ 4α2(x2α + x−2α) − (β + γ)2(x2α + x−2α) − (β − γ)2(x2α + x−2α)  ≥ 2(2α2 − β2 − γ2)(x2α + x−2α) ≥ 0,  since |2α| ≥ |β + γ| and |2α| ≥ |β − γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' It follows that H′(x) ≥ 0 for all  x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Since H(1) = 0, we must have H(x) ≥ 0 for all x ≥ 1 and H(x) ≤ 0  for all 0 < x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Therefore, G′(x) ≥ 0 for all x ≥ 1 and G′(x) ≤ 0 for all  0 < x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Since G(1) = 0, it follows that G(x) ≥ 0 for all x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' The  equality occurs if and only if x = 1 or |2α| = |β + γ| or |2α| = |β − γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Therefore, the equality occurs if and only if x = 1 or |α| = |β| = |γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  For the converse, suppose that G(x) ≥ 0 for all x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' From the calcu-  lations above, we have G′(1) = 0 and G′′(1) = 4(2α2 − β2 − γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Since G  attains a minimum at x = 1, we must have G′′(1) ≥ 0, which implies that  2α2 ≥ β2 + γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  A function f(x) is said to be concave on an interval [a, b], if  f(tx1 + (1 − t)x2) ≥ tf(x1) + (1 − t)f(x2),  4  for all x1, x2 ∈ [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' A function f(x) is said to be midpoint-concave on an  interval [a, b], if  f  �x1 + x2  2  �  ≥ f(x1) + f(x2)  2  ,  for all x1, x2 ∈ [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' We are now ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Let fx : (0, ∞) → R be defined by  fx(α) = ln T√α(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  We show that fx is a midpoint-concave function of α > 0 for any fixed value  of x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' It follows from Lemma 4 that  fx(α) + fx(β) = ln T√α(x) + ln T√β(x) = ln T√α(x)T√β(x) ≤ ln T√γ(x)2,  where γ = (α+β)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Therefore, fx(α)+fx(β) ≤ 2fx((α+β)/2), which means  that fx is a midpoint-concave function on (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' A theorem of Jensen states  that if a function is continuous and midpoint-concave, then it is concave [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  It follows that fx is concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' By Jensen’s inequality [3], we conclude that  n  �  i=1  tifx(α2  i ) ≤ fx  � n  �  i=1  tiα2  i  �  ,  which implies the inequality (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  For the converse, by replacing x with (x + x−1)/2 in (4) and taking the  natural logarithm of both sides, suppose that  G(x) = ln(xα + x−α) −  n  �  i=1  ti ln(xαi + x−αi) ≥ 0,  for all x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' It is straightforward to show that G(1) = G′(1) = 0 and  G′′(1) = 1  2  �  α2 −  �  i=1  tiα2  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Since G attains a minimum at x = 1, we conclude that G′′(1) ≥ 0, and the  claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  □  3  A related cyclic homogeneous inequality  In this section, we study the cyclic homogeneous inequality (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' We first  consider the case of n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  5  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Let a, b ≥ 0 and c = (a + b + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' If (c − 1)2 ≤ 2ab, then  (xc + yc)2 ≥ (x + y)(xayb + yaxb),  (8)  for all x, y ≥ 0, and the equality occurs if and only if x = y or {a, b} = {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' With α = c/2, β = 1/2, and γ = (a − b)/2, Lemma 4 implies that  ((x/y)c/2 + (x/y)−c/2)2 ≥ ((x/y)1/2 + (x/y)−1/2)((x/y)(a−b)/2 + (x/y)(b−a)/2),  for all x ≥ 1, if 2α2 ≥ β2 + γ2 or equivalently (c − 1)2 ≤ 2ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' The equality  occurs if and only if x/y = 1 or |α| = |β| = |γ|, or equivalently, if and only  if x = y or {a, b} = {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Next, we consider the following general homogeneous cyclic inequality  � n  �  i=1  xc  i  �2  ≥  n  �  i=1  xi  n  �  i=1  xa  i xb  i+1,  (9)  where c = (a + b + 1)/2 and a, b, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , xn ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  One asks that under  what conditions on a, b, c, the inequality (9) holds for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , xn ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  It is straightforward to see that if a + b = 1, then the inequality (9) for  all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , xn ≥ 0, follows from the Rearrangement inequality [2, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  However, if a+b ̸= 1, then the inequality (9) fails to hold if n is large enough  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' In other words, the validity of the inequality (9) for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , xn ≥ 0  for fixed values of a, b ≥ 0 with a + b ̸= 1 depends on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Similarly, given a  fixed value of n, the inequality (9) holds for a specific subset On of values  (a, b) ∈ [0, ∞)×[0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Theorem 5 describes O2 completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' For n > 2, such  a complete description seems to be difficult to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' However, in the following  theorem, we derive a sufficient condition for the inequality (9) in the case of  n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' If 2a + 1 ≥ b ≥ (a − 1)/2 ≥ −b/2, then  (xc + yc + zc)2 ≥ (x + y + z)(xayb + yazb + zaxb),  for all x, y, z ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' The equality occurs if and only if x = y = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Without loss of generality, we assume that a ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' If b ≥ a − 1, then  the claim follows from [4, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Thus, suppose that a − b − 1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Let  x, y, z ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' By Jensen’s inequality [2, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' 7]:  a + 1 − b  2c  x2c + b  cxcyc ≥ xa+1yb,  b  2cx2c + 2b − a + 1  2c  y2c + a − b  c  xcyc ≥ xayb+1,  a − b − 1  2c  x2c + 1  cxczc + b  cxcyc ≥ xaybz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  6  Adding these inequalities yields  2a − b  2c  x2c + 2b − a + 1  2c  y2c + a + b  c  xcyc + 1  cxczc ≥ (x + y + z)xayb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  (10)  Similarly  2a − b  2c  y2c + 2b − a + 1  2c  z2c + a + b  c  yczc + 1  cxcyc ≥ (x + y + z)yazb,  (11)  2a − b  2c  z2c + 2b − a + 1  2c  x2c + a + b  c  xczc + 1  cyczc ≥ (x + y + z)zaxb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  (12)  The claim follows from adding inequalities (10)-(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  4  The case of a = b = 1  In this section, we prove Theorem 2 which states that the inequality (5) holds  in the case of a = b = 1 if and only if n ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' To show that the inequality  (5) does not hold for n ≥ 9, it is sufficient to find a counterexample for the  inequality with 9 variables, since if the inequality (5) holds for n variables,  then it holds for n − 1 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Here is one such counterexample [4]:  x1 = x9 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='5, x2 = x8 = 9, x3 = x7 = 10, x4 = x6 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='5, x5 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  It is then left to prove that the inequality (5) holds with a = b = 1 for all  x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , x8 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' We first need a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , x8 be nonnegative real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Then  8  �  i=1  x3  i ≥ 1  8  �  8  �  i=1  xi  � �  8  �  i=1  x2  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' By the Power Mean Inequality [1, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' III], one has  �  1  8  8  �  i=1  x3  i  �1/3  ≥ 1  8  8  �  i=1  xi and  �  1  8  8  �  i=1  x3  i  �1/3  ≥  �  1  8  8  �  i=1  x2  i  �1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  The claim follows from these inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Now, we are ready to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  7  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Equivalently, we show that the maximum value of  the function f : U → R defined by  f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , x8) =  �8  i=1 x4  i x4  i+1  ��8  i=1 x6  i  �2 ,  is 1, where  U =  �  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , x8) :  8  �  i=1  x4  i = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  one has (�8  i=1 x6  i /8)1/6 ≥ (�8  i=1 x4  i /8)1/4 by the Power Mean Inequality [1,  Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' III].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Therefore, �8  i=1 x6  i ≥  �  1/8 and so the function f is bounded from  above on U, hence it attains a positive absolute maximum on the compact set  U, say at (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , x8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Without loss of generality, we can assume x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , x8 ≥  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' By the method of Lagrange multipliers, there exists a real number λ such  that  1  A4  �  4x3  i (x4  i−1 + x4  i+1)A2 − 2AB(6x5  i )  �  = λ(4x3  i ), ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , 8,  (13)  where A = x6  1 + · · · + x6  8 and B = x4  1x4  2 + · · · + x4  8x4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Therefore,  x4  i (x4  i−1 + x4  i+1)A − 3Bx6  i = λA3x4  i , ∀1 ≤ i ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  (14)  By summing the equations (14), we have λ = −B/A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' We need to show  that A2 ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  On the contrary, suppose B > A2, and we will derive a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Equations (13) imply that, if xi ̸= 0, then  3Bx2  i = A(x4  i−1 + x4  i+1) + AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  (15)  First, we show that xi ̸= 0 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , 8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' On the contrary, and  without loss of generality, suppose x8 = 0 and x7 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Given ǫ ∈ (0, x7), let  δ = δ(ǫ) = (x4  7 − (x7 − ǫ)4)1/4,  such that (x7 − ǫ)4 + δ4 = x4  7, and so δ3δ′ = (x7 − ǫ)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' We define  F(ǫ) = f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , x6, x7 − ǫ, δ) = Bǫ  A2ǫ  ,  and compute  F ′(ǫ) = 4(x7 − ǫ)3  A4ǫ  �  A2  ǫ(−x4  6 − δ4 + (x7 − ǫ)4 + x4  1) + 3AǫBǫ((x7 − ǫ)2 − δ2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  8  It follows that  lim  ǫ→0+ F ′(ǫ) = x3  7  A4(A2(+x4  7 + x4  1) − A2x2  6 + 3ABx2  7)  = x3  7  A4(A2(x4  1 + x4  7) + A2B) > 0,  (16)  where we have used equation (15) with i = 7 to obtain −A2x2  6 + 3ABx2  7 =  A2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' The inequality (16) is a contradiction with the assumption that F(ǫ)  attains a maximum as ǫ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' We conclude that xi > 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  In particular, equations (15) hold for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' In the rest of the proof,  we let yi = x2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Hence, with C = B/A, the equations (15) turn into  3yiC = y2  i−1 + y2  i+1 + B, ∀1 ≤ i ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  (17)  It follows that  3(yi + yi+4)C = 2B + y2  i−1 + y2  i+1 + y2  i+3 + y2  i+5,  3(yi+2 + yi+6)C = 2B + y2  i+1 + y2  i+3 + y2  i+5 + y2  i+7,  which imply that yj + yj+4 = yj+2 + yj+6 for all j, since yi−1 = yi+7 as  the indices are computed modulo 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Therefore, there exist nonnegative real  numbers r, s such that  y1 + y5 = y3 + y7 = r,  (18)  y2 + y6 = y4 + y8 = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  (19)  Equations (17) imply that  3(y1 − y3)C = y2  8 − y2  4  3(y2 − y4)C = y2  1 − y2  5  3(y3 − y5)C = y2  2 − y2  6  3(y4 − y6)C = y2  3 − y2  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Let ¯yi = yi − r/2 if i is odd, and ¯yi = yi − s/2 if i is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' It follows that  ¯yi + ¯yi+4 = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Moreover, for i odd, we have  3C(¯yi − ¯yi+4) = 3C(yi − yi+4) = y2  i−1 − y2  i+1 + y2  i+3 − y2  i+5  = (yi−1 − yi+3)(yi−1 + yi+3) + (yi+1 − yi+5)(yi+1 + yi+5)  = 2¯yi−1s + 2¯yi+1s,  which implies that ¯yi = (¯yi−1 + ¯yi+1)s/(3C) for odd i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Similarly, ¯yi = (¯yi−1 +  ¯yi+1)r/(3C) for even i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' It then follows that  ¯yi = s  3C (¯yi−1 + ¯yi+1) = rs  9C2(¯yi−2 + ¯yi + ¯yi + ¯yi+2) = 2rs  9C2 ¯yi,  9  for odd i, and similarly for even i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' We claim that 9C2 ̸= 2rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' On the contrary,  suppose 9C2 = 2rs, and so, since C = B/A > A, we must have  6  √  2A < 6  √  2C ≤ 4√rs ≤ 2(r + s) ≤  8  �  i=1  yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  (20)  However, by Lemma 7, we have 8A ≥ �8  i=1 yi which contradicts (20), since  6  √  2 > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Thus 9C2 ̸= 2rs, and so ¯yi = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Therefore, y1 = y3 =  y5 = y7 = r/2, and y2 = y4 = y6 = y8 = s/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  So we have r2 + s2 =  4((r/2)2 + (s/2)2) = �8  i=1 y2  i = 1 and  f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' , x8) =  8(r/2)2(s/2)2  (4(r/2)3 + 4(s/2)3)2 =  2r2s2  (r3 + s3)2 ≤ 1,  (21)  since it follows from r2 + s2 = 1 that  r3 + s3 ≥ 2  �r2 + s2  2  �3/2  ≥ 2  �1  2  �3/2  ≥ r2 + s2  √  2  ≥  √  2rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  The equality occurs in (21) if and only if r = s (and so y1 = y2 = · · · = y8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  hence, the equality in (6) occurs if and only if x1 = x2 = · · · = x8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  □  References  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Bullen, Handbook of Means and Their Inequalities, Springer (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  [2] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Cvetkovski, Inequalities, Theorems, Techniques, and Selected Prob-  lems, Springer, Berlin (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Hardy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Littlewood, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Pólya, Inequalities , Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  Press (1934).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Javaheri, A new arrangement inequality, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Ineq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  7(5) (2006), Article 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Jensen, Sur les fonctions convexes et les inégalités entre  les valeurs moyennes, Acta Mathematica (Institut Mittag-Leffler) 1906,  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' 30, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' 1, pp 175–193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Niculescu and L-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content=' Persson, Convex Functions and Their Applica-  tions, Springer International Publishing, (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
+page_content='  10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AyT4oBgHgl3EQfyPkN/content/2301.00679v1.pdf'}
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+ 
+1 
+ 
+CaraNet: Context Axial Reverse Attention Network for Segmentation 
+of Small Medical Objects  
+Ange Lou1, Shuyue Guan2, Murray Loew2 
+Vanderbilt University1, Nashville TN, USA 
+ange.lou@vanderbilt.edu 
+The George Washington University2, Washington DC, USA 
+{frankshuyueguan, loew}@gwu.edu 
+Abstract 
+Purpose: Segmenting medical images accurately and reliably is important for disease diagnosis and treatment. It is a 
+challenging task because of the wide variety of objects’ sizes, shapes, and scanning modalities. Recently, many 
+convolutional neural networks (CNN) have been designed for segmentation tasks and achieved great success. Few studies, 
+however, have fully considered the sizes of objects, and thus most demonstrate poor performance for small objects 
+segmentation. This can have a significant impact on the early detection of diseases.  
+Approach: This paper proposes a Context Axial Reverse Attention Network (CaraNet) to improve the segmentation 
+performance on small objects compared with several recent state-of-the-art models. CaraNet applies axial reserve attention 
+(ARA) and channel-wise feature pyramid (CFP) module to dig feature information of small medical object. And we 
+evaluate our model by six different measurement metrics. 
+Results: We test our CaraNet on brain tumor (BraTS 2018) and polyp (Kvasir-SEG, CVC-ColonDB, CVC-ClinicDB, 
+CVC-300, and ETIS-LaribPolypDB) segmentation datasets. Our CaraNet achieves the top-rank mean Dice segmentation 
+accuracy, and results show a distinct advantage of CaraNet in the segmentation of small medical objects. 
+Conclusion: We proposed CaraNet to segment small medical objects and outperform other state-of-the-art methods. 
+Codes available: https://github.com/AngeLouCN/CaraNet 
+Keywords: Small object segmentation; Brain tumor; Colonoscopy; Attention; Context axial reverse 
+1. INTRODUCTION  
+Deep learning has had a tremendous impact on various fields in science. Our focus of the current study in deep learning 
+is on one of the most critical areas of computer vision: medical image segmentation. Recently, various convolutional neural 
+networks (CNNs) have shown great performance on medical image segmentation [1,2,3,4]. Those CNNs have been 
+introduced for various medical imaging modalities, including X-ray, visible-light imaging, magnetic resonance imaging 
+(MRI), positron emission tomography (PET), and computerized tomography (CT). They all achieved excellent 
+performance on medical image segmentation challenges from different modalities, like BraTS [5,6,7], KiTS19 [8], and 
+COVID19-20 [9,10]. To obtain more accurate segmentation results, many works introduced improvements of network 
+architectures. Those improvements are mostly attributed to exploring new neural architectures by designing networks with 
+varying depths (ResNet [11]), widths (ResNeXt [12]), connectivity (DenseNet [13] and GoogLeNet [14]), or new types of 
+components (pyramid scene [15] and atrous convolution [16]). Although those new architectures improve the overall 
+segmentation results, they are less sensitive to detecting small medical objects. And it is very common in medical image 
+segmentation that the anatomy of interest occupies only a very small portion of the image [17]. Most extracted features 
+belong to the background, while these small lesion areas are important for early detection and diagnosis. For example, the 
+survival rate decreases with the growing size of a brain tumor [18]. Thus, it has clinical significance to build an 
+effective network to detect tiny medical objects. 
+The attention mechanism plays a dominant role in neural network research. It can effectively use information 
+transferred from several subsequent feature maps to detect the salience features [19]. Many attention methods such 
+as self-attention and multi-head attention have been verified to have high performance in applications of natural 
+language processing [20] and computer vision [21]. Those attention methods also have been successfully used for 
+medical image segmentation; for example, the Medical Transformer [22] (MedT) used a gated axial self-attention 
+layer to build a Local-Global (LoGo) network for ultrasound and microscopy image segmentation, TransUNet [23] 
+
+ 
+2 
+ 
+stacked self-attention as a transformer in the encoder for CT image segmentation, and CoTr [24] bridged two CNN 
+encoder and decoder by the transformer encoder for multi-organ segmentation. All those attention-based 
+segmentations achieve significant improvement compared with purely convolutional neural networks like U-Net [1] 
+and FCN [25]. 
+Although those new types of neural networks show good performance on many medical segmentation tasks, 
+they seldom consider the small object segmentation, especially in the medical image area. We propose here a novel 
+attention-based deep neural network, called Context Axial Reverse Attention Network (CaraNet). The contribution 
+of the paper can be summarized as follows: 
+1) 
+We propose a novel neural network – CaraNet -- to solve the problem of segmentation of small medical objects. 
+2) 
+We introduce a method to evaluate the network’s performance on small medical objects. 
+3) 
+Our experiments show that CaraNet outperforms most current models (e.g., DS-TransUNet from TMI ’22, 
+CCBANet from MICCAI ‘21and PraNet from MICCAI ’20) and advances the state-of-the-art by a large margin, 
+both overall and on small objects, in segmentation performance on polyps. 
+2. METHOD  
+Figure 1 shows the architecture of our CaraNet, which uses a parallel partial decoder [26] to generate the high-
+level semantic global map and a set of context and axial reverse attention operations to detect global and local feature 
+information. We introduce main components of the CaraNet architecture in the following subsections with 
+explanation of the motivation, purpose, or effectiveness to add these components. 
+ 
+Figure 1. Overview of CaraNet, which contains pretrained backbone, partial decoder, channel-wise feature pyramid (CFP) module and 
+axial reverse attention (A-RA) module. 
+2.1 Backbone  
+Transfer learning provides a feasible method for alleviating the challenge of data-hunger, and it has been widely 
+applied to the field of computer vision [27]. Benefiting from the strong visual knowledge distributed in ImageNet 
+[28], the pre-trained CNNs can be fine-tuned with a small amount of task-specific data and can perform well on 
+
+f1
+f2
+PD
+Sg
+f3
+f4
+fs
+CFP
+CFP
+CFP
+Down-sampling
+Axial-attention
+fi
+8=P
+8=p
+8=P
+Height
+Width
+f1
+f2
+f3
+axis
+axis
+Si
+A-RA
+A-RA
+A-RA
+S
+ARA3
+ARA4
+ARA5
++
++
++
+Up-sampling
+Deep supervision
+Partial decoder
+Featureflow
+Mapflow
+GT
+Prediction
+S3
+S4
+I Ss
+S
+Sigmoid 
+3 
+ 
+downstream tasks. Since Res2Net [29] can construct hierarchical residual-like connections within one single residual 
+block that has stronger multi-scale representation ability, we applied the pre-trained Res2Net as the backbone of 
+CaraNet.  
+2.2 Partial decoder  
+Existing state-of-the-art segmentation networks rely on aggregating multi-level features from the encoder (e.g., U-
+Net aggregates all level features extracted from an encoder). Compared to the high-level features, however, low-level 
+features contribute less to performance but have higher computational cost because of their larger spatial resolution [30]. 
+Thus, we applied the parallel partial decoder [26] as shown in Figure 2 to aggregate high-level features. We feed the original 
+image which size is ℎ × 𝑤 × 𝑐 (ℎ, 𝑤, and 𝑐 represent the height, width, and channel) into Res2Net, and we can get five 
+different level features {𝑓𝑖, 𝑖 = 1, … , 5}  with resolution {
+ℎ
+2𝑖−1 ,
+𝑤
+2𝑖−1} . We aggregated the high-level features {𝑓3, 𝑓4, 𝑓5} 
+from Res2Net by using the partial decoder with a parallel connection. Then, we can get a global map 𝑆𝑔 = 𝑃𝐷(𝑓3, 𝑓4, 𝑓5). 
+ 
+Figure 2. Overview of partial decoder with parallel connection. 
+2.3 Channel-wise feature pyramid module 
+The Feature Pyramid (FP) has been widely used in deep learning models for computer vision tasks due to its ability 
+to represent multi-scale features. For example, PSPNet [31] builds a pyramid pooling module with different sizes’ pooling 
+layers to extract multi-scale features, and the Feature Pyramid Network (FPN) [32] takes different strides with convolution 
+kernels to obtain a FP. Although those FP-based methods perform well in the computer vision area, they cannot avoid using 
+large numbers of parameters, which consume a large amount of computation resources. In addition, their receptive fields 
+are usually small and do not perform well in datasets with sharply varying object sizes [33]. Alternatively, our previous 
+works [33, 34] proposed a lightweight Channel-wise Feature Pyramid (CFP) module and successfully applied it to both 
+nature and medical image segmentation. The architecture of this CFP module is shown in Figure 3.  
+Figure 3(a) shows the architecture of the CFP module; it contains total 𝐾 channels and each channel has its own 
+dilation rate 𝑟𝐾 . Typically, we choose the 𝐾 = 4  for CaraNet and the dilation rates for each channel {𝑟1, 𝑟2, 𝑟3, 𝑟4} =
+{1,2,4,8}  which has been verified as the best dilation rates combination for CaraNet in Table 4; thus, each channel’s 
+dimension is 𝑀/4. Simple feature fusion method sometimes introduces some unwanted checkboard or gridding artifacts 
+that greatly influence the accuracy and quality of segmentation masks [33]. Thus, we applied hierarchical feature fusion 
+[35] (HFF) to sum the outputs of all channels step by step. For the FP channel, we provide two versions with regular 
+convolution and asymmetric convolution as shown in Figure 3(b) and (c). We connected the outputs of each convolutional 
+module by using skip connection, and thus each channel can be considered as a sub-pyramid. We selected the regular 
+convolution as FP channel for CaraNet. The overall FP is obtained from concatenating those sub-pyramids from the 
+hierarchical feature fusion operation. The final FP contains four levels of feature stacks as shown in Figure 4. These four 
+levels of feature stacks {𝑙𝑒𝑣𝑒𝑙𝑖, 𝑖 = 1, … ,4} are computed by: 
+
+Aggregation
+Aggregation
+13
+Node l-1
+Node3-2
+Aggregation
+Aggregation
+Node 1-2
+Node2-2
+Aggregation
+f5
+Node1-3
+Upsampling 
+4 
+ 
+{
+𝑙𝑒𝑣𝑒𝑙1 =  𝑜𝑢𝑡𝐹𝑃1  
+𝑙𝑒𝑣𝑒𝑙2 = 𝑙𝑒𝑣𝑒𝑙1 + 𝑜𝑢𝑡𝐹𝑃2
+𝑙𝑒𝑣𝑒𝑙3 = 𝑙𝑒𝑣𝑒𝑙2 + 𝑜𝑢𝑡𝐹𝑃3
+𝑙𝑒𝑣𝑒𝑙4 = 𝑙𝑒𝑣𝑒𝑙3 + 𝑜𝑢𝑡𝐹𝑃4
+(1) 
+And the final FP is computed by ∑ 𝑙𝑒𝑣𝑒𝑙𝑖
+𝑖
+. Based on our split-merge feature pyramid strategy, the receptive fields of a 
+single CFP module with dilation rates {𝑟1, 𝑟2, 𝑟3, 𝑟4} = {1,2,4,8}  varies from 3 × 3  to 55 × 55 , which successfully 
+overcomes the challenge from sharply varying object sizes. 
+    
+    
+ 
+(a)                                   (b)                 (c) 
+Figure 3. (a) CFP module, (b) FP channel with regular convolution, (c) FP channel with asymmetric convolution 
+ 
+Figure 4. Final feature pyramid obtained from CFP module. 
+2.4 Axial reverse attention module 
+The previous partial decoder that generates the global map 𝑆𝑔 (Sec. 2.2) could roughly locate the position of medical 
+objects, and the CFP module extracted only multi-scale features from the pre-trained model. To obtain more accurate 
+feature information, we designed the Axial Reverse Attention (A-RA) module to refine localization information and multi-
+scale features efficiently. The overview and detail of the A-RA module can be seen in Figure 1 and Figure 5, respectively. 
+The input of the top line is the multi-scale feature maps 𝑓𝑖
+′ from the CFP module and we used axial attention to analyze 
+the salience information. The axial attention is based on self-attention, which maps a query and a set of key-value pairs to 
+an output and the operation: 
+
+Input
+M.1×1.M/K
+FP channel, r
+FP channel, r2
+FP channel, rk.
+FP channel, rk
+Add
+PpV
+HFF
+Add
+Concatenation
+M. 1x1.M
+Add3x3 convolution
+3x3convolution
+3x3convolution
+concatenation3x1convolution
+1x3convolution
+3x1 convolution
+1x3convolution
+3x1 convolution
+1x3convolution
+concatenationLevel l:
+Level 3:
+Level2:
+Level4:
+Output: 
+5 
+ 
+ 
+𝐴𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(𝑄, 𝐾, 𝑉) = 𝑆𝑜𝑓𝑡𝑚𝑎𝑥 (
+𝑄𝐾𝑇
+√𝑑𝐾)
+(2) 
+where 𝑄, 𝐾, 𝑉, and 𝑑𝐾 represent query, key, value, and dimension of key, respectively. However, self-attention consumes 
+great computational resources, especially when the spatial dimension of the input is large [36]. Therefore, we applied axial 
+attention, which factorizes 2D attention into two 1D attention along height and width axes. Here we replace the softmax 
+activation function with a sigmoid, based on the experiments. For the second line, we applied the reverse operation [37] to 
+detect the salience features from the side-output 𝑆𝑖, which is obtained from the output of the previous A-RA module. The 
+reverse operation is:  
+𝑅𝑖 =  1 − 𝑆𝑖𝑔𝑚𝑜𝑖𝑑(𝑆𝑖).
+(3) 
+The total axial reverse attention operation is: 
+𝐴𝑅𝐴𝑖 = 𝐴𝐴𝑖⨀𝑅𝑖
+(4) 
+where ⊙ is element-wise multiplication, and the 𝐴𝐴𝑖 is feature from the axial attention route. 
+ 
+Figure 5. Structure of Axial Reverse Attention module. 
+2.5 Deep supervision 
+We apply weighted intersection over union (IoU) and weighted binary cross-entropy (BCE) in our loss function: ℒ =
+ℒ𝐼𝑜𝑈
+𝑤
++ ℒ𝐵𝐶𝐸
+𝑤
+  to calculate the global loss and local (pixel-level) loss, respectively. To train CaraNet, we apply deep 
+supervision for the three side-outputs (𝑆1, 𝑆2, 𝑆3) and the global map 𝑆𝑔. Before calculating the loss, we up-sampled them 
+to the same size as ground truth 𝐺. Thus, the total loss: 
+ℒ𝑡𝑜𝑡𝑎𝑙 = ℒ(𝐺, 𝑆𝑔
+𝑢𝑝) + ∑ ℒ(𝐺, 𝑆𝑖
+𝑢𝑝)
+5
+𝑖=3
+(5) 
+2.6 Small object segmentation analysis 
+Since the size of all images input to the segmentation models must be fixed, the size of an object is determined by the 
+number of pixels in the object m and the number of total pixels in the image N. Thus, we consider the object’s size using 
+the size ratio (proportion) = m/N. Then, we evaluate the performance of segmentation models according to the sizes of 
+objects. Especially, we mainly focus on the small areas whose size ratios are smaller than 5%. To define the watershed of 
+small size is a question; it depends on data and model performance. We further discuss this question in Discussion section. 
+To evaluate the performance of segmentation models according to the sizes of objects, we first obtain the mean-Dice 
+coefficients and size ratios of segmentations from the test dataset. Similar to computing the histogram, we plot the results 
+in a curve whose y-axis is mean-Dice coefficients and x-axis is increasingly sorted size ratios. To smooth the curve, we 
+
+Axialattcntion
+Height
+Wcight
+Convolutionalfeature
+Weighted convolutionalfeature
+S;
+Sigmoid
+Reverse weight
+W=1-Sigmoid(Si)
+Up-sampledfeature 
+6 
+ 
+take interval-averaged mean-Dice coefficients by sorted size ratios: we divide the entire range of size ratios into a 
+consecutive, non-overlapping, and of equal length series of intervals, and then calculate the average mean-Dice coefficients 
+of size ratios in each interval. The interval-averaged coefficients have a smooth curve and are more stable in the presence 
+of noise. 
+3. EXPERIMENT 
+3.1 Implementation details 
+We implemented our model in PyTorch accelerated by the NVIDIA RTX 2070Ti GPU. We resized input images to 
+352 × 352 for polyp segmentation and 256 × 256 for brain tumor segmentation and employed a multi-scale training 
+strategy {0.75, 1.0, 1.25} instead of data augmentation. We used Adam optimizer with the initial learning rate 1𝑒−4. 
+3.2 Dataset 
+We test our CaraNet on five polyp segmentation datasets: ETIS [38], CVC-ClinicDB [39], CVC-ColonDB [40], 
+EndoScene [41], and Kvasir [42]. The first four are standard benchmarks, and the last one is the largest dataset, which was 
+released recently. We also test our model on the Brain Tumor Segmentation 2018 (BraTS 2018) dataset [48, 49], which 
+contains more extremely small medical objects. Table 1 shows the details of these datasets: image size, scale of testing set, 
+and size ratios of medical objects. 
+The data of brain tumor segmentation is from the multimodal brain tumor segmentation challenge 2018 (BraTS 2018) 
+built by the Section for Biomedical Image Analysis (SBIA) at the University of Pennsylvania. It contains the multimodal 
+brain MRI scans and manual ground truth labels of glioblastoma from 285 cases (patients). Each case includes four scan-
+modals: 1) native (T1), 2) T1 contrast enhanced (T1ce), 3) T2-weighted (T2), and 4) T2 Fluid Attenuated Inversion 
+Recovery (FLAIR). And each case includes three types of ground truth labels: necrotic and non-enhancing tumor core 
+(NET), gadolinium-enhancing tumor (ET), and peritumoral edema (Ed). In this study, T1ce is selected as our input images 
+and ground truth labels use NET type because it delineates the minimum areas for small object segmentation. The MRI 
+scans for each case are sliced to 2-D images. As shown in Table 1, the test samples are chosen by the sizes of objects in 
+images (by examining the areas of truth labels) ranging in 0.01% - 4.91%. 
+Table 1. Details of datasets 
+ 
+Image size 
+Number of test samples 
+Object size ratio 
+ETIS 
+966 × 1225 
+196 
+0.11% - 29.05% 
+CVC-ClinicDB 
+288 × 384 
+62 
+0.34 % - 45.88% 
+CVC-ColonDB 
+500 × 574 
+380 
+0.30% - 63.15% 
+CVC-300 
+500 × 574 
+60 
+0.55% - 18.42% 
+Kvasir 
+1070 × 1348 
+100 
+0.79% - 62.13% 
+BraTS 2018 
+256 × 256 
+3231 
+0.01% - 4.91% 
+3.3 Baseline 
+We compared CaraNet with six medical image segmentation models, including state-of-the-art models: U-Net [1], U-
+Net++ [2], ResUNet-mod [43], ResUNet++ [3], SFA [44], PraNet [27], CCBANet [50] and DS-TransUNet [51].  
+3.4 Training and measurement metrics 
+We randomly split 80% of images from Kvasir and CVC-ClinicDB as training set and the remainder as a testing 
+dataset. In addition to mean Dice and mean IoU, we also apply four other measurement metrics: weighted dice metric 𝐹𝛽
+𝑤, 
+MAE, enhanced alignment metric Ε𝜙
+𝑚𝑎𝑥  [45], and structural measurement 𝑆𝛼  [46]. Table 2 shows the polyp 
+segmentation on the five datasets. The weighted dice metric 𝐹𝛽
+𝑤 is used to amend the “equal importance flaw” in dice. 
+
+ 
+7 
+ 
+The MAE is used to measure the pixel-to-pixel accuracy. The recently released enhanced alignment metric Ε𝜙
+𝑚𝑎𝑥  is 
+utilized to evaluate the pixel-level and global-level similarity. And 𝑆𝛼 is used to measure the structure similarity between 
+predictions and ground truth. 
+3.5 Results 
+We first report the polyp segmentation results and compare with state-of-the-arts such as DS-TransUNet, CCBANet 
+and PraNet in Table 2. Among all five public endoscopy segmentation datasets, our proposed CaraNet achieves best 
+performance, especially in the ETIS dataset which contains more small polyps. We also show some polyp segmentation 
+results in Figure 6. For the five polyp datasets, CaraNet not only outperforms the compared models in overall performance, 
+but also on samples with small polyps. Figure 7 shows the segmentation performance of CaraNet and PraNet for small 
+objects (proportions ≤ 5%). For the extremely small object segmentation analysis on the BraTS 2018 dataset, we compare 
+only CaraNet with PraNet because PraNet has the closest performance to ours, and the overall accuracies of the other 
+segmentation models are clearly lower than those of CaraNet and PraNet. (Note: the fluctuations with size in colonoscopy 
+datasets are caused by types and boundary of polyps, and quality of imaging) 
+Table 2. Quantitative results on Kvasir, CVC-ClinicDB, CVC-ColonDB, ETIS, and CVC-T (test dataset of EndoScene). Note: mDice: 
+mean Dice, mIoU: mean IoU, 𝐹𝛽
+𝑤: weighted dice, 𝑆𝛼: structural measurement [46], Ε𝜙
+𝑚𝑎𝑥: enhanced alignment [45] and MAE: mean 
+absolute error. ↑ denotes higher the better and ↓ denotes lower the better.  
+     Methods 
+mDice ↑ 
+mIoU ↑ 
+𝑭𝜷
+𝒘 ↑ 
+𝑺𝜶 ↑ 
+𝚬𝝓
+𝒎𝒂𝒙 ↑ 
+MAE ↓ 
+Kvasir 
+UNet 
+0.818 
+0.746 
+0.794 
+0.858 
+0.893 
+0.055 
+UNet++ 
+0.821 
+0.743 
+0.808 
+0.862 
+0.910 
+0.048 
+ResUNet-mod 
+0.791 
+n/a 
+n/a 
+n/a 
+n/a 
+n/a 
+ResUNet++ 
+0.813 
+0.793 
+n/a 
+n/a 
+n/a 
+n/a 
+SFA 
+0.723 
+0.611 
+0.670 
+0.782 
+0.849 
+0.075 
+PraNet 
+0.898 
+0.840 
+0.885 
+0.915 
+0.948 
+0.030 
+CCBANet 
+0.902 
+0.845 
+0.887 
+0.916 
+0.952 
+0.032 
+DS-TransUNet 
+0.913 
+0.857 
+0.902 
+0.923 
+0.963 
+0.023 
+CaraNet 
+0.918 
+0.865 
+0.909 
+0.929 
+0.968 
+0.023 
+CVC-ClinicDB 
+UNet 
+0.823 
+0.755 
+0.811 
+0.889 
+0.954 
+0.019 
+UNet++ 
+0.794 
+0.729 
+0.785 
+0.873 
+0.931 
+0.022 
+ResUNet-mod 
+0.779 
+n/a 
+n/a 
+n/a 
+n/a 
+n/a 
+ResUNet++ 
+0.796 
+0.796 
+n/a 
+n/a 
+n/a 
+n/a 
+SFA 
+0.700 
+0.607 
+0.647 
+0.793 
+0.885 
+0.042 
+PraNet 
+0.899 
+0.849 
+0.896 
+0.936 
+0.979 
+0.009 
+CCBANet 
+0.909 
+0.856 
+0.903 
+0.939 
+0.971 
+0.010 
+DS-TransUNet 
+0.912 
+0.859 
+0.908 
+0.936 
+0.976 
+0.007 
+CaraNet 
+0.936 
+0.887 
+0.931 
+0.954 
+0.991 
+0.007 
+ColonDB 
+UNet 
+0.512 
+0.444 
+0.498 
+0.712 
+0.776 
+0.061 
+UNet++ 
+0.483 
+0.410 
+0.467 
+0.691 
+0.760 
+0.064 
+SFA 
+0.469 
+0.347 
+0.379 
+0.634 
+0.765 
+0.094 
+PraNet 
+0.709 
+0.640 
+0.696 
+0.819 
+0.869 
+0.045 
+CCBANet 
+0.758 
+0.675 
+0.736 
+0.842 
+0.880 
+0.042 
+DS-TransUNet 
+0.762 
+0.682 
+0.738 
+0.829 
+0.872 
+0.053 
+CaraNet 
+0.773 
+0.689 
+0.729 
+0.853 
+0.902 
+0.042 
+ETIS 
+UNet 
+0.398 
+0.335 
+0.366 
+0.684 
+0.740 
+0.036 
+UNet++ 
+0.401 
+0.344 
+0.390 
+0.683 
+0.776 
+0.035 
+SFA 
+0.297 
+0.217 
+0.231 
+0.557 
+0.633 
+0.109 
+PraNet 
+0.628 
+0.567 
+0.600 
+0.794 
+0.841 
+0.031 
+CCBANet 
+0.677 
+0.610 
+0.640 
+0.800 
+0.838 
+0.028 
+DS-TransUNet 
+0.675 
+0.592 
+0.625 
+0.802 
+0.859 
+0.023 
+CaraNet 
+0.747 
+0.672 
+0.709 
+0.868 
+0.894 
+0.017 
+CVC-T 
+UNet 
+0.710 
+0.627 
+0.684 
+0.843 
+0.876 
+0.022 
+UNet++ 
+0.707 
+0.624 
+0.687 
+0.839 
+0.898 
+0.018 
+SFA 
+0.467 
+0.329 
+0.341 
+0.640 
+0.817 
+0.065 
+PraNet 
+0.871 
+0.797 
+0.843 
+0.925 
+0.972 
+0.010 
+CCBANet 
+0.903 
+0.833 
+0.881 
+0.933 
+0.986 
+0.007 
+DS-TransUNet 
+0.880 
+0.798 
+0.854 
+0.920 
+0.978 
+0.007 
+CaraNet 
+0.903 
+0.838 
+0.887 
+0.940 
+0.989 
+0.007 
+
+ 
+8 
+ 
+ 
+ 
+Figure 6. Polyp segmentation results 
+ 
+ 
+
+CVC-300
+CVC-ClinicDB
+CVC-ColonDB
+ETIS
+Kvasir
+Images
+Ground
+truth
+UNet
+UNet++
+SFA
+PraNet
+CCBANet
+DSTransUNet
+CaraNet 
+9 
+ 
+ 
+      
+ 
+   
+ 
+ 
+Figure 7. The performance of CaraNet and PraNet for small object segmentation. More discussion about small object segmentation 
+analysis is in Sec. 2.6. For each subfigure, the x-axis is the proportion size (%) of polyp and the y-axis is the averaged mean Dice 
+coefficient. Subfigures show performance vs. size on the five polyp datasets, which are Kvasir, CVC-ClinicDB, CVC-ColonDB, ETIS, 
+and CVC-300. Blue line is for our CaraNet and orange is for the PraNet, showing only the results of small polyp sizes (<6%). We can 
+find that our CaraNet overperforms the PraNet for most cases of small size polyps from the five datasets. 
+Table 3. Quantitative results on brain tumor (BraTS 2018) dataset. 
+Methods 
+mean Dice 
+mean IoU 
+𝑭𝜷
+𝒘 
+𝑺𝜶 
+𝚬𝝓
+𝒎𝒂𝒙 
+MAE 
+CaraNet (Ours) 
+0.631 
+0.619 
+0.507 
+0.629 
+0.786 
+0.927 
+0.003 
+PraNet (MICCAI’20) 
+0.494 
+0.606 
+0.776 
+0.920 
+0.003 
+
+Kvasir
+1.0
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+0
+1
+2
+3
+4
+5
+6CVC-ClinicDB
+1.0
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+0
+1
+2
+3
+4
+5
+6CVC-ColonDB
+1.0
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+0
+2
+3
+4
+5ETIS
+1.0
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+：
+2
+3
+4
+6CVC-300
+1.0
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+0
+1
+2
+3
+4
+5
+6
+-CaraNet (Ours)
+-PraNet 
+10 
+ 
+ 
+ 
+ 
+Figure 8. Performance vs. Size on brain tumor datasets. The x-axis is the proportion size (%) of tumor. Upper figure: y-axis is the 
+mean Dice coefficient results of our CaraNet and PraNet. For the very small tumor sizes (≤1%), almost all results of CaraNet are 
+better than PraNet. Lower figure: the y-axis is the difference between the averaged mean Dice coefficient results of CaraNet and 
+PraNet. Red indicates the Dice value of CaraNet is greater than that of PraNet; blue shows the opposite. 
+
+BrainTumor
+0.9
+0.8
+0.7
+0.6
+Dice
+0.5
+0.4
+0.3
+0.2
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Tumor Sizes (%)
+-CaraNet (Ours)
+--PraNet0.03
+0.02
+ice
++0.7630
+0.01
+0
+Car
+-0.2362
+-0.02
+-0.03
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+TumorSizes(%) 
+11 
+ 
+ 
+ 
+Figure 9. Brain tumor segmentation results 
+To further evaluate the effectiveness of CaraNet for small-object segmentation, we conducted another experiment 
+using the brain tumor dataset (BraTS 2018). The polyp datasets lack extremely small objects (the minimum is about 0.11%) 
+and do not have enough small samples (like Kvasir and CVC-ClinicDB, in Figure 7, there are fewer samples in the range 
+of small sizes). The brain tumor dataset was created from the BraTS 2018 database by slicing 2D images from the “T1ce” 
+source with “NET” type labels. We randomly select 60% of the images as the training set and the remainder as the testing 
+dataset. Altogether, 3231 images with proportions of tumor sizes ranging from 0.01% – 4.91% were in the testing dataset. 
+Table 3, Figure 8, and Figure 9 show the comparison result. We compared CaraNet with only PraNet for the same reason 
+stated above. Clearly, our CaraNet performed better, especially for the extremely small cases (range 0.01% – 0.1% in 
+Figure 8 and the red area indicates that the Dice value of CaraNet is greater than that of PraNet; blue area shows the 
+opposite. Values on the right show the summations of all red and blue differential values). 
+3.5 Ablation study 
+Table 4. Quantitative results on Kvasir for different dilation rate 
+Dilation Rate 
+Mean Dice 
+0 
+0.909 
+4 
+0.908 
+8 
+0.918 
+16 
+0.913 
+32 
+0.907 
+To search the best dilation rate for our CFP module, we set up experiments and choose different dilation rates from 0 
+to 32. The performance of CaraNet with different dilation rate are testing on Kvasir testing set as shown in Table 4. When 
+we choose small dilation rates like 0 and 4, the dilations rates for each channel are {𝑟1, 𝑟2, 𝑟3, 𝑟4} = {0,0,0,0}  and 
+{𝑟1, 𝑟2, 𝑟3, 𝑟4} = {0,1,2,4}. The CFP module focuses on local information but ignore the global one, thus the accuracy is 
+about 1% lower than the best one. When large dilation rates are chosen like 16 and 32, the dilation rates for each channel 
+are {𝑟1, 𝑟2, 𝑟3, 𝑟4} = {0,4,8,16} and {𝑟1, 𝑟2, 𝑟3, 𝑟4} = {0,8,16,32}. There only one channel focus on local features which is 
+unreasonable for small medical object detection. When we choose a fairish dilation rate like 8 which {𝑟1, 𝑟2, 𝑟3, 𝑟4} =
+{0,2,4,8}, the weights of local and global information can be balanced and then achieves best performance. 
+We further conduct ablation studies to demonstrate the effectiveness of our proposed CFP module and Axial Reverse 
+Attention (A-RA) module five public endoscopy segmentation datasets, and we choose same six measurement metrics as 
+in Section 3.4 for evaluation. 
+
+Images
+Ground truth
+PraNet
+CaraNet 
+12 
+ 
+We first conduct an experiment to evaluate CaraNet without CFP module. As shown in Table 5, the performance 
+without CFP module drops sharply on five public datasets. In particular, the mDice achieves about 8% reduction on ETIS 
+dataset, which strongly verifies that CFP module can effectively detect local-to-global feature due to its various sizes 
+receptive field. Then, we evaluate the CaraNet without both CFP module and A-RA module. The mDice continued to 
+decrease about 2%-3%, indicating that our A-RA module enables our CaraNet to accurately distinguish true polyp tissues.  
+Table 5. Ablation study for CaraNet on Kvasir, CVC-ClinicDB, CVC-ColonDB, ETIS, and CVC-T (test dataset of EndoScene). Note: 
+mDice: mean Dice, mIoU: mean IoU, 𝐹𝛽
+𝑤: weighted dice, 𝑆𝛼: structural measurement [46], Ε𝜙
+𝑚𝑎𝑥: enhanced alignment [45] and 
+MAE: mean absolute error. ↑ denotes higher the better and ↓ denotes lower the better.  
+Dataset 
+CFP 
+A-RA 
+mDice ↑ 
+mIoU ↑ 
+𝑭𝜷
+𝒘 ↑ 
+𝑺𝜶 ↑ 
+𝚬𝝓
+𝒎𝒂𝒙 ↑ 
+MAE ↓ 
+Kvasir 
+ 
+ 
+0.870 
+0.798 
+0.836 
+0.899 
+0.938 
+0.043 
+ 
+✓ 
+0.888 
+0.830 
+0.878 
+0.911 
+0.940 
+0.032 
+✓ 
+✓ 
+0.918 
+0.865 
+0.909 
+0.929 
+0.968 
+0.023 
+CVC-ClinicDB 
+ 
+ 
+0.862 
+0.789 
+0.841 
+0.908 
+0.949 
+0.021 
+ 
+✓ 
+0.887 
+0.830 
+0.874 
+0.928 
+0.966 
+0.012 
+✓ 
+✓ 
+0.936 
+0.887 
+0.931 
+0.954 
+0.991 
+0.007 
+ColonDB 
+ 
+ 
+0.681 
+0.586 
+0.595 
+0.797 
+0.865 
+0.063 
+ 
+✓ 
+0.707 
+0.636 
+0.689 
+0.815 
+0.867 
+0.048 
+✓ 
+✓ 
+0.773 
+0.689 
+0.729 
+0.853 
+0.902 
+0.042 
+ETIS 
+ 
+ 
+0.646 
+0.570 
+0.587 
+0.790 
+0.833 
+0.055 
+ 
+✓ 
+0.662 
+0.580 
+0.604 
+0.805 
+0.869 
+0.032 
+✓ 
+✓ 
+0.747 
+0.672 
+0.709 
+0.868 
+0.894 
+0.017 
+CVC-T 
+ 
+ 
+0.839 
+0.765 
+0.802 
+0.909 
+0.943 
+0.022 
+ 
+✓ 
+0.870 
+0.803 
+0.839 
+0.917 
+0.965 
+0.009 
+✓ 
+✓ 
+0.903 
+0.838 
+0.887 
+0.940 
+0.989 
+0.007 
+ 
+4. DISCUSSION 
+We propose a novel deep-learning based segmentation model – CaraNet, by combining the Axial Reverse Attention 
+and Channel-wise Feature Pyramid (CFP) modules. This new method can help improve the performance of the 
+segmentation of small medical objects. Through the experiments, we show that CaraNet outperforms the most famous 
+models by a large margin overall for six measurement metrics. As shown by the polyp segmentation results, CaraNet not 
+only produces high quality segmentation on samples of large polyps, but also performs well for small and multi small-
+object segmentation. Figure 9 shows some results of extremely small tumor segmentation from the BraTS 2018 dataset. 
+The advantage of the CaraNet in segmenting small single- and multi-objects is evident. In addition, compared with the 
+recent state-of-the-art network, PraNet, CaraNet provides a more precise prediction for the most-challenging cases.  
+We also introduce the process to evaluate segmentation models according to the size of objects. We consider the 
+object’s size using the size ratio including the sizes of objects and the whole image. In this study, we assume the size ratios 
+of “small objects” are less than 5%. However, the definition of “small objects” is not specified. Since few studies have 
+fully considered the sizes of objects and the small-object problems in medical imaging, we could further study this question 
+in future work. Preliminarily, the small size (watershed) could be defined by the Performance vs. Size curves. Figure 10 
+shows an example on ETIS datasets. The watershed of small size could be defined at 9.8%. After that point, the performance 
+generally increase/change slower and is more stable. The definition of small area may depend on datasets, segmentation 
+models, and object shapes; but if these conditions are fixed, it is feasible to make fair comparisons. The definition of small 
+area discussed here may not be perfect, but it is worth paying attention to the model’s performance on small size cases 
+besides the whole dataset. Figure 10 also indicates that the overall performance of segmentation depends on size ranges. 
+If we remove the small size cases, the overall performance will be greatly improved. 
+
+ 
+13 
+ 
+ 
+Figure 10. Performance vs. Size on ETIS datasets. The x-axis is the proportion size (%) of polyp; y-axis is the mean Dice coefficient. 
+Blue is for our CaraNet and orange is for the SFA model. Unlike Figure 7, it shows the results of all sizes. 
+Although CaraNet achieves good improvement on medical image segmentation tasks, there are still some limitations 
+and potentials to optimize the model. For example, using bilinear interpolation to up-sample feature maps cannot avoid the 
+loss of some useful information and lead to a coarse boundary. It can be improved by applying a deconvolutional layer. In 
+addition, the backbone of CaraNet is pre-trained on ImageNet, which contains natural images that are very different from 
+medical images. Moreover, the sliced brain MRI data also cause loss of spatial information between the voxels; that may 
+influence the accuracy of small tumor detection. In our future work, we will use the Model Genesis [47] as a 3-D backbone 
+to replace the Res2Net (2-D backbone) and adjust the CaraNet to employ it on 3-D medical imaging segmentation to build 
+a 3-D version CaraNet for more accurate CT or MRI image segmentation. Segmentation of 3-D medical images is of 
+growing interest; we believe CaraNet will address that problem successfully. 
+5. CONCLUSION 
+We have proposed a novel neural network, CaraNet, for small medical object segmentation. We use a partial decoder 
+to roughly localize polyp position, improve the adaptability of the CaraNet to detect different sizes objects by using a CFP 
+module and finally refine the polyp segmentation by A-RA module. From the overall segmentation accuracy, we can find 
+that CaraNet outperforms all state-of-the-art approaches by at least 2% (mean dice accuracy). For the early diagnosis 
+dataset (ETIS) which contains many small polyps, however, CaraNet can reach 74.7% mean dice accuracy, which is about 
+12% higher than PraNet. For the extremely small object segmentation dataset (BraTS 2018), CaraNet can achieve 3% 
+higher than PraNet. When evaluating segmentation models according to the size of objects, CaraNet outperforms PraNet 
+for small objects in all six datasets we used. In addition to the average accuracy through the whole dataset, to 
+evaluate/compare segmentation models according to sizes of objects could show more characteristics of models; especially, 
+their performance on small areas. And this method can find or verify models that are good for small objects segmentation. 
+DISCLOSURES 
+No conflicts of interest. 
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+
diff --git a/kdFQT4oBgHgl3EQfmjaf/content/tmp_files/load_file.txt b/kdFQT4oBgHgl3EQfmjaf/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf,len=1252
+page_content='1  CaraNet: Context Axial Reverse Attention Network for Segmentation  of Small Medical Objects  Ange Lou1, Shuyue Guan2, Murray Loew2  Vanderbilt University1, Nashville TN, USA  ange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='lou@vanderbilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='edu  The George Washington University2, Washington DC, USA  {frankshuyueguan, loew}@gwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='edu  Abstract  Purpose: Segmenting medical images accurately and reliably is important for disease diagnosis and treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' It is a  challenging task because of the wide variety of objects’ sizes, shapes, and scanning modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Recently, many  convolutional neural networks (CNN) have been designed for segmentation tasks and achieved great success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Few studies,  however, have fully considered the sizes of objects, and thus most demonstrate poor performance for small objects  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' This can have a significant impact on the early detection of diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Approach: This paper proposes a Context Axial Reverse Attention Network (CaraNet) to improve the segmentation  performance on small objects compared with several recent state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' CaraNet applies axial reserve attention  (ARA) and channel-wise feature pyramid (CFP) module to dig feature information of small medical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' And we  evaluate our model by six different measurement metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Results: We test our CaraNet on brain tumor (BraTS 2018) and polyp (Kvasir-SEG, CVC-ColonDB, CVC-ClinicDB,  CVC-300, and ETIS-LaribPolypDB) segmentation datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Our CaraNet achieves the top-rank mean Dice segmentation  accuracy, and results show a distinct advantage of CaraNet in the segmentation of small medical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Conclusion: We proposed CaraNet to segment small medical objects and outperform other state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Codes available: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='com/AngeLouCN/CaraNet  Keywords: Small object segmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Brain tumor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Colonoscopy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Attention;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Context axial reverse  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' INTRODUCTION  Deep learning has had a tremendous impact on various fields in science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Our focus of the current study in deep learning  is on one of the most critical areas of computer vision: medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Recently, various convolutional neural  networks (CNNs) have shown great performance on medical image segmentation [1,2,3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Those CNNs have been  introduced for various medical imaging modalities, including X-ray, visible-light imaging, magnetic resonance imaging  (MRI), positron emission tomography (PET), and computerized tomography (CT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' They all achieved excellent  performance on medical image segmentation challenges from different modalities, like BraTS [5,6,7], KiTS19 [8], and  COVID19-20 [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' To obtain more accurate segmentation results, many works introduced improvements of network  architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Those improvements are mostly attributed to exploring new neural architectures by designing networks with  varying depths (ResNet [11]), widths (ResNeXt [12]), connectivity (DenseNet [13] and GoogLeNet [14]), or new types of  components (pyramid scene [15] and atrous convolution [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Although those new architectures improve the overall  segmentation results, they are less sensitive to detecting small medical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' And it is very common in medical image  segmentation that the anatomy of interest occupies only a very small portion of the image [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Most extracted features  belong to the background, while these small lesion areas are important for early detection and diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' For example, the  survival rate decreases with the growing size of a brain tumor [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Thus, it has clinical significance to build an  effective network to detect tiny medical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  The attention mechanism plays a dominant role in neural network research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' It can effectively use information  transferred from several subsequent feature maps to detect the salience features [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Many attention methods such  as self-attention and multi-head attention have been verified to have high performance in applications of natural  language processing [20] and computer vision [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Those attention methods also have been successfully used for  medical image segmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' for example, the Medical Transformer [22] (MedT) used a gated axial self-attention  layer to build a Local-Global (LoGo) network for ultrasound and microscopy image segmentation, TransUNet [23]  2  stacked self-attention as a transformer in the encoder for CT image segmentation, and CoTr [24] bridged two CNN  encoder and decoder by the transformer encoder for multi-organ segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' All those attention-based  segmentations achieve significant improvement compared with purely convolutional neural networks like U-Net [1]  and FCN [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Although those new types of neural networks show good performance on many medical segmentation tasks,  they seldom consider the small object segmentation, especially in the medical image area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We propose here a novel  attention-based deep neural network, called Context Axial Reverse Attention Network (CaraNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The contribution  of the paper can be summarized as follows:  1)  We propose a novel neural network – CaraNet -- to solve the problem of segmentation of small medical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  2)  We introduce a method to evaluate the network’s performance on small medical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  3)  Our experiments show that CaraNet outperforms most current models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=', DS-TransUNet from TMI ’22,  CCBANet from MICCAI ‘21and PraNet from MICCAI ’20) and advances the state-of-the-art by a large margin,  both overall and on small objects, in segmentation performance on polyps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' METHOD  Figure 1 shows the architecture of our CaraNet, which uses a parallel partial decoder [26] to generate the high-  level semantic global map and a set of context and axial reverse attention operations to detect global and local feature  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We introduce main components of the CaraNet architecture in the following subsections with  explanation of the motivation, purpose, or effectiveness to add these components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Overview of CaraNet, which contains pretrained backbone, partial decoder, channel-wise feature pyramid (CFP) module and  axial reverse attention (A-RA) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='1 Backbone  Transfer learning provides a feasible method for alleviating the challenge of data-hunger, and it has been widely  applied to the field of computer vision [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Benefiting from the strong visual knowledge distributed in ImageNet  [28], the pre-trained CNNs can be fine-tuned with a small amount of task-specific data and can perform well on  f1  f2  PD  Sg  f3  f4  fs  CFP  CFP  CFP  Down-sampling  Axial-attention  fi  8=P  8=p  8=P  Height  Width  f1  f2  f3  axis  axis  Si  A-RA  A-RA  A-RA  S  ARA3  ARA4  ARA5  +  +  +  Up-sampling  Deep supervision  Partial decoder  Featureflow  Mapflow  GT  Prediction  S3  S4  I Ss  S  Sigmoid  3  downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Since Res2Net [29] can construct hierarchical residual-like connections within one single residual  block that has stronger multi-scale representation ability, we applied the pre-trained Res2Net as the backbone of  CaraNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='2 Partial decoder  Existing state-of-the-art segmentation networks rely on aggregating multi-level features from the encoder (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=', U-  Net aggregates all level features extracted from an encoder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Compared to the high-level features, however, low-level  features contribute less to performance but have higher computational cost because of their larger spatial resolution [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Thus, we applied the parallel partial decoder [26] as shown in Figure 2 to aggregate high-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We feed the original  image which size is ℎ × 𝑤 × 𝑐 (ℎ, 𝑤, and 𝑐 represent the height, width, and channel) into Res2Net, and we can get five  different level features {𝑓𝑖, 𝑖 = 1, … , 5}  with resolution {  ℎ  2𝑖−1 ,  𝑤  2𝑖−1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We aggregated the high-level features {𝑓3, 𝑓4, 𝑓5}  from Res2Net by using the partial decoder with a parallel connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Then, we can get a global map 𝑆𝑔 = 𝑃𝐷(𝑓3, 𝑓4, 𝑓5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Overview of partial decoder with parallel connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='3 Channel-wise feature pyramid module  The Feature Pyramid (FP) has been widely used in deep learning models for computer vision tasks due to its ability  to represent multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' For example, PSPNet [31] builds a pyramid pooling module with different sizes’ pooling  layers to extract multi-scale features, and the Feature Pyramid Network (FPN) [32] takes different strides with convolution  kernels to obtain a FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Although those FP-based methods perform well in the computer vision area, they cannot avoid using  large numbers of parameters, which consume a large amount of computation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' In addition, their receptive fields  are usually small and do not perform well in datasets with sharply varying object sizes [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Alternatively, our previous  works [33, 34] proposed a lightweight Channel-wise Feature Pyramid (CFP) module and successfully applied it to both  nature and medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The architecture of this CFP module is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Figure 3(a) shows the architecture of the CFP module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' it contains total 𝐾 channels and each channel has its own  dilation rate 𝑟𝐾 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Typically, we choose the 𝐾 = 4  for CaraNet and the dilation rates for each channel {𝑟1, 𝑟2, 𝑟3, 𝑟4} =  {1,2,4,8}  which has been verified as the best dilation rates combination for CaraNet in Table 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' thus, each channel’s  dimension is 𝑀/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Simple feature fusion method sometimes introduces some unwanted checkboard or gridding artifacts  that greatly influence the accuracy and quality of segmentation masks [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Thus, we applied hierarchical feature fusion  [35] (HFF) to sum the outputs of all channels step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' For the FP channel, we provide two versions with regular  convolution and asymmetric convolution as shown in Figure 3(b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We connected the outputs of each convolutional  module by using skip connection, and thus each channel can be considered as a sub-pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We selected the regular  convolution as FP channel for CaraNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The overall FP is obtained from concatenating those sub-pyramids from the  hierarchical feature fusion operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The final FP contains four levels of feature stacks as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' These four  levels of feature stacks {𝑙𝑒𝑣𝑒𝑙𝑖, 𝑖 = 1, … ,4} are computed by:  Aggregation  Aggregation  13  Node l-1  Node3-2  Aggregation  Aggregation  Node 1-2  Node2-2  Aggregation  f5  Node1-3  Upsampling  4  {  𝑙𝑒𝑣𝑒𝑙1 =  𝑜𝑢𝑡𝐹𝑃1  𝑙𝑒𝑣𝑒𝑙2 = 𝑙𝑒𝑣𝑒𝑙1 + 𝑜𝑢𝑡𝐹𝑃2  𝑙𝑒𝑣𝑒𝑙3 = 𝑙𝑒𝑣𝑒𝑙2 + 𝑜𝑢𝑡𝐹𝑃3  𝑙𝑒𝑣𝑒𝑙4 = 𝑙𝑒𝑣𝑒𝑙3 + 𝑜𝑢𝑡𝐹𝑃4  (1)  And the final FP is computed by ∑ 𝑙𝑒𝑣𝑒𝑙𝑖  𝑖  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Based on our split-merge feature pyramid strategy, the receptive fields of a  single CFP module with dilation rates {𝑟1, 𝑟2, 𝑟3, 𝑟4} = {1,2,4,8}  varies from 3 × 3  to 55 × 55 , which successfully  overcomes the challenge from sharply varying object sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  (a)                                   (b)                 (c)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' (a) CFP module, (b) FP channel with regular convolution, (c) FP channel with asymmetric convolution  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Final feature pyramid obtained from CFP module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='4 Axial reverse attention module  The previous partial decoder that generates the global map 𝑆𝑔 (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='2) could roughly locate the position of medical  objects, and the CFP module extracted only multi-scale features from the pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' To obtain more accurate  feature information, we designed the Axial Reverse Attention (A-RA) module to refine localization information and multi-  scale features efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The overview and detail of the A-RA module can be seen in Figure 1 and Figure 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  The input of the top line is the multi-scale feature maps 𝑓𝑖  ′ from the CFP module and we used axial attention to analyze  the salience information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The axial attention is based on self-attention, which maps a query and a set of key-value pairs to  an output and the operation:  Input  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='1×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='M/K  FP channel, r  FP channel, r2  FP channel, rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  FP channel, rk  Add  PpV  HFF  Add  Concatenation  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' 1x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='M  Add3x3 convolution  3x3convolution  3x3convolution  concatenation3x1convolution  1x3convolution  3x1 convolution  1x3convolution  3x1 convolution  1x3convolution  concatenationLevel l:  Level 3:  Level2:  Level4:  Output:  5  𝐴𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(𝑄, 𝐾, 𝑉) = 𝑆𝑜𝑓𝑡𝑚𝑎𝑥 (  𝑄𝐾𝑇  √𝑑𝐾)  (2)  where 𝑄, 𝐾, 𝑉, and 𝑑𝐾 represent query, key, value, and dimension of key, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' However, self-attention consumes  great computational resources, especially when the spatial dimension of the input is large [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Therefore, we applied axial  attention, which factorizes 2D attention into two 1D attention along height and width axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Here we replace the softmax  activation function with a sigmoid, based on the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' For the second line, we applied the reverse operation [37] to  detect the salience features from the side-output 𝑆𝑖, which is obtained from the output of the previous A-RA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The  reverse operation is:  𝑅𝑖 =  1 − 𝑆𝑖𝑔𝑚𝑜𝑖𝑑(𝑆𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  (3)  The total axial reverse attention operation is:  𝐴𝑅𝐴𝑖 = 𝐴𝐴𝑖⨀𝑅𝑖  (4)  where ⊙ is element-wise multiplication, and the 𝐴𝐴𝑖 is feature from the axial attention route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Structure of Axial Reverse Attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='5 Deep supervision  We apply weighted intersection over union (IoU) and weighted binary cross-entropy (BCE) in our loss function: ℒ =  ℒ𝐼𝑜𝑈  𝑤  + ℒ𝐵𝐶𝐸  𝑤  to calculate the global loss and local (pixel-level) loss, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' To train CaraNet, we apply deep  supervision for the three side-outputs (𝑆1, 𝑆2, 𝑆3) and the global map 𝑆𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Before calculating the loss, we up-sampled them  to the same size as ground truth 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Thus, the total loss:  ℒ𝑡𝑜𝑡𝑎𝑙 = ℒ(𝐺, 𝑆𝑔  𝑢𝑝) + ∑ ℒ(𝐺, 𝑆𝑖  𝑢𝑝)  5  𝑖=3  (5)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='6 Small object segmentation analysis  Since the size of all images input to the segmentation models must be fixed, the size of an object is determined by the  number of pixels in the object m and the number of total pixels in the image N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Thus, we consider the object’s size using  the size ratio (proportion) = m/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Then, we evaluate the performance of segmentation models according to the sizes of  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Especially, we mainly focus on the small areas whose size ratios are smaller than 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' To define the watershed of  small size is a question;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' it depends on data and model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We further discuss this question in Discussion section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  To evaluate the performance of segmentation models according to the sizes of objects, we first obtain the mean-Dice  coefficients and size ratios of segmentations from the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Similar to computing the histogram, we plot the results  in a curve whose y-axis is mean-Dice coefficients and x-axis is increasingly sorted size ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' To smooth the curve, we  Axialattcntion  Height  Wcight  Convolutionalfeature  Weighted convolutionalfeature  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Sigmoid  Reverse weight  W=1-Sigmoid(Si)  Up-sampledfeature  6  take interval-averaged mean-Dice coefficients by sorted size ratios: we divide the entire range of size ratios into a  consecutive, non-overlapping, and of equal length series of intervals, and then calculate the average mean-Dice coefficients  of size ratios in each interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The interval-averaged coefficients have a smooth curve and are more stable in the presence  of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' EXPERIMENT  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='1 Implementation details  We implemented our model in PyTorch accelerated by the NVIDIA RTX 2070Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We resized input images to  352 × 352 for polyp segmentation and 256 × 256 for brain tumor segmentation and employed a multi-scale training  strategy {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='25} instead of data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We used Adam optimizer with the initial learning rate 1𝑒−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='2 Dataset  We test our CaraNet on five polyp segmentation datasets: ETIS [38], CVC-ClinicDB [39], CVC-ColonDB [40],  EndoScene [41], and Kvasir [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The first four are standard benchmarks, and the last one is the largest dataset, which was  released recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We also test our model on the Brain Tumor Segmentation 2018 (BraTS 2018) dataset [48, 49], which  contains more extremely small medical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Table 1 shows the details of these datasets: image size, scale of testing set,  and size ratios of medical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  The data of brain tumor segmentation is from the multimodal brain tumor segmentation challenge 2018 (BraTS 2018)  built by the Section for Biomedical Image Analysis (SBIA) at the University of Pennsylvania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' It contains the multimodal  brain MRI scans and manual ground truth labels of glioblastoma from 285 cases (patients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Each case includes four scan-  modals: 1) native (T1), 2) T1 contrast enhanced (T1ce), 3) T2-weighted (T2), and 4) T2 Fluid Attenuated Inversion  Recovery (FLAIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' And each case includes three types of ground truth labels: necrotic and non-enhancing tumor core  (NET), gadolinium-enhancing tumor (ET), and peritumoral edema (Ed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' In this study, T1ce is selected as our input images  and ground truth labels use NET type because it delineates the minimum areas for small object segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The MRI  scans for each case are sliced to 2-D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' As shown in Table 1, the test samples are chosen by the sizes of objects in  images (by examining the areas of truth labels) ranging in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='01% - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='91%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Details of datasets  Image size  Number of test samples  Object size ratio  ETIS  966 × 1225  196  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='11% - 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='05%  CVC-ClinicDB  288 × 384  62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='34 % - 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='88%  CVC-ColonDB  500 × 574  380  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='30% - 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='15%  CVC-300  500 × 574  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='55% - 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='42%  Kvasir  1070 × 1348  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='79% - 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='13%  BraTS 2018  256 × 256  3231  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='01% - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='91%  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='3 Baseline  We compared CaraNet with six medical image segmentation models, including state-of-the-art models: U-Net [1], U-  Net++ [2], ResUNet-mod [43], ResUNet++ [3], SFA [44], PraNet [27], CCBANet [50] and DS-TransUNet [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='4 Training and measurement metrics  We randomly split 80% of images from Kvasir and CVC-ClinicDB as training set and the remainder as a testing  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' In addition to mean Dice and mean IoU, we also apply four other measurement metrics: weighted dice metric 𝐹𝛽  𝑤,  MAE, enhanced alignment metric Ε𝜙  𝑚𝑎𝑥  [45], and structural measurement 𝑆𝛼  [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Table 2 shows the polyp  segmentation on the five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The weighted dice metric 𝐹𝛽  𝑤 is used to amend the “equal importance flaw” in dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  7  The MAE is used to measure the pixel-to-pixel accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The recently released enhanced alignment metric Ε𝜙  𝑚𝑎𝑥  is  utilized to evaluate the pixel-level and global-level similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' And 𝑆𝛼 is used to measure the structure similarity between  predictions and ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='5 Results  We first report the polyp segmentation results and compare with state-of-the-arts such as DS-TransUNet, CCBANet  and PraNet in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Among all five public endoscopy segmentation datasets, our proposed CaraNet achieves best  performance, especially in the ETIS dataset which contains more small polyps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We also show some polyp segmentation  results in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' For the five polyp datasets, CaraNet not only outperforms the compared models in overall performance,  but also on samples with small polyps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Figure 7 shows the segmentation performance of CaraNet and PraNet for small  objects (proportions ≤ 5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' For the extremely small object segmentation analysis on the BraTS 2018 dataset, we compare  only CaraNet with PraNet because PraNet has the closest performance to ours, and the overall accuracies of the other  segmentation models are clearly lower than those of CaraNet and PraNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' (Note: the fluctuations with size in colonoscopy  datasets are caused by types and boundary of polyps, and quality of imaging)  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Quantitative results on Kvasir, CVC-ClinicDB, CVC-ColonDB, ETIS, and CVC-T (test dataset of EndoScene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Note: mDice:  mean Dice, mIoU: mean IoU, 𝐹𝛽  𝑤: weighted dice, 𝑆𝛼: structural measurement [46], Ε𝜙  𝑚𝑎𝑥: enhanced alignment [45] and MAE: mean  absolute error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' ↑ denotes higher the better and ↓ denotes lower the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Methods  mDice ↑  mIoU ↑  𝑭𝜷  𝒘 ↑  𝑺𝜶 ↑  𝚬𝝓  𝒎𝒂𝒙 ↑  MAE ↓  Kvasir  UNet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
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+page_content=' For each subfigure, the x-axis is the proportion size (%) of polyp and the y-axis is the averaged mean Dice  coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Subfigures show performance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' size on the five polyp datasets, which are Kvasir, CVC-ClinicDB, CVC-ColonDB, ETIS,  and CVC-300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Blue line is for our CaraNet and orange is for the PraNet, showing only the results of small polyp sizes (<6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We can  find that our CaraNet overperforms the PraNet for most cases of small size polyps from the five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Quantitative results on brain tumor (BraTS 2018) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Methods  mean Dice  mean IoU  𝑭𝜷  𝒘  𝑺𝜶  𝚬𝝓  𝒎𝒂𝒙  MAE  CaraNet (Ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
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+page_content='003  PraNet (MICCAI’20)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
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+page_content='0  0  1  2  3  4  5  6  CaraNet (Ours)  PraNet  10  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Performance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Size on brain tumor datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The x-axis is the proportion size (%) of tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Upper figure: y-axis is the  mean Dice coefficient results of our CaraNet and PraNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' For the very small tumor sizes (≤1%), almost all results of CaraNet are  better than PraNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Lower figure: the y-axis is the difference between the averaged mean Dice coefficient results of CaraNet and  PraNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Red indicates the Dice value of CaraNet is greater than that of PraNet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' blue shows the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  BrainTumor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
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+page_content='5  5  TumorSizes(%)  11  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Brain tumor segmentation results  To further evaluate the effectiveness of CaraNet for small-object segmentation, we conducted another experiment  using the brain tumor dataset (BraTS 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The polyp datasets lack extremely small objects (the minimum is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='11%)  and do not have enough small samples (like Kvasir and CVC-ClinicDB, in Figure 7, there are fewer samples in the range  of small sizes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The brain tumor dataset was created from the BraTS 2018 database by slicing 2D images from the “T1ce”  source with “NET” type labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We randomly select 60% of the images as the training set and the remainder as the testing  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Altogether, 3231 images with proportions of tumor sizes ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='01% – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='91% were in the testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Table 3, Figure 8, and Figure 9 show the comparison result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We compared CaraNet with only PraNet for the same reason  stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Clearly, our CaraNet performed better, especially for the extremely small cases (range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='01% – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='1% in  Figure 8 and the red area indicates that the Dice value of CaraNet is greater than that of PraNet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' blue area shows the  opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Values on the right show the summations of all red and blue differential values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='5 Ablation study  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Quantitative results on Kvasir for different dilation rate  Dilation Rate  Mean Dice  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
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+page_content='907  To search the best dilation rate for our CFP module, we set up experiments and choose different dilation rates from 0  to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The performance of CaraNet with different dilation rate are testing on Kvasir testing set as shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' When  we choose small dilation rates like 0 and 4, the dilations rates for each channel are {𝑟1, 𝑟2, 𝑟3, 𝑟4} = {0,0,0,0}  and  {𝑟1, 𝑟2, 𝑟3, 𝑟4} = {0,1,2,4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The CFP module focuses on local information but ignore the global one, thus the accuracy is  about 1% lower than the best one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' When large dilation rates are chosen like 16 and 32, the dilation rates for each channel  are {𝑟1, 𝑟2, 𝑟3, 𝑟4} = {0,4,8,16} and {𝑟1, 𝑟2, 𝑟3, 𝑟4} = {0,8,16,32}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' There only one channel focus on local features which is  unreasonable for small medical object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' When we choose a fairish dilation rate like 8 which {𝑟1, 𝑟2, 𝑟3, 𝑟4} =  {0,2,4,8}, the weights of local and global information can be balanced and then achieves best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  We further conduct ablation studies to demonstrate the effectiveness of our proposed CFP module and Axial Reverse  Attention (A-RA) module five public endoscopy segmentation datasets, and we choose same six measurement metrics as  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='4 for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Images  Ground truth  PraNet  CaraNet  12  We first conduct an experiment to evaluate CaraNet without CFP module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' As shown in Table 5, the performance  without CFP module drops sharply on five public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' In particular, the mDice achieves about 8% reduction on ETIS  dataset, which strongly verifies that CFP module can effectively detect local-to-global feature due to its various sizes  receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Then, we evaluate the CaraNet without both CFP module and A-RA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The mDice continued to  decrease about 2%-3%, indicating that our A-RA module enables our CaraNet to accurately distinguish true polyp tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Ablation study for CaraNet on Kvasir, CVC-ClinicDB, CVC-ColonDB, ETIS, and CVC-T (test dataset of EndoScene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Note:  mDice: mean Dice, mIoU: mean IoU, 𝐹𝛽  𝑤: weighted dice, 𝑆𝛼: structural measurement [46], Ε𝜙  𝑚𝑎𝑥: enhanced alignment [45] and  MAE: mean absolute error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' ↑ denotes higher the better and ↓ denotes lower the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Dataset  CFP  A-RA  mDice ↑  mIoU ↑  𝑭𝜷  𝒘 ↑  𝑺𝜶 ↑  𝚬𝝓  𝒎𝒂𝒙 ↑  MAE ↓  Kvasir  \uf0fb  \uf0fb  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
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+page_content=' DISCUSSION  We propose a novel deep-learning based segmentation model – CaraNet, by combining the Axial Reverse Attention  and Channel-wise Feature Pyramid (CFP) modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' This new method can help improve the performance of the  segmentation of small medical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Through the experiments, we show that CaraNet outperforms the most famous  models by a large margin overall for six measurement metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' As shown by the polyp segmentation results, CaraNet not  only produces high quality segmentation on samples of large polyps, but also performs well for small and multi small-  object segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Figure 9 shows some results of extremely small tumor segmentation from the BraTS 2018 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  The advantage of the CaraNet in segmenting small single- and multi-objects is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' In addition, compared with the  recent state-of-the-art network, PraNet, CaraNet provides a more precise prediction for the most-challenging cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  We also introduce the process to evaluate segmentation models according to the size of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We consider the  object’s size using the size ratio including the sizes of objects and the whole image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' In this study, we assume the size ratios  of “small objects” are less than 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' However, the definition of “small objects” is not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Since few studies have  fully considered the sizes of objects and the small-object problems in medical imaging, we could further study this question  in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Preliminarily, the small size (watershed) could be defined by the Performance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Size curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Figure 10  shows an example on ETIS datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The watershed of small size could be defined at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' After that point, the performance  generally increase/change slower and is more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The definition of small area may depend on datasets, segmentation  models, and object shapes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' but if these conditions are fixed, it is feasible to make fair comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The definition of small  area discussed here may not be perfect, but it is worth paying attention to the model’s performance on small size cases  besides the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Figure 10 also indicates that the overall performance of segmentation depends on size ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  If we remove the small size cases, the overall performance will be greatly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  13  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Performance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Size on ETIS datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' The x-axis is the proportion size (%) of polyp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' y-axis is the mean Dice coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Blue is for our CaraNet and orange is for the SFA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Unlike Figure 7, it shows the results of all sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  Although CaraNet achieves good improvement on medical image segmentation tasks, there are still some limitations  and potentials to optimize the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' For example, using bilinear interpolation to up-sample feature maps cannot avoid the  loss of some useful information and lead to a coarse boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' It can be improved by applying a deconvolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' In  addition, the backbone of CaraNet is pre-trained on ImageNet, which contains natural images that are very different from  medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Moreover, the sliced brain MRI data also cause loss of spatial information between the voxels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' that may  influence the accuracy of small tumor detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' In our future work, we will use the Model Genesis [47] as a 3-D backbone  to replace the Res2Net (2-D backbone) and adjust the CaraNet to employ it on 3-D medical imaging segmentation to build  a 3-D version CaraNet for more accurate CT or MRI image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Segmentation of 3-D medical images is of  growing interest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' we believe CaraNet will address that problem successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' CONCLUSION  We have proposed a novel neural network, CaraNet, for small medical object segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' We use a partial decoder  to roughly localize polyp position, improve the adaptability of the CaraNet to detect different sizes objects by using a CFP  module and finally refine the polyp segmentation by A-RA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' From the overall segmentation accuracy, we can find  that CaraNet outperforms all state-of-the-art approaches by at least 2% (mean dice accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' For the early diagnosis  dataset (ETIS) which contains many small polyps, however, CaraNet can reach 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='7% mean dice accuracy, which is about  12% higher than PraNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' For the extremely small object segmentation dataset (BraTS 2018), CaraNet can achieve 3%  higher than PraNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' When evaluating segmentation models according to the size of objects, CaraNet outperforms PraNet  for small objects in all six datasets we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' In addition to the average accuracy through the whole dataset, to  evaluate/compare segmentation models according to sizes of objects could show more characteristics of models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' especially,  their performance on small areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' And this method can find or verify models that are good for small objects segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  DISCLOSURES  No conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
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+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  [4]  Fan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=', Tran-Dinh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=', Nguyen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=', & Tran, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' (2021, September).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Ccbanet:  16  Cascading context and balancing attention for polyp segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' In International Conference on Medical Image  Computing and Computer-Assisted Intervention (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' 633-643).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Springer, Cham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content='  [51] Lin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=', Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=', Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=', Lu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=', & Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' Ds-transunet: Dual swin transformer u-net for  medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
+page_content=' IEEE Transactions on Instrumentation and Measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFQT4oBgHgl3EQfmjaf/content/2301.13366v1.pdf'}
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf,len=1147
+page_content='1  Deep Learning for Short-Latency Epileptic  Seizure Detection with Probabilistic Classification  Yankun Xu, Jie Yang, Wenjie Ming, Shuang Wang, and Mohamad Sawan, Fellow, IEEE  Abstract—In this manuscript, we propose a novel deep learning  (DL)-based framework intended for obtaining short latency in  real-time electroencephalogram-based epileptic seizure detection  using multiscale 3D convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' We pioneer  converting seizure detection task from traditional binary classifi-  cation of samples from ictal and interictal periods to probabilistic  classification of samples from interictal, ictal, and crossing peri-  ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' We introduce a crossing period from seizure-oriented EEG  recording and propose a labelling rule using soft-label for samples  from the crossing period to build a probabilistic classification  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' A novel multiscale short-time Fourier transform feature  extraction method and 3D convolution neural network architec-  ture are proposed to accurately capture predictive probabilities  of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Furthermore, we also propose rectified weighting  strategy to enhance predictive probabilities, and accumulative  decision-making rule to achieve short detection latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' We  implement leave-one-seizure-out cross validation on two prevalent  datasets – CHB-MIT scalp EEG dataset and SWEC-ETHZ  intracranial EEG dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Eventually, the proposed algorithm  achieved 94 out of 99 seizures detected during the crossing period,  averaged 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='84% rectified predictive ictal probability (RPIP)  errors of crossing samples, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3 s detection latency, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='32/h false  detection rate on CHB-MIT dataset, meanwhile 84 out of 89  detected seizures, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='17% RPIP errors, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7 s detection latency,  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='75/h FDR are achieved on SWEC-ETHZ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The  obtained detection latencies are at least 50% faster than state-  of-the-art results reported in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Index  Terms—Epilepsy,  seizure  detection,  EEG,  brain-  computer interface, detection latency, probabilistic classification,  deep learning  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' INTRODUCTION  E  PILEPTIC patients suffering from unprovoked recurrent  seizures occupy approximately 1% population worldwide  [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Seizure is originated from abnormal discharged neu-  ron inside small brain region, then discharging current is  spread to other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Two main seizure types, named focal  and generalized seizure, depend on seizures begin in one area  then spread to other areas or seizures begin throughout brain  cortex simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Long-term drug therapy is one of the  major treatment intended for epileptic patients, however, one-  third of patients face drug-resistant epilepsy [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  This work was supported in part by the Westlake University, in part by  the Zhejiang Key R&D Program under Grant 2021C03002, and in part by  the Zhejiang Leading Innovative and Entrepreneur Team Introduction under  Grant 2020R01005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' (Corresponding authors: Jie Yang;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Mohamad Sawan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=')  Yankun Xu, Jie Yang and Mohamad Sawan are with the Center of  Excellence in Biomedical Research on Advanced Integrated-on-chips Neu-  rotechnologies (CenBRAIN Neurotech), School of Engineering, Westlake  University, Hangzhou, Zhejiang, China, (e-mail: yangjie@westlake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  sawan@westlake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Wenjie Ming and Shuang Wang are with the Epilepsy Center, Department  of Neurology, Second Affiliated Hospital, School of Medicine, Zhejiang  University, Hangzhou, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  EEG Onset  Clinical Onset  Many  seconds  Samples  …  …  …  1  0  Times (s）  0  Times (s）  EEG  Recordings  : True Label  : Binary Detection  Labelling  Decision  1  Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  : ∑ ������������������������ (Sum of Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=')  : ������������������������ (Predictive Ictal Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=')  Decision Threshold  Interictal  Crossing  Ictal  Latency  ������������������������  ������������������������  ������������������������ : Expected Detection Latency  ������������������������ : Latency in Binary Way  Alarm  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Schematic figure for illustrating EEG recordings, segmented samples,  traditional seizure detection challenges, and expected decision-making system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Recently, brain-computer interface (BCI) has been broadly  applied to healthcare and neuroscience domain, such as re-  habilitation, brain stimulation, prosthetic control, and brain  activity diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' [5]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' A BCI-based closed-loop seizure  detection system that consists of recording, detection, stim-  ulation elements, can heavily support epileptic patients [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Especially for patients with tonic-clonic seizures because  electrical stimulation intervenes can be operated in time prior  to the beginning of convulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Hence, an accurate real-time  epileptic seizure onset detection algorithm becomes a critical  factor to alarm the occurrence of seizure promptly [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  As typical BCI monitoring modalities, scalp electroen-  cephalogram (EEG) and intracranial EEG (iEEG) are widely  applied to supervise epileptic patients by recording brain activ-  ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Usually, patients suffer from a couple of seizures per day,  and seizure onsets and endings are identified by an experienced  epileptologist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Clinically the period between seizure onset and  seizure end is defined as ictal period usually lasting 30 s to  2 min, followed by a postictal period usually lasting 5-30  min [11]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The other period appearing in the health state  is defined as interictal period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Medical experts annotate the  EEG onset time according to the aberrant signs occurred in  the EEG recordings from epileptic patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' However, patients  do not behave abnormally at EEG onset time immediately,  usually unequivocal EEG onset precedes clinical onset by  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='03465v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='SP]  4 Jan 2023  2  several seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Litt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' [14], [15] announced this gap  would be 7-10 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The clinical onset refers to the appearance  of relevant symptoms, such as convulsion and jerking, that  can be obviously reflected on the EEG recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Hence, a  reliable seizure detection algorithm is required to obtain short  enough detection latency, thereby being capable of recognizing  the seizure occurred during the period between EEG onset and  clinical onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' As shown in the top of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 1, EEG Recordings  part displays a real EEG recording example of a patient around  the time of EEG onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Over the past decades, numerous algorithm-based seizure  detection studies have been published, and most of works an-  nounced they achieved high sensitivity and low false detection  rate (FDR) at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' However, high sensitivity alone is  still far away from actual seizure intervention usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Because  in terms of practical epileptic patients supervision scenarios,  short enough detection latency is crucial to guarantee the risk  alarms promptly and intervenes can be effectively operated  prior to serious onset symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Unfortunately, most previous  studies overlooked this important metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' According to our best  knowledge, most previous seizure detection studies trained the  machine learning (ML)-based or DL-based seizure detection  algorithm as a binary classification model to distinguish the  segmented interictal and ictal samples extracted from corre-  sponding periods, which is shown in Samples part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  However, this strategy remains a drawback that the trained  binary classification model cannot correctly detect the crossing  samples consisting of partial interictal and ictal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  As shown in Labelling part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 1, interictal and ictal  samples are labelled as 0 or 1 in traditional binary way  for training and trained binary model can recognize them as  0 or 1 correctly, but trained binary model would wrongly  detect the crossing samples as 0 or 1 randomly according  to our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The reason is that crossing samples are  significantly different from the major part of ictal period  because they are closed to the interictal period, if crossing  samples are directly considered as ictal periods in binary way,  the binary classification model would only learn the ictal  samples with obvious oscillation characteristics mainly instead  of crossing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' It should be noted that the time of each  detected dot in the figure is consistent with the end of detected  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  An accurate and prompt seizure detection system is ex-  pected to detect crossing samples in linearly increasing prob-  abilistic format according to the corresponding percentage of  ictal component, meanwhile to keep the complete interictal  or ictal samples in binary format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Then, an accumulative  decision-making rule can be used to alarm the seizure oc-  currence in short latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In Decision part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 1, gray  dots represent predictive probabilities of samples in real-time,  and the blue line shows accumulative probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' When the  accumulative probability reaches the decision threshold, the  detection system would alarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Therefore, as shown in Latency  part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 1, the expected detection latency 𝐿1 can be  shorter than the length of segmented samples, while the binary  classification model can only achieve the latency 𝐿2 at least  longer than the length of segmented samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  In this manuscript, we propose a novel DL-based framework  intended for real-time detection of epileptic seizures with short  latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The main innovative contributions aiming to address  the challenges and limitations mentioned above are as follows:  We pioneer introducing crossing period and convert  seizure detection task from traditional binary classifica-  tion to probabilistic classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  We propose a novel DL model based on multiscale 3D  convolution neural networks (M-3D-CNN) to accurately  capture predictive probabilities of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  A rectified probability weighting strategy is proposed to  further enhance the probabilistic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  And an accumulative decision-making rule is proposed to  achieve short latency and low FDR simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  The remaining content of this manuscript is organized as:  We describe in Section II recent related works about seizure  detection field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Section III elaborates on utilized materials, data  processing, and the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Section IV illustrates  experimental settings and achieved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Sections V  and VI are the objects of discussion and conclusion, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' RELATED WORK  Over the past decades, algorithm-based seizure detection  study is a hot topic attracted by numerous researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Shoeb  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' [16] initially detected 131 of 139 seizure events with 8  s detection latency and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='25/h FDR on 36 clinical pediatric  subjects in 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Then they published significant performance  that 96% detection sensitivity with averaged 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6 s latency on  24 pediatric subjects in 2010 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' They took advantage of  spectral and spatial features combined with support vector  machine (SVM) classifier to achieve advanced results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' As  pioneers of this field, Shoeb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' [17] also published CHB-  MIT scalp dataset, it has become one of the most famous and  important resource intended for seizure-related study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In 2011,  from the same group, researchers applied similar approaches to  the clinical iEEG dataset from 10 patients and achieved 97%  detection sensitivity, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='03/h FDR and 5 s detection latency  [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  In 2017, Vidyaratne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' [19] used an improved wavelet  method known as harmonic wavelet packet transform to extract  higher frequency information as features, then SVM is used  as classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' They achieved 96% sensitivity, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1/h FDR, and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='89 s detection latency on CHB-MIT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In [20], authors  applied statistical and morphological features combined with  an adaptive distance-based change point detector to achieve  96% sensitivity, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='12/h FDR, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='21 detection latency,  respectively on CHB-MIT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The empirical mode de-  composition method is a prevalent technique broadly applied  to seizure detection applications, numerous authors utilized  it and its variation to achieve satisfactory performance over  the past decade [21]–[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The short-time Fourier transform  (STFT) is an effective method widely used to extract seizure-  related features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' [25] proposed a multi-view  deep learning framework for EEG seizure detection based on  STFT and convolution neural network (CNN), they achieved  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='37% accuracy using 5-fold patient-specific cross validation  on CHB-MIT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Different from traditional convolutional  3  operation that considers EEG signal or STFT features as 2D  image-like features, 1D-CNN architecture is also applied to  seizure-related studies to [26]–[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In [27], authors achieved  sensitivity, FDR, and detection latency of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='31%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2/h, and  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1 s from event-based level on CHB-MIT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  [30] proposed a novel channel-embedding spectral-temporal  squeeze-and-excitation network using wavelet features to rec-  ognize epileptic EEG signals, they achieved 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='41% sensitivity  and 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='05% specificity on CHB-MIT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Burrello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' [31]–[33] from Swiss research group pub-  lished a seizure-oriented iEEG database, known as SWEC-  ETHZ database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' They achieved 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='84% specificity and 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='42%  accuracy on short-term dataset, and detected 79 out of 92  unseen seizures without any false alarms across all the patients  on long-term dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Afterwards, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' [27] and Sun  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' [34] obtained sensitivity, FDR, detection latency of  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='52%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='07/h FDR, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2 s and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='06/h FDR, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7  s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  From our point of view, there is a remained controversy in  measuring detection latency metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' As we depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 1,  binary classification model cannot achieve the latency shorter  than the length of segmented samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' According to prior-art  publications, they segmented EEG samples in at least 5 s,  however, most of them announced their proposed algorithms  can achieve less than 5 s detection latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Hence, we doubt  that previous researchers overlooked the crossing samples  from real-time perspectives, leading to compute the detection  latency by measuring the distance between EEG onset and  begin of detected sample instead of end of detected sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  In this manuscript, we pioneer introducing crossing period and  probabilistic classification in real-time seizure detection task  to achieve short latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' METHODOLOGY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Materials  In this work, EEG and iEEG datasets are both considered to  validate the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The CHB-MIT scalp EEG  dataset is one of the most prevalent open access datasets  intended for seizure-related research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' There are 23 pediatric  patients monitored by 22 electrode channels with 256 Hz  sampling rate, and each subject obtains various numbers of  seizure and non-seizure EEG recording files in 1h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The earliest  change associated with the clinical seizure is annotated as  EEG onset by clinical experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' We ignored the subjects with  changing electrode placement, thereby 19 subjects are selected  for the following experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  The SWEC-ETHZ dataset is an emerging and open-access  iEEG dataset collected from pre-surgical evaluations of pa-  tients with pharmacoresistant epilepsies, it was published in  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' This database contains two versions according to differ-  ent recording durations – long-term and short-term versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  In this work, we use short-term version to test our proposed  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In terms of short-term dataset, each patient obtains  several seizure files, and each file consists of a 3-min interictal  period immediately followed by an ictal period and a 3-  min postictal period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The EEG onset time is identified by  an experienced epileptologist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Because there is insufficient  TABLE I  SUMMARY OF TWO DATASETS USED IN THIS WORK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  dataset  EEG  type  # of  selected  patients  # of  channels  # of  seizures  Interictal  duration  CHB-MIT  sEEG  19  22  99  335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5h  SWEC-ETHZ  iEEG  11  42∼100  89  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='45h  sEEG: scalp EEG  iEEG: intracranial EEG  interictal duration for model training on short-term dataset, we  only select patients with no less than 4 seizures in this dataset,  so that 11 of 16 patients are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Furthermore, the number  of implanted electrode channels used in these patients varies  from 42 to 100, and 512 Hz sampling rate is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Table  I summarizes the characteristics of two datasets used in this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Definitions of different periods  DL algorithms cannot train the model directly on the suc-  cessive EEG recordings, so that we need to extract successive  segmented samples from the recording in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Firstly,  we need to identify the interictal, ictal and crossing periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  According to the seizure file of CHB-MIT dataset, beginning  and ending time of each seizure is provided, and we manually  set a 30 min postictal period following seizure ending, then  left part is belong to interictal period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In terms of short-term  version of SWEC-ETHZ dataset, each patient obtains several  seizure recording files, each file consists of a 3 min interictal  segment immediately followed by an ictal segment and a 3min  postictal segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  The schematic figure for definition of crossing period is  shown as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 2, where  𝐿𝑖𝑐𝑡𝑎𝑙  𝐿𝑐𝑟𝑜𝑠𝑠 denotes the length of ictal  component occupying the whole length of crossing sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  The crossing period is defined as ending of extracted samples  begins at EEG onset time (the last time for  𝐿𝑖𝑐𝑡𝑎𝑙  𝐿𝑐𝑟𝑜𝑠𝑠 = 0) and  ends at a duration of extracted sample after EEG onset (the  first time for 𝐿𝑖𝑐𝑡𝑎𝑙  𝐿𝑐𝑟𝑜𝑠𝑠 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In experiments, samples are extracted  in 5 s and 10 s segments for CHB-MIT dataset and SWEC-  ETHZ dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Data preparation  Furthermore, duration of interictal period is much longer  than duration of ictal period on CHB-MIT dataset, so that we  overcome this unbalanced data issue by extracting interictal  samples without any overlaps, and ictal samples with 80%  overlaps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' And we data-point-wisely extracted  crossing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Because the duration of crossing period for  each seizure equals to the length of segmented samples, the  duration of crossing period should be 5 s, and the number  of extracted crossing samples for each seizure is computed  by duration of segmented samples multiplying sampling rate  (5 (s) × 256 (Hz) = 1280 (samples)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' As for SWEC-ETHZ  dataset, we directly extracted samples of all three periods with  80% overlaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Explanation of crossing period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In crossing period, extracted sample  consists of partial interictal and ictal component simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The duration  of crossing period depends on the length of segmented samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Probabilistic labelling rule  In terms of traditional binary classification, interictal and  ictal samples are annotated by [1, 0] and [0, 1] according  to the one-hot encoding rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' However, the label of crossing  sample should contain probabilistic information rather than  simple one-hot information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The intuitive operation is to  directly label crossing samples as a single probability as a  regression task required, but this type of labelling would cause  crossing samples cannot be trained with interictal and ictal  samples together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' We aim to train the M-3D-CNN model  using interictal, ictal, and crossing samples together, because  crossing samples are comprised of partial interictal and ictal  parts, and M-3D-CNN model is expected to learn the unique  crossing samples characteristics from complete interictal and  ictal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  In this work, we keep labelling ictal and interictal sam-  ples as binary format, and take soft-label strategy to an-  notate the crossing samples in probabilistic format [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Compared to traditional regression task training, soft-label  annotation replace the output vector with the shape of  (1, ) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=', 𝑃𝑖𝑐𝑡𝑎𝑙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' ) with the output vec-  tor with the shape of (1, 2) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=', [𝑃𝑖𝑛𝑡𝑒𝑟𝑖𝑐𝑡𝑎𝑙, 𝑃𝑖𝑐𝑡𝑎𝑙]  =  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3], [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6], [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='8], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' There are several advan-  tages of soft-label strategy compared to the regression training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Firstly, the soft-label strategy makes networks can train the  crossing samples, complete interictal and ictal samples simul-  taneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Secondly, soft-label enables usage of cross entropy  loss function, which takes both interictal and ictal information  into account rather than only ictal probability provided with  simple mean square error loss function in regression task .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  In terms of experimental training setting, we keep la-  bels of interictal and ictal samples in the traditional way  (𝐿𝑎𝑏𝑒𝑙𝑖𝑛𝑡𝑒𝑟𝑖𝑐𝑡𝑎𝑙 = [1, 0], 𝐿𝑎𝑏𝑒𝑙𝑖𝑐𝑡𝑎𝑙 = [0, 1]) and label cross-  ing samples into 20 probability pairs as following rules:  𝐿𝑎𝑏𝑒𝑙𝑐𝑟𝑜𝑠𝑠 = [𝑃𝑖𝑛𝑡𝑒𝑟𝑖𝑐𝑡𝑎𝑙, 𝑃𝑖𝑐𝑡𝑎𝑙]  𝑤ℎ𝑒𝑟𝑒, 𝑃𝑖𝑐𝑡𝑎𝑙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='05𝑝 𝑖 𝑓  𝐿𝑖𝑐𝑡𝑎𝑙  𝐿𝑐𝑟𝑜𝑠𝑠  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='05𝑝  𝑃𝑖𝑛𝑡𝑒𝑟𝑖𝑐𝑡𝑎𝑙 = 1 − 𝑃𝑖𝑐𝑡𝑎𝑙,  𝑝 = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='19  (1)  where,  𝐿𝑖𝑐𝑡𝑎𝑙  𝐿𝑐𝑟𝑜𝑠𝑠 ranges from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In terms of loss function  utilized for model training, we take binary cross entropy loss  function which is defined as:  L(𝑃(𝑡)  𝑖𝑐𝑡𝑎𝑙, ˆ𝑃(𝑡)  𝑖𝑐𝑡𝑎𝑙) = − (𝑃(𝑡)  𝑖𝑐𝑡𝑎𝑙 · 𝑙𝑜𝑔( ˆ𝑃(𝑡)  𝑖𝑐𝑡𝑎𝑙)  + (1 − 𝑃(𝑡)  𝑖𝑐𝑡𝑎𝑙) · 𝑙𝑜𝑔(1 − ˆ𝑃(𝑡)  𝑖𝑐𝑡𝑎𝑙))  (2)  where, ˆ𝑃𝑖𝑐𝑡𝑎𝑙 denotes predictive 𝑃𝑖𝑐𝑡𝑎𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' A Sigmoid activation  function is used to scale the  ˆ𝑃𝑖𝑐𝑡𝑎𝑙 from 0 to 1 before  computing the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Multiscale 3D convolution neural network architecture  As a multiscale architecture, proposed model aims to ad-  dress the challenge that recognition of target samples in  probabilistic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 3 shows the detailed architecture of  M-3D-CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Each segmented sample input is a period  of multichannel EEG signals, we implement multiscale STFT  feature extraction at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' As a prevalent and effective signal  analysis tool in frequency domain, STFT method is widely  used in seizure detection studies [36]–[38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Different from  previous works, we propose to extract channel-wise STFT  features in different scales simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The computation  method is elaborated as follows:  𝑋𝑠𝑐𝑎𝑙𝑒[𝜔, 𝑚] =  𝑊 𝐿−1  ∑︁  𝑘=0  𝑤[𝑘] · 𝑥[𝑚 × 𝑊𝐿  2  + 𝑘] · 𝑒− 𝑗2𝜋𝜔𝑘  𝑊𝐿 = 𝐿𝑠𝑖𝑔𝑛𝑎𝑙  𝑠𝑐𝑎𝑙𝑒 , 𝑠𝑐𝑎𝑙𝑒 = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  (3)  where 𝑚 and 𝜔 are the index of the sliding window and  frequency coefficient, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 𝑊𝐿 and 𝐿𝑠𝑖𝑔𝑛𝑎𝑙 denote  window length and length of EEG signal, and 𝑊 𝐿  2  refers to  the overlapping length is set to half of window length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Various scales of STFT in M-3D-CNN model stand for  the chosen length of sliding window in STFT indeed, mean-  while we set 50% overlapping for sliding window, thus the  number of detecting windows in time axis of STFT result is  𝑁𝑤𝑖𝑛𝑑𝑜𝑤 = 2𝑛 − 1 at scale 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Meanwhile, the size of fast  Fourier transform (FFT) is set to 64, so that 32 informative  coefficients are left in the frequency dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 4 displays  STFT result of the single-channel signal at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  In M-3D-CNN, we take 5 different scales from 1 to 5, each  scale would generate a 3D STFT feature tensor (channel ×  frequency × time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Previously, most studies considered this  type of feature as a 2D image in the shape of H (time) ×  W (frequency) with depth whose size is equal to the number  of channels, then implemented 2D convolution operation [39]–  [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' It is an intuitive operation, however, this way cannot take  time step information into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In M-3D-CNN model, we  consider 3D STFT feature as a 2D image in the shape of H  (channel) × W (frequency) with time step Depth, then 3D  convolution operation is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 5 shows the difference  between 2D and 3D convolution operations for STFT features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  The values from achieved 3D STFT result need to be channel-  wisely normalized from 0 to 1 at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Then, extracted 3D features in different scales are fed to 3  same 3D convolution blocks, and each block contains a 3D  convolution module with kernel size in the shape of 3 × 3 ×  3 and a max-pooling module with kernel size in 2 × 2 × 1 for  the first two scales, and 2 × 2 × 2 for the rest three scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The  5  Multichannel  Sample  Scale 1  Scale 2  Scale 3  Scale 4  Scale 5  3×3×1  Convolution  ReLU  2×2×1  Pooling  3×  3×3×3  Convolution  ReLU  2×2×2  Pooling  3×  1×512  +  5×512  5×5  Conv  ReLU  2×2  Pooling  3×  3D Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Blocks  W(Chs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=')×H(Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' )×D(Time)  Multiscale STFT  Feature Extraction  Probabilistic Output:  ������������������������������������������������������������������������������������������������������������������������������������ , ������������������������������������������������������������������������  chs×32×31  chs×32×1  chs×32×3  chs×32×7  chs×32×15  Normalization in Frequency Dimension  FC Layer  Flatten  Flatten  Flatten  Flatten  Flatten  FC  FC  FC  Flatten  1024  256  64  Sigmoid  2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Architecture of proposed multiscale 3D convolution neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The input is a multichannel EEG sample, we extract short-time Fourier transform  (STFT) features of the input at scale 1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Then a FreqNorm layer is used to channel-wisely normalize 3D STFT values from 0 to 1 in frequency dimension  at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The 3D STFT feature is fed to 3× same 3D convolution (Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=') blocks comprising of 3D Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=', ReLU activation function, and 3D  max-pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The last layer generated by Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' block is flattened and connected to a fully connected (FC) layer with 512 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 5 vectors obtained from 5  different scales are concatenated to a 5×512 matrix, then 3× traditional 2D Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' blocks and 3× FC layers with ReLU are used to make classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The  last output FC layer with 2 nodes utilizes a sigmoid activation function to guarantee the output in probability ranged from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Multiscale STFT feature extraction scheme for single-channel sample  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Schematic figure for difference between traditional 2D and proposed  3D convolution operation for multichannel STFT feature at each scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  output generated after 3× convolution blocks is flattened, then  a multilayer perceptron (MLP) layer with a shape of 1 × 512 is  connected to this flattened output to achieve a 1D vector with  the same shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' This operation aims to unify the shape, thereby  eliminating the effects of inconsistent vector dimension due  to input multichannel EEG signals with different numbers of  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' These 5 vectors originated from 3D feature tensors  at 5 different scales are concatenated together to build a 2D  matrix in the shape of 5 × 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Then successive three same  2D convolution blocks with 5 × 5 kernel size convolution  operation and 2 × 2 kernel size max-pooling operation, are  connected to the 2D matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The output is also flattened to a  vector, and 3 MLP layers are followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Eventually, we achieve  the output in the last MLP layer in the shape of 1 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' It  is important to note that except for the last layer Sigmoid  activation function is used to generate probabilistic output,  ReLU activation function is used in the M-3D-CNN model  everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  This novel architecture is inspired by the fact that probabilis-  tic crossing samples are comprised of short complete interictal  and ictal periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 𝑃𝑖𝑐𝑡𝑎𝑙 is the percentage of a complete ictal  period occupying the crossing sample, rather than uniformly  distributed in the crossing sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' However, traditional feature  extraction approaches either implement FFT for the whole  duration or STFT with a single specific scale, which cannot  meet the situation that the probability pairs of crossing samples  are dynamically changing in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Furthermore, 5 scales  setting is applicable for most situations, because we already set  50% overlapping for the STFT time window and the duration  of segmented samples is usually less than 10 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Rectified weighting strategy  According to the Labelling part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' (1),  𝑃𝑖𝑐𝑡𝑎𝑙 of crossing samples is expected to be linearly increased  from 0 to 1 along with the linearly increasing percentage of  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='.6  ictal period occupying the whole crossing sample ( 𝐿𝑖𝑐𝑡𝑎𝑙  𝐿𝑐𝑟𝑜𝑠𝑠 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In  practical experiments, however, predictive ictal probabilities  (PI) cannot be always precise or perfectly linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Thus, we  propose a rectified weighting strategy to enhance the predic-  tive ictal probability (PIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The principle of this strategy is  that previously achieved PIPs are used to rectify the current  achieved PIP at time 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Due to real-time scenario, we store the  PIPs every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1 s generated from previous 5 s, then we utilize  PIPs from previous 5 s, 3 s, and 1 s to fit three linear regression  (LR) functions, in order to generate three new PIPs (PIP𝐿𝑅5𝑠,  PIP𝐿𝑅3𝑠, PIP𝐿𝑅1𝑠) only based on previous PIPs from different  durations instead of current PIP𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Eventually, we can achieve  rectified PIP at time 𝑡 (RPIP𝑡) that is computed as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  The weights 𝜆1, 𝜆2, 𝜆3, 𝜆4 are experimentally set to adjust the  weighting of different PIPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  In short, rectified weighting strategy aims to enhance the  current PIP by rendering it more relevant to previous PIPs,  this can help reduce the impact of abnormal PIPs which are  might be generated by noises, artifacts, or model limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  𝑅𝑃𝐼𝑃𝑡 =  � 𝜆1  𝜆2  𝜆3  𝜆4  � ��������  𝑃𝐼𝑃𝐿𝑅5𝑠  𝑃𝐼𝑃𝐿𝑅3𝑠  𝑃𝐼𝑃𝐿𝑅1𝑠  𝑃𝐼𝑃𝑡  ��������  (4)  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Decision-making rule  We do not make decision only based on a single PIP because  it is difficult to significantly shorten the detection latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The  reason is that if the decision threshold is high, the detection  latency is inevitably longer than the duration of segmented  samples, meanwhile FDR would be high if we set a low  decision threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Therefore, in this work, we also propose  an accumulative decision-making rule, whose schematic figure  is shown in the Decision part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Same as rectified weighting strategy, we store the RPIPs  from previous 5 s with detection rate 𝑟, which means we  store the RPIPs in a time step of  1  𝑟 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Then we compute  the accumulative probability (AP) at current time 𝑡 (𝐴𝑃𝑡) as  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In short, we only accumulate increased RPIPs during  the period of previous 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' And the detection system would  alarm at time 𝑡𝑑 when 𝐴𝑃𝑡𝑑 ≥ 𝑇ℎ𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=', where 𝑇ℎ𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' represents  the decision threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Eventually, Algorithm 1 illustrates the  detailed decision-making rule of proposed framework intended  to detect seizures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  𝐴𝑃𝑡 =  �𝑡  𝑖=𝑡−5 𝑅𝑃𝐼𝑃𝑖+1  𝑟  (if 𝑅𝑃𝐼𝑃𝑖+1 > 𝑅𝑃𝐼𝑃𝑖)  (5)  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Performance metrics  In this work, there are four metrics - sensitivity, errors, de-  tection latency, and FDR, used to investigate the performance  of proposed DL-based framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' (6) reveals how we  compute these four metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The sensitivity is computed as the  number of seizures detected during the crossing period over  the total number of seizures in each patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' RPIP errors are  computed as the second equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' (6), we only consider  RPIP errors of crossing samples to figure out the capacity of  Algorithm  1: Decision-making rule of proposed  framework intended to detect seizures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Input: Sample at time 𝑡: 𝑆𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Output: Detection time: 𝑡𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Initialize: Detection rate: 𝑟;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Decision threshold: 𝑇ℎ𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='.  Stacking vector with zeros for previous 5s RPIPs:  ℙ = [𝑃𝑡−5, 𝑃𝑡−5+ 1  𝑟 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=', 𝑃𝑡− 1  𝑟 , 𝑃𝑡] = 𝟘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  while True do  [1 − 𝑃𝐼𝑃𝑡, 𝑃𝐼𝑃𝑡] ← M-3D-CNN(𝑆𝑡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  𝑅𝑃𝐼𝑃𝑡 ← 𝑅𝑊𝑆(𝑃𝐼𝑃𝑡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  ℙ ← 𝑎𝑝𝑝𝑒𝑛𝑑(ℙ[𝑃𝑡−5+ 1  𝑟 : 𝑒𝑛𝑑], 𝑅𝑃𝐼𝑃𝑡)  𝐴𝑃𝑡 ← �𝑡  𝑖=𝑡−5 ℙ (if 𝑅𝑃𝐼𝑃𝑖+1 > 𝑅𝑃𝐼𝑃𝑖)  if 𝐴𝑃𝑡𝑑 > 𝑇ℎ𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' then  Alarm at time 𝑡𝑑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Refresh ℙ = 𝟘  end  end  Abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' :  RPIP: Rectified Predictive Ictal Probability  M-3D-CNN: Multiscale 3D Convolution Neural Networks  RWS: Rectified Weighting Strategy  AP: Accumulative Probability  M-3D-CNN model combined with rectified weighting strategy  to recognize crossing samples in probabilistic way, the 𝑡  denotes the sample detected at time 𝑡 and the 𝑇 refers to  the duration of crossing period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In terms of detection latency,  we implement Algorithm 1 and mark the time (𝑡𝑑) when  𝐴𝑃𝑡𝑑 larger than and equal to decision threshold (𝑇ℎ𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' ), then  compute the detection latency by calculating distance between  𝑡𝑑 and EEG onset time (𝑡𝑜𝑛𝑠𝑒𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The last metric is false  detection rate (FDR), we directly count the number of false  detection according to Algorithm 1 during the interictal period  per hour as FDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Sensitivity =  𝑁𝐷𝐶  𝑁𝑇 𝑜𝑡𝑎𝑙  RPIP Errors =  �𝑇 −1  𝑡=0  √︃  (𝑃(𝑡)  𝑖𝑐𝑡𝑎𝑙 − ˆ𝑃(𝑡)  𝑖𝑐𝑡𝑎𝑙)2  𝑇  Detection Latency = 𝑡𝑑 − 𝑡𝑜𝑛𝑠𝑒𝑡  if 𝐴𝑃𝑡𝑑 ≥ 𝑇ℎ𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  FDR = 𝑁𝐹𝐷/ℎ  (6)  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' EXPERIMENTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Experimental setting  The experiments were implemented by Python with Pytorch  deep learning framework, and M-3D-CNN model training  and inference works are carried out on the single NVIDIA  2080Ti GPU machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In this work, we trained patient-specific  model training and implemented the leave-one-seizure-out  cross validation (LOSOCV) scheme, which means we select  one seizure for validation, and the rest seizures are used to train  the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' LOSOCV is meaningful from clinical perspective  because the selected validated seizure can be regarded as a  fresh seizure unseen by the model yet, if the model performs  well in this scheme, we can believe that the trained model can  also accurately and promptly alarm seizures in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Experimentally, all interictal, ictal and crossing samples are  used to train the model, and only errors of crossing samples are  7  (a)  (b)  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Performance of rectified probability weighting strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Here are 3 representative examples, each figure refers to crossing period of a seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Red,  blue and orange dots stand for true, predictive, and rectified probabilities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The original errors achieved by PIPs and rectified errors achieved by  RPIPs are also highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  TABLE II  PERFORMANCE OF PROPOSED ALGORITHM ON TWO DATASETS IN THE PATIENT-SPECIFIC AND LEAVE-ONE-SEIZURE-OUT CROSS VALIDATION SCHEME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  RECTIFIED PREDICTIVE ICTAL PROBABILITY (RPIP) IS GENERATED BY M-3D-CNN MODEL FOLLOWED BY RECTIFIED WEIGHTING STRATEGY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  DETECTION LATENCY AND FALSE DETECTION RATE ARE (FDR) OBTAINED BY ACCUMULATIVE DECISION-MAKING RULE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' THERE ARE 19 AND 11 CASES  IN CHB-MIT AND SWEC-ETHZ DATASETS, RESPECTIVELY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' WE PROVIDE MEAN AND STANDARD DEVIATION VALUES FROM EVERY SEIZURE OF EACH  PATIENT FOR RPIP ERRORS, DETECTION LATENCY, AND FDR METRICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  CHB-MIT dataset  SWEC-ETHZ dataset  Patient  ID  Sensitivity  (NDC / NT)  RPIP Errors of  Crossing Samples  (%)  Detection  Latency  (s)  FDR  (/h)  Patient  ID  Sensitivity  (NDC / NT)  RPIP Errors of  Crossing Samples  (%)  Detection  Latency  (s)  FDR  (/h)  chb01  7/7  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='85 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6  0  ID1  13/13  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='18 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='27  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='44  chb02  2/3  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='76 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='49  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='10 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='82  ID2  4/4  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='90 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='98  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='94 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='62  chb03  7/7  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='10 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='74  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='4  0  ID4  14/14  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='35 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='29  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='81  chb04  3/4  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='09 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='29  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='80 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='60  ID5  10/10  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='63 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='30  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='94  chb05  5/5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='33 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='32  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5  0  ID6  4/4  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='76 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='56  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='11 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='28  chb06  7/7  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='85 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='22  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3  0  ID9  9/9  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='42 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='53  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='74 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='11  chb07  3/3  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='57 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='71  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1  0  ID10  4/5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='31 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='46  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='50  chb08  4/5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='44 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='72  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='80 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='26  ID12  10/10  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='86 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='70  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='35  chb09  4/4  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='31 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='51  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='50 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='02  ID13  6/7  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='96 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='53  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='54  chb10  7/7  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='80 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='19  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='44 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='20  ID14  5/7  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='07 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='95  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='76  chb11  3/3  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='82 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='59  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='82  ID16  5/6  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='75 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='44  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='57  chb14  8/8  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='46 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='97  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3  0  chb17  2/3  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='86 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='90  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6  0  chb18  5/6  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='05 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='51  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2  0  chb19  3/3  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='03 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='4  0  chb20  8/8  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='15 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='33  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='72 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='04  chb21  4/4  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='93 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='45  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='4  0  chb22  3/3  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='26 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='01  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='24 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='92  chb23  7/7  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='60 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='70  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='9  0  Overall  94/99  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='84 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='80  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='34 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='11  Overall  84/89  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='17 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='26  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='75 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='04  RPIP: Rectified Predictive Ictal Probability  FDR: False Detection Rate  NDC: Number of Seizures Detected during the Crossing Period  NT: Number of Total Seizures  considered as the most crucial metric to quantify the quality of  the model because the trained model can perfectly recognize  accurate probabilities of complete interictal and ictal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  During the phase of model training, we set 20 training epochs  and used optimizer is Nesterov-accelerated Adam, known as  Nadam, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='0001 learning rate, 𝛽1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='9, 𝛽2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='999  [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' We implement the LOSOCV scheme and only save the  best model performing the lowest RPIP errors of crossing  samples on the validated seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In terms of rectified weight-  ing strategy, weights 𝜆1, 𝜆2, 𝜆3, 𝜆4 are experimentally set to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The detection rate and decision threshold  used to make decision are 10 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='5 for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Original Errors :-6%  Rectified Errors : 4%  TrueProb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Predictive Prob  Rectified ProbOriginal Errors : 23%  Rectified Errors : 7%  True Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Predictive Prob  Rectified ProbOriginal Errors :-7%  Rectified Errors : 9%  TrueProb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Predictive Prob  Rectified Prob8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Boxplot for rectified predictive ictal probability of each seizure on two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' (RPIP: Rectified Predictive Ictal Probability)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Boxplot for detection latency of each seizure on two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The averaged latencies of two datasets are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3 s and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2 s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Results  At first, we need prove the effectiveness of rectified weight-  ing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 6 shows 3 representative examples of PIP  and RPIP performance from CHB-MIT dataset, where red,  blue, and orange dots stand for true, predictive, and rectified  probabilities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The x-axis represents percentage of  ictal period in crossing sample, and y-axis refers to probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  As mentioned in Section III-B and III-C, there are 1280  extracted crossing samples in crossing period for each seizure,  and there are divided into 20 probability pairs annotation as  true labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Thus, every 5% ictal period contains 64 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  We can see that even though original PIPs perform well  according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 4(a), rectified weighting strategy still can  enhance the results from 6% to 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 4(b), original  PIPs show worse fitting result with 23% errors, while rectified  weighting strategy can significantly decrease the errors to 7%,  and the overall probabilities are increasing more linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  4(c) is another type of representative example that RPIPs seem  to achieve increased errors compared to original PIPs (from  7% to 9%), but we can see that RPIPs increase more linearly  than PIPs, so that we still keep the results of RPIPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' According  to these three representative examples, we can conclude that  rectified weighting strategy is effective to rectify the PIPs  by decreasing errors further and making PIPs increase more  linearly as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' This operation aims to help detection  system can recognize samples more accurately and meanwhile  decrease FDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Table II shows performance of proposed M-3D-CNN model  on CHB-MIT and SWEC-ETHZ datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In this table, we  only consider RPIPs after implementing rectified weighting  strategy, then compute detection latency and FDR based on  RPIPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' We achieved overall 94 of 99 and 84 of 89 seizures  detected during the crossing period, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='84% ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='80% and  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='17% ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='26% RPIP errors of crossing samples, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3 s  ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7 s and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7 s ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='0 s detection latency and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='34/h ±  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='11/h and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='75/h ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='04/h FDR on CHB-MIT scalp dataset  and SWEC-ETHZ iEEG dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' These mean  and standard deviation values are calculated based on results  attained from various numbers of seizures in each patient  according to LOSOCV scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In terms of performance on  CHB-MIT dataset, M-3D-CNN model performs well on 8  CHB-MIT  SWEC-ETHZ  10  T  ９987  S  Latency  6５4321  DB-  cr  PatientIDCHB-MIT  SWEC-ETHZ  50  45  40  (%  535250  Errors(  RPIP  5  chl  ch  PatientID9  TABLE III  PERFORMANCE COMPARISON BETWEEN THIS WORK AND PRIOR-ART STUDIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Year  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Dataset  EEG  type  # of  pat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Feature  extraction  method  Len.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' of  Sample  Model  Sen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  (%)  FDR  (/h)  Detection  Latency  LOSO-  CV  scheme  Probabi-  listic  task  2010  [17]  CHB-MIT  sEEG  24  FFT  6s  SVM  96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='08  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6s (+6s)  ✓  –  2011  [18]  Clinical  iEEG  10  FFT  3s  SVM  97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='03  5s (+3s)  ✓  –  2017  [19]  CHB-MIT  sEEG  22  Wavelet  6s  SVM  96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='89s (+6s)  ✓  –  2018  [20]  CHB-MIT  sEEG  22  Statistical  6s  ADCD  96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='21s (+3s)  ✓  –  2019  [25]  CHB-MIT  sEEG  22  STFT  3s  2D-CNN  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='9  –  –  –  –  2019  [26]  CHB-MIT  sEEG  23  Raw  2s  2D-CNN  90  –  –  –  –  2020  [30]  CHB-MIT  sEEG  21  Wavelet  –  2D-CNN  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='4  –  –  –  –  2020  [33]  SWEC-ETHZ  iEEG  11  LBP  + LGP  6-bit  SVM  MLP  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='8  0  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='9s  –  –  2021  [24]  CHB-MIT  sEEG  24  Wavelet  + EMD  4s  SVM  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='64  –  –  –  2021  [27]  CHB-MIT  SWEC-ETHZ  sEEG  iEEG  24  18  Raw  2s  1D-CNN  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='07  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1s (+2s)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2s (+2s)  –  –  2021  [43]  SWEC-ETHZ  iEEG  16  Statistical  2s  1D-CNN  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='4  –  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='8s (+2s)  –  –  2023  This  work  CHB-MIT  SWEC-ETHZ  sEEG  iEEG  19  11  Multiscale  STFT  5s  10s  3D-CNN  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='0  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3s  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7s  �  �  –: N/A  Len.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' : Length  Sen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=': Sensitivity  FDR: False Detection Rate  LOSOCV: Leave-One-Seizure-Out Cross Validation  sEEG: scalp EEG  iEEG: intracranial EEG  FFT: Fast Fourier Transform  SVM: Support Vector Machine  STFT: Short-Time Fourier Transform  ADCD: Adaptive Distance-based Change Point Detector  LBP: Local Binary Pattern  LGP: Local Gradient Pattern  MLP: Multilayer Perception  EMD: Empirical Mode Decomposition  cases (chb01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' chb03,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' chb05,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' chb06,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' chb07,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' chb14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' chb19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  chb23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' all seizures are detected during the crossing period,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  low RPIP errors of crossing samples (≤15%),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' short detection  latency (≤2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7 s) and none false detection are attained on these  patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The rest subjects show slight drawbacks on one or  two performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' There are 5 patients (chb02, chb04,  chb08, chb17, chb18) containing 1 seizure do not be detected  by M-3D-CNN model during the crossing period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' However,  these miss-detected seizures still can be detected after crossing  period, so that we set the detection latency of them to 5 s and  10 s which equals to the length of crossing period for CHB-  MIT and SWEC-ETHZ datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' We can see from  table that 1 miss-detected seizure leads to high RPIP and  FDR, except for chb18, M-3D-CNN model obtains larger than  20% even close to 30% mean and larger than 10% standard  deviation RPIP errors, and larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='8/h FDR on the rest 4  patients (chb02, chb04, chb08, chb17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' As for chb22 patient,  even though there is no miss-detected seizure, we still achieved  slightly higher RPIP errors and FDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In terms of SWEC-  ETHZ dataset, except for ID13 and ID14 subjects, proposed  model performs well on the rest 8 patients, where ≤20%  RPIP errors of crossing samples and ≤5 s detection latency  are achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' There are 4 patients (ID10, ID13, ID14, ID16)  containing miss-detected seizures during the crossing period,  correspondingly worse performance metrics are achieved on  these patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Especially for ID14 case where M-3D-CNN  model performs worst, there are two miss-detected seizures  and the highest RPIP errors, detection latency, and FDR are  attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 8 display boxplots specifying every seizure  performance of RPIP errors and detection latency on two  datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 7, it is obvious that  most seizures achieved less than 30% RPIP errors and the  majority is less than 20%, meanwhile there only 8 out of 99  seizures achieved larger than 30%, and 4 of them achieved  larger than 40% on CHB-MIT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Each of these seizures  obtaining higher RPIP errors appears in different subjects,  this phenomenon indicates that worse RPIP errors are not  generated by the poor model or abnormal patient, this may be  caused by a single abnormal seizure among these correspond-  ing patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' And these possible abnormal seizures lead to a  large standard deviation as shown in reverent cases from Table  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In terms of SWEC-ETHZ dataset, all seizures obtained less  than 30% RPIP errors except for the aforementioned worst-  performing case ID14, we suspect ID14 is an abnormal patient  different from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' As for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' 8, we can see from CHB-  MIT dataset that except for 5 miss-detected seizures during  the crossing period set to 5 s latency, all rest seizures achieve  less than 4 s latency, and the averaged latency is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' From  SWEC-ETHZ dataset, there are also 5 miss-detected seizures  set to 10 s latency, among the rest seizures, around 10%  seizures are larger than 8 s, and the majority is less than 6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  The averaged detection latency among 89 seizures is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Table III shows performance comparisons between prior-art  publications and this work, we list several characteristics of the  used dataset and proposed method to make comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' There  10  are 10 previous highly cited works with prior-art performance  selected to prove the advantages of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' It should be  noted that in terms of detection latency item, we add the  length of detected sample to previously reported latencies  because we doubt previous works did not compute the latency  by measuring the distance between EEG onset and the end  of detected sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' According prior publications, the state-  of-the-art detection latencies on two datasets are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2 s and  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1 s respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The latencies obtained by our proposed  algorithm are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='3 s and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='7 s which are significantly faster  than prior-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' And we can see that several previous  studies even if utilized the naive FFT feature extraction method  and SVM classifier, they still achieved good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  However, numerous recent emerging studies took advantage  of advanced feature extraction methods and deep learning  models, they cannot significantly enhance the seizure detection  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Furthermore, less recent studies focused on  testing meaningful metrics from clinical perspectives, such  as FDR, detection latency, and LOSOCV scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In the last  three columns of the table, we highlight three innovations of  this work, which are whether or not implementing LOSOCV  scheme and probabilistic classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' DISCUSSION  In this section, we discuss several issues, including further  clarification of achieved performance, model comparison from  hardware and performance perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Performance clarification  Firstly, we will clarify the sensitivity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In this work,  we only consider the number of seizures detected during  the crossing period as effectively detected seizures instead  of seizures detected at any time as previous works did [16],  [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Because we think the seizure can be detected during the  crossing period means detection latency of this seizure is short  enough to guarantee detection time can precede clinical onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Experimentally, M-3D-CNN model can detect all seizures after  crossing period (during the ictal period), so that sensitivity  would be 100% as the way previous researchers measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  But we think such 100% sensitivity would not be clinically  beneficial for epileptic patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Secondly, in Table III we only displayed RPIP errors  of crossing samples instead of all three kinds of samples  (interictal, ictal, and crossing samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The reason is that  proposed M-3D-CNN model is capable of recognizing com-  plete interictal and ictal samples perfectly, and the errors  are less than 3% stably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' We believe that such good results  on complete samples cannot lead to short detection latency,  only accurately predicted interictal samples can help us to  reduce FDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Therefore, we showed errors of crossing samples,  detection latency, and FDR to investigate the performance of  proposed M-3D-CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Thirdly, the lengths of extracted samples for two datasets  are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' We empirically and experimentally set 5 s and 10  s for CHB-MIT and SWEC-ETHZ datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' We  initially learn from previous studies that achieved detection  latency of two datasets were around 5 s and larger than 10  TABLE IV  PERFORMANCE COMPARISONS OF VARIOUS MODELS ON CHB03 SUBJECT  WITH 7 SEIZURES FROM CHB-MIT DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Model Name  PIP Errors of  Crossing Samples  (%)  Number of  Parameters  Model  Size  M-3D-CNN  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='14 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='38  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='8M  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='46MB  M-2D-CNN  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='41 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='36  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='6M  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='42MB  5-2D-CNN  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='07 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='34  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='2M  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='65MB  M-LSTM  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='53 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='61  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1M  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='56MB  5-LSTM  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='13 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='20  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='1M  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='56MB  M-ViT  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='87 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='28  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='9M  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='32MB  5-ViT  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='65 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='00  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='9M  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='32MB  PIP: Predictive Ictal Probability  M: Multiscale  5: Scale 5  ViT: Vision Transformer  CNN: Convolution Neural Network  LSTM: Long Short-Term Memory  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Then we experimentally tuned this parameter, we found  that longer extracted samples would bring longer detection  latency and lower FDR, while shorter extracted samples would  cause shorter detection latency and higher FDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Thus, this  is a kind of trade-off parameter tuning work, eventually the  length of extracted sample making the first ictal sample (or  the last crossing sample) just contain the obvious EEG signal  oscillations is expected, thereby we set 5 s and 10 s for two  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Fourthly, achieved detection latency on SWEC-ETHZ is ob-  viously longer than the latency achieved on CHB-MIT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  There are two possible reasons to answer this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  The first is that the criterion of annotating EEG onset time is  implemented by different clinical experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The second is that  iEEG modality is more sensitive to the abnormal discharging  inside brain, thereby can detect the slight EEG abnormality  more earlier than scalp EEG [44], [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' However, such earlier  slight abnormality appearing in EEG signal is difficult to be  detected by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Model comparison  In this manuscript, we proposed a novel M-3D-CNN model  deep learning architecture based on multiscale STFT features  and 3D convolution operation in CNN backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' In Table  IV, we change three innovative parameters - multiscale or  not, 3D or 2D, CNN or other DL backbone architectures,  then generate several variant DL models to make comparisons  with proposed M-3D-CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' The original PIP errors  of crossing samples and the model size achieved on the  chb03 patient from CHB-MIT dataset are used to compare the  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' According to the table, M-3D-CNN architecture  performs best among all variant models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' We can also conclude  that multiscale is better than single-scale STFT features, and  3D-CNN outperforms 2D-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Although 5-2D-CNN model  shows advantages in model size, it obtains the worst PIP er-  rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Furthermore, long short-term memory (LSTM) recurrent  neural networks and vision transformer (ViT) also achieve  satisfactory performance, but their model sizes are quite large  compared to the M-3D-CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' ViT is an emerging and  11  powerful deep learning model applied to various applications,  but ViT model is too large to fit this real-time seizure detection  application even if we already tried our best to shrink the  model parameters, and eventually model size is still doubled as  M-3D-CNN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' It should be noted that we flatten the  time axis of all STFT features as various time steps for both  LSTM and ViT models, which makes multiscale or single-  scale features only change the length of time steps, and would  not change the number of trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' CONCLUSION  The proposed framework uses a deep M-3D-CNN model  intended to address several challenges and limitations in the  field of seizure detection study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' It consists of a novel proba-  bilistic classification concept to accurately recognize crossing  samples simultaneously containing partial interictal and ictal  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Then we also propose rectified weighting strategy  and accumulative decision-making rule to significantly shorten  the detection latency of seizure onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content='  Furthermore, although proposed framework is intended for  seizure detection application, the concept of probabilistic  classification, rectified weighting strategy and accumulative  decision-making rule can be applied to other electrophys-  iological signal based real-time BCI applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Also, it  can benefit other detection systems to make decisions more  promptly and accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
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+page_content=' Thome-Souza, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
+page_content=' Jackson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E1T4oBgHgl3EQf1QUs/content/2301.03465v1.pdf'}
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diff --git a/n9E0T4oBgHgl3EQfZgDd/content/tmp_files/2301.02323v1.pdf.txt b/n9E0T4oBgHgl3EQfZgDd/content/tmp_files/2301.02323v1.pdf.txt
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+Ab initio calculation of carrier mobility in semiconductors
+including ionized-impurity scattering
+Joshua Leveillee,1, 2 Xiao Zhang,3 Emmanouil Kioupakis,3 and Feliciano Giustino1, 2
+1Oden Institute for Computational Engineering and Sciences,
+The University of Texas at Austin, Austin, Texas 78712, USA
+2Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA∗
+3Department of Materials Science and Engineering,
+University of Michigan, Ann Arbor, Michigan, 48109, USA
+(Dated: January 9, 2023)
+The past decade has seen the emergence of ab initio computational methods for calculating
+phonon-limited carrier mobilities in semiconductors with predictive accuracy. More realistic calcu-
+lations ought to take into account additional scattering mechanisms such as, for example, impurity
+and grain-boundary scattering. In this work, we investigate the effect of ionized-impurity scattering
+on the carrier mobility. We model the impurity potential by a collection of randomly distributed
+Coulomb scattering centers, and we include this relaxation channel into the ab initio Boltzmann
+transport equation, as implemented in the EPW code. We demonstrate this methodology by con-
+sidering silicon, silicon carbide, and gallium phosphide, for which detailed experimental data are
+available. Our calculations agree reasonably well with experiments over a broad range of tempera-
+tures and impurity concentrations. For each compound investigated here, we compare the relative
+importance of electron-phonon scattering and ionized-impurity scattering, and we critically assess
+the reliability of Matthiessen’s rule. We also show that an accurate description of dielectric screening
+and carrier effective masses cam improve quantitative agreement with experiments.
+I.
+INTRODUCTION
+The ability to predict the charge transport properties
+of semiconductors using non-empirical ab initio meth-
+ods is of paramount importance for the design of next-
+generation electronics, neuromorphic computing, energy-
+efficient lighting, and energy conversion and storage. For
+example, as beyond-silicon materials for next-generation
+field-effect transistors are being explored, such as wide-
+gap semiconductors like GaN [1], SiC [2], and Ga2O3 [3],
+or high-mobility materials such as GaAs [4], ab initio
+methods for calculating transport properties with pre-
+dictive accuracy are acquiring an increasingly important
+role.
+The past decade has seen numerous developments
+in first-principles calculations of phonon-limited charge
+transport coefficients such as the electrical conductivity
+in metals, and the drift and Hall mobilities in semicon-
+ductors [5–11]. More recently, several groups turned their
+attention to ab initio calculations of additional scattering
+mechanisms [5, 12–16]. Among the various mechanisms,
+impurity scattering is of particular interest since ionized
+donors and acceptors are ubiquitous in high-purity doped
+semiconductors, and intrinsic point defects are unavoid-
+able in all other materials [17–19]. In this work we fo-
+cus on ionized-impurity scattering, which is expected to
+provide the most significant contribution to the carrier
+relaxation rates beyond phonons, given the long-ranged
+nature of the Coulomb potential.
+Ionized-impurity scattering in semiconductors has first
+been studied via the Conwell-Weisskopf model. In this
+∗ fgiustino@oden.utexas.edu
+model, the scattering potential of the impurity is de-
+scribed using a Coulomb monopole immersed in the di-
+electric background of the semiconductor [20]. The long-
+range nature of this potential makes it ill-behaved at
+long-wavelength, and the singularity at long wavelengths
+is removed using an ad hoc infrared cutoff.
+A better
+handling of this singularity is achieved in the Brooks-
+Herring model by considering free-carrier screening [21].
+This latter model proved very successful [22], and is still
+widely used owing to its simplicity as it only requires
+the electronic density of states, the carrier effective mass,
+the high-frequency dielectric constant, and the impurity
+concentration. Further improvements upon these models
+were subsequently introduced, e.g., carrier statistics, dis-
+persive electronic screening, two-impurity scattering, and
+atomic form factors [23]. While this class of models en-
+joyed considerable success with calculations of the carrier
+mobility of silicon, they do not perform as well with other
+semiconductors [24, 25]. These and similar other empir-
+ical adjustments make it harder to quantify the role of
+each scattering channels, and most importantly decrease
+the transferability of the models and ultimately their use-
+fulness in materials design.
+During the past decade, considerable progress has been
+achieved in ab initio calculations of charge carrier mo-
+bilities [5–8, 14, 26].
+These approaches are based on
+the use of electronic band structures from density func-
+tional theory (DFT) [27, 28], as well as phonon disper-
+sion relations and electron-phonon matrix elements from
+supercell calculations or from density-functional pertur-
+bation theory (DFPT) [29–32]. To achieve a numerically
+converged sampling of the Brillouin zone, most calcula-
+tions by now employ Wannier-Fourier interpolation [33–
+35]. Mobilities are then obtained by solving the ab initio
+arXiv:2301.02323v1  [cond-mat.mtrl-sci]  5 Jan 2023
+
+2
+Boltzmann transport equation (aiBTE) [26]. The first
+study of ionized-impurity scattering from first principles
+was reported by Restrepo and Pantelides [5], and more
+recent, state-of-the-art calculations have been reported
+by Lu and coworkers [14]. In this latter work, the au-
+thors find good agreement between calculated mobilities
+and experimental data for silicon. Additional work using
+a semi-empirical approach combining DFT calculations
+and models was also reported recently [36, 37].
+In this work, we investigate from first principles the
+effect of ionized-impurity scattering on the carrier mobil-
+ity of semiconductors. To this aim, we take into account
+both carrier-phonon and carrier-impurity scattering on
+the same footing, within the aiBTE formalism as imple-
+mented in the EPW code. [38] Given that the shape of
+the impurity potential depends on the details of the crys-
+tal structure and its evaluation would require thermody-
+namic calculations of defects and defect levels [39], we
+limit ourselves to consider the monopole term of the scat-
+tering potential and a random distribution of impurities.
+This simplification allows us to achieve an elegant and
+compact formalism, and to compute carrier mobilities by
+using solely the concentration of ionized impurities as
+input. To validate our methodology, we perform calcu-
+lations for three test systems: Si, 3C-SiC, and GaP. For
+Si there is an abundance of experimental data and previ-
+ous calculations to compare with. 3C-SiC, which is also
+referred to as cubic SiC or β-SiC in the literature, is con-
+sidered a promising candidate for next-generation power
+electronics [40–42]. Several experimental data sets are
+available for carrier mobility in 3C-SiC, especially for n-
+type (N) doping and less so for p-type doping (Al). GaP
+is a standard optoelectronic semiconductor which is of in-
+terest in non-linear optical switching [43–45]; experimen-
+tal mobility data for GaP are available both for n-type
+doping (Sn) and p-type doping (Zn). For each of these
+compounds we calculate the temperature-dependent car-
+rier mobility at variable impurity concentration. We in-
+vestigate the relative importance of carrier-phonon and
+carrier-impurity scattering, and we examine the validity
+of the classic Matthiessen’s rule [46].
+The manuscript is organized as follows. In Sec. II we
+briefly summarize the aiBTE formalism, we provide a
+detailed derivation of the matrix elements for carrier-
+impurity scattering, and we discuss the key approxima-
+tions involved. In this section we also discuss free-carrier
+screening, and we examine under which conditions the
+Matthiessen rule can reliably be used in transport cal-
+culations.
+Section III is devoted to the implementa-
+tion details and the calculation parameters used in this
+work.
+In Sec. IV we discuss our results for Si, SiC,
+and GaP. In particular, in Sec. IV B we present our cal-
+culated temperature- and concentration-dependent mo-
+bilities and compare our data with experiments.
+In
+Sec. IV C we analyze the relative importance of phonon-
+and impurity-mediated scattering processes in the carrier
+relaxation rates. In Sec. IV D we test Matthiessen’s rule
+by comparing full aiBTE calculations with the results
+of separate calculations including only phonon-limited or
+impurity-limited mobilities. In Sec. IV E we investigate
+how the DFT dielectric screening and carrier effective
+masses influence calculated mobilities, and we test sim-
+ple correction schemes along the lines of Ref. [10].
+In
+Sec. V we summarize our findings and offer our conclu-
+sions. Additional details on the calculation procedure are
+discussed in the Appendices.
+II.
+THEORETICAL APPROACH
+A.
+Carrier mobility from the ab initio Boltzmann
+transport equation
+A detailed derivation of the aiBTE formalism is given
+in Ref. [38]. Here we limit ourselves to summarize the key
+equations in order to keep this manuscript self-contained.
+Within the linearized Boltzmann transport equation, the
+carrier mobility tensor is obtained as:
+µαβ = −
+2
+Ωucnc
+1
+Nuc
+�
+nk
+vα
+nk∂Eβfnk,
+(1)
+where the factor of 2 is for the spin degeneracy, Greek
+indices indicate Cartesian directions, Eβ indicate the
+Cartesian components of the electric field, and ∂Eβfnk
+is the linear variation of the electronic occupation of the
+state with band index n and wavevector k in response
+to the applied field. vα
+nk represents the expectation value
+of the velocity operator along the direction α, for the
+Kohn-Sham state nk. e, nc, Ωuc, and Nuc indicate the
+electron charge, the carrier density, the volume of the
+unit cell, and the number of unit cells in the Born-von
+K´arm´an (BvK) supercell, respectively. The n-summation
+extends over all Kohn-Sham states, although in practice
+only those states near the chemical potential contribute
+to the mobility. The k-summation is over a uniform Bril-
+louin zone grid.
+The variation ∂Eβfnk is obtained from the self-
+consistent solution of the equation:
+−evβ
+nk
+∂f 0
+nk
+∂ϵnk
+=
+�
+mq
+�
+τ −1
+mk+q→nk ∂Eβfmk+q
+−τ −1
+nk→mk+q ∂Eβfnk
+�
+,
+(2)
+where f 0
+nk denotes the Fermi-Dirac occupation of the
+state nk in the absence of electric field. The quantity
+τ −1
+nk→mk+q is the partial scattering rate from the Kohn-
+Sham state nk to the state mk + q. In many-body per-
+turbation theory, this rate is derived from the imaginary
+parts of the electron self-energy, therefore different scat-
+tering mechanisms simply add up to the lowest order in
+perturbation theory. In this work, we write the scattering
+rate as the sum of the rates of carrier-phonon scattering
+(ph) and carrier-impurity (imp) scattering:
+1
+τnk→mk+q
+=
+1
+τ ph
+nk→mk+q
++
+1
+τ imp
+nk→mk+q
+.
+(3)
+
+3
+The partial carrier-phonon scattering rate is given
+by [26]:
+1
+τ ph
+nk→mk+q
+=
+1
+Nuc
+�
+ν
+2π
+¯h |gmnν(k, q)|2
+×
+�
+(nqν + 1 − f 0
+mk+q)δ(ϵnk−ϵmk+q − ¯hωqν)
++(nqν + f 0
+mk+q)δ(ϵnk−ϵmk+q + ¯hωqν)
+�
+,
+(4)
+where ϵnk denote Kohn-Sham eigenstates,
+and ωqν
+stands for the frequency of a phonon with branch in-
+dex ν, wavevector q, and Bose-Einstein occupation nqν.
+The matrix elements gmnν(k, q) indicate the probability
+amplitude for the scattering of an electron from state nk
+to state mk+q via a phonon qν [35]. The partial rate in
+Eq. (4) can be obtained either from Fermi’s golden rule or
+from many-body perturbation theory [35]. The carrier-
+impurity scattering rate required in Eq. (3) is derived in
+the next section and is given by Eq. (17).
+Together, Eqs. (1)-(4) and (17) define the aiBTE
+framework employed in this work. This approach consis-
+tently captures back-scattering and Umklapp processes,
+with a computational cost that is similar to more ap-
+proximate approaches based on various relaxation-time
+approximations. We refer the reader to Ref. [26] for a
+comprehensive review of common approximations to the
+Boltzmann transport equation.
+B.
+Scattering of Carriers by ionized impurities in
+the monopole approximation
+To
+obtain
+the
+carrier-impurity
+scattering
+rate
+1/τ imp
+nk→mk+q we proceed as follows: (i) We derive the
+matrix element of the scattering potential for a single
+impurity in a periodic BvK supercell of the crystal unit
+cell; (ii) We generalize the matrix element to consider
+a number Nimp of impurities in the BvK supercell; (iii)
+From this matrix element, we obtain the scattering rate
+corresponding to the Nimp impurities by using the first
+Born approximation; (iv) We average the resulting rate
+over a random uniform distribution of impurity positions
+using a method due to Kohn and Luttinger.
+1.
+Scattering potential and matrix element for single
+impurity
+We employ the monopole approximation to describe
+the potential of an impurity of charge Ze located at the
+position r0 in the BvK supercell. A more refined choice
+would entail explicitly calculating the impurity poten-
+tial in DFT and its matrix elements. This approach was
+pursued in Refs. [5] and [14], but it carries the disadvan-
+tage that one needs to compute defect energetics prior
+to mobility calculations, and then perform rotational av-
+erages to account for the randomness of the impurity
+orientation. Our simpler approach is useful for system-
+atic transport calculations when detailed knowledge of
+the atomic-scale structure of impurities is lacking, and
+can be made more accurate by incorporating dipole and
+quadrupole terms along the lines of Refs. [47–49].
+By solving the Poisson equation in the BvK supercell
+and considering a background anisotropic static dielec-
+tric constant tensor ε0 = ε0
+αβ, the potential of this point
+charge is found to be [see Eq. (S3) of Ref. [47]]:
+φ(r; r0) = 4π
+Ωsc
+Ze
+4πε0
+�
+q
+�
+G̸=−q
+ei(q+G)·(r−r0)
+(q + G)·ε0· (q + G),
+(5)
+modulo an inessential constant that reflects the compen-
+sating background charge. In this expression, ε0 is the
+vacuum permittivity, G is a reciprocal lattice vector,
+and the wavevector q belongs to a uniform Brillouin-
+zone grid. Here an in the following, we consider that the
+BvK cell consists of Nuc unit cells, so that its volume is
+Ωsc = NucΩuc, and that the Brillouin zone is discretized
+in a uniform grid of Nuc points. The potential φ(r, r0) is
+periodic over the BvK supercell.
+The perturbation potential resulting from this impu-
+rity is V = ∓eφ for electrons and holes, respectively.
+For definiteness, we consider electrons in the following.
+The matrix elements of the perturbation V between the
+Kohn-Sham states ψnk and ψmk+q is given by:
+gimp
+mn (k, q; r0) = ⟨ψmk+q|V (r; r0)|ψnk⟩sc,
+(6)
+where the integral is over the supercell. The states can
+be written as ψnk = N −1/2
+uc
+eik·runk, where unk is the
+Bloch-periodic part and is normalized in the unit cell.
+The combination of Eqs. (5) and (6) yields:
+gimp
+mn (k, q; r0) = −e2
+4πε0
+4πZ
+Ωsc
+�
+G̸=−q
+e−i(q+G)·r0Bmn,G(k, q)
+(q + G)·ε0· (q + G) ,
+(7)
+having defined the overlap integral:
+Bmn,G(k, q) = ⟨umk+q|eiG·r|unk⟩uc,
+(8)
+which is evaluated over the unit cell.
+2.
+Scattering rate from multiple impurities within the first
+Born approximation
+We now consider N sc
+imp impurities located at the po-
+sitions r1, r2, · · · , rNimp in the BvK supercell.
+The
+corresponding perturbation potential is the sum of
+the potentials obtained in the previous section, V
+=
+�N sc
+imp
+I=1 V (r; rI), therefore the generalization of Eq. (7)
+to the case of multiple identical impurities reads:
+gimp
+mn (k, q; {rI}) = −e2
+4πε0
+4πZ
+Ωsc
+�
+G̸=−q
+Bmn,G(k, q)
+(q + G)·ε0· (q + G)
+×
+�N sc
+imp
+I=1 e−i(q+G)·rI.
+(9)
+
+4
+The total scattering rate out of state nk associated
+with this matrix element can be written using the first
+Born approximation for the scattering matrix [50] [Eqs.
+(6.1.16) and (6.1.32)]:
+1
+τ imp
+nk
+=
+�
+mq
+2π
+¯h |gimp
+mn (k, q; {rI})|2δ(ϵnk − ϵmk+q).
+(10)
+We note that this expression is an intensive quantity,
+as expected, i.e. it does not scale with the size of the
+BvK supercell [see discussion after Eq. (17)]. The partial
+scattering rate needed in Eq. (3) is then defined as:
+1
+τ imp
+nk→mk+q
+= 2π
+¯h |gimp
+mn (k, q; {rI})|2δ(ϵnk−ϵmk+q). (11)
+Unlike Eq. (4), in this expressions we do not have the
+Fermi-Dirac occupations. These occupations drop out in
+the linearized Boltzmann transport equation, as it can
+be verified, for example, by setting nqν = 0 and ωqν = 0
+in Eq. (4). In Eq. (11) the Dirac delta function ensures
+energy conservation, consistent with the fact that we are
+considering the scattering by a fixed potential, i.e. we
+are neglecting the recoil of the impurity upon collision.
+By combining Eqs. (9) and (11) we find:
+1
+τ imp
+nk→mk+q
+({ri}) = 2π
+¯h
+� e2
+4πε0
+4πZ
+Ωsc
+�2
+δ(ϵnk − ϵmk+q)
+×
+�
+G,G′̸=−q
+Bmn,G(k, q)B∗
+mn,G′(k, q)
+(Q ·ε0· Q)(Q′ ·ε0· Q′)
+�N sc
+imp
+I,J=1ei(Q′·rJ−Q·rI),
+(12)
+having defined Q = q + G and Q′ = q + G′ for conve-
+nience.
+3.
+Kohn-Luttinger ensamble averaging of the scattering rate
+In order to account for the randomness in the dis-
+tribution of impurities, we perform a configuration av-
+erage of the scattering rate in Eq. (12) by considering
+a uniform probability distribution, following the Kohn-
+Luttinger approach [51]:
+1
+τ imp,ave
+nk→mk+q
+=
+�
+sc
+dr1 · · · drN sc
+imp
+Ω
+N sc
+imp
+sc
+1
+τ imp
+nk→mk+q
+({ri}). (13)
+The only term that depends on the impurity positions in
+Eq. (12) is the sum over I, J on the second line. Below we
+evaluate the ensemble average of this sum by separating
+the I = J and I ̸= J terms:
+�
+sc
+dr1 · · · drN sc
+imp
+ΩNimp
+sc
+�N sc
+imp
+I,J=1ei(Q′·rJ−Q·rI)
+= N sc
+imp
+Ωsc
+�
+sc
+dr ei(Q′−Q)·r
++N sc
+imp(N sc
+imp − 1)
+Ω2sc
+��
+sc
+dr eiQ′·r
+���
+sc
+dr e−iQ·r
+�
+. (14)
+Both terms on the r.h.s. require the evaluation of an in-
+tegral of the type:
+�
+sc
+dr eiQ·r.
+(15)
+This integral equals Ωsc for Q = 0; for finite Q, we note
+that the integral becomes the Fourier representation of
+the Dirac delta when Nuc → ∞, therefore it vanishes. In
+this limit, Eq. (14) reduces to:
+�
+sc
+dr1 · · · drNimp
+Ω
+N sc
+imp
+sc
+�N sc
+imp
+I,J=1ei(Q′·rJ−Q·rI)
+= N sc
+imp δG,G′ + N sc
+imp(N sc
+imp − 1) δG,−qδG′,−q,
+(16)
+Using Eqs. (16) and (12) inside Eq. (13), we obtain:
+1
+τ imp,ave
+nk→mk+q
+=
+1
+Nuc
+N uc
+imp
+2π
+¯h
+� e2
+4πε0
+4πZ
+Ωuc
+�2
+×
+�
+G̸=−q
+|Bmn,G(k, q)|2
+|(q + G) ·ε0· (q + G)|2 δ(ϵnk − ϵmk+q), (17)
+where we use N uc
+imp = N sc
+imp/Nuc to denote the number
+of impurities per unit cell; N uc
+imp is a dimensionless quan-
+tity. We note that, in practical calculations, the prefac-
+tor 1/Nuc in Eq. (17), which also appears in the partial
+carrier-phonon scattering rate in Eq. (4), is included as
+a k-point weight in Brillouin zone summations, so that
+the sum in Eq. (2) becomes N −1
+uc
+�
+q and is independent
+of the size of the BvK supercell.
+The scattering rate given in Eq. (17) is similar but not
+identical to alternative forms used in previous work. For
+example, it differs from classic approaches such as the
+Conwell-Weisskopf formula [20] and the Brooks-Herring
+formula [21] in that here the details of band structures,
+Kohn-Sham orbital overlaps, and anisotropic dielectric
+screening are fully taken into account. Furthermore, it
+differs from more recent ab initio approaches such as
+Ref. [5] in that the long-range nature of the Coulomb in-
+teraction is taken into account from the start, as opposed
+to being included as an ad hoc correction. Our expression
+is similar to the formula provided in Ref. [14], except that
+here we take into account the periodicity of the impurity
+potential over the BvK supercell and the anisotropy of
+the dielectric tensor. The fact that we reached a similar
+expression as in Ref. [14] starting from a rather different
+viewpoint involving the Kohn-Luttinger ensemble aver-
+age lends support to both approaches.
+C.
+Free-carrier screening of the impurity potential
+The carrier-impurity scattering rate given by Eq. (17)
+contains a singular q−4 term that is not integrable (with
+q = |q|), and leads to incorrect results when used in the
+aiBTE of Eq. (2). This problem was already identified by
+
+5
+Conwell and Weisskopf [20], who introduced an infrared
+cutoff to suppress the Coulomb singularity.
+The formal way to overcome this difficulty is to observe
+that ionized impurities are accompanied by free-carriers,
+which introduce metallic-like screening of the impurity
+potentials. In the Thomas-Fermi model, free-carriers in-
+troduce an additional screening
+εTF(q) = 1 + q2
+TF
+q2 ,
+(18)
+where qTF is the Thomas-Fermi wavevector. When used
+in combination with the impurity potential appearing in
+Eq. (17), this additional screening lifts the Coulomb sin-
+gularity. In fact, by temporarily ignoring the G vectors
+and the anisotropy of the dielectric tensor, free-carrier
+screening modifies the denominator of Eq. (17) as fol-
+lows:
+1
+(ε0q2)2
+−−→
+1
+[εTF(q)ε0q2]2 =
+1
+[ε0(q2 + q2
+TF)]2 , (19)
+which tends to the finite value 1/(ε0q2
+TF)2 at long wave-
+length.
+To incorporate free-carrier screening in our calcula-
+tions, while taking into account all details of band struc-
+tures and effective masses, we employ the Lindhard di-
+electric function instead of the Thomas-Fermi model, fol-
+lowing Ref. [52]. The same approach was employed in
+Ref. [14]. The Lindhard dielectric function is given by:
+εL(q) = 1 −
+e2
+4πε0
+4π
+q2
+2
+NucΩuc
+�
+nk
+f 0
+nk+q − f 0
+nk
+ϵnk+q − ϵnk
+.
+(20)
+Since the density of free-carriers is typically low in doped
+semiconductors, we only need the long wavelength limit
+of this expression. In this limit, (f 0
+nk+q − f 0
+nk)/(ϵnk+q −
+ϵnk) = ∂f 0
+nk/∂ϵnk, therefore we can write:
+εL(q) = 1 + q2
+TF
+q2 ,
+(21)
+having introduced the effective Thomas-Fermi vector:
+qTF =
+e2
+4πε0
+2 · 4π
+NucΩuc
+�
+nk
+����
+∂f 0
+nk
+∂ϵnk
+���� .
+(22)
+For parabolic bands, Eq. (21) reduces to the Thomas-
+Fermi or Debye model in the respective temperature lim-
+its.
+The free-carrier screening provides an additional
+screening mechanism to the dielectric screening of the
+insulating semiconductors, and is included in our calcu-
+lations by replacing ε0 in Eq. (17) by the total dielectric
+function:
+ε0
+−−→
+ε0 + 1q2
+TF
+q2 ,
+(23)
+where 1 denotes the 3 × 3 identity matrix. We note that
+this improved description of the screening includes tem-
+perature effects via the Fermi-Dirac occupations entering
+the definition of the effective Thomas-Fermi wavevector,
+Eq. (22).
+D.
+Matthiessen’s Rule
+Matthiessen’s rule [46] is widely employed to interpret
+transport measurements. In the context of carrier trans-
+port in semiconductors, this rule can be stated as follows:
+the contributions of different scattering channels to the
+mobility can be obtained by adding the reciprocals of the
+individual mobilities. In the case of carrier-phonon and
+impurity-phonon scattering, we would have:
+1
+µ =
+1
+µph
++
+1
+µimp
+.
+(24)
+In Sec. IV we proceed to quantify the reliability of this
+approximation by comparing mobility data calculated us-
+ing the complete aiBTE including both phonons and im-
+purities with the prediction of Eq. (24) obtained by cal-
+culating the mobility with these two scattering channels
+taken individually. We will show that this rule does not
+carry predictive power for the examples considered in this
+work.
+From a formal standpoint, the rule expressed by
+Eq. (24) is obviously related to the choice of expressing
+the total scattering rates as the sum or the individual
+rates, see Eq. (3).
+That choice was motivated by the
+observation that, to first order in perturbation theory,
+different scattering channels do not mix. However, it is
+easy to see that, even when Eq. (3) is a good approx-
+imation, the additivity of the rates does not imply the
+Matthiessen rule as expressed by Eq. (24). To appreciate
+this point, we observe that the aiBTE in Eq. (2) can be
+recast as a linear system of the type:
+A × {∂Eβfnk} = b,
+(25)
+where the matrix A contains the partial scattering rates
+τ −1
+nk→n′k′, the vector b contains the drift term on the left
+hand side of Eq. (2), and {∂Eβfnk} denotes the vector of
+solutions. If we break down the matrix A into its contri-
+butions from carrier-phonon and carrier-impurity scat-
+tering, Aph and Aimp respectively, we see immediately
+that
+{∂Eβfnk} = (Aph + Aimp)−1b ̸= A−1
+ph b + A−1
+impb,
+(26)
+therefore the additivity of the scattering rates does not
+imply the Matthiessen rule.
+This point can be made
+even more explicit by considering the self-energy relax-
+ation time approximation to the aiBTE. The approxi-
+mation consists of neglecting the first term on the r.h.s.
+of Eq. (2), and yields the following expression for the
+mobility:
+µαβ = −
+e
+Ωucnc
+2
+Nuc
+�
+nk
+∂f 0
+nk
+∂ϵnk
+vα
+nkvβ
+nk
+×
+1
+1
+τ ph
+nk
++
+1
+τ imp
+nk
+.
+(27)
+
+6
+TABLE I. Calculation parameters used in this work: Experi-
+mental lattice constant, plane wave kinetic energy cutoff, and
+non-vanishing elements of the quadrupole tensor are chosen
+to be consistent with Ref. [58].
+Si 3C-SiC GaP
+Lattice constant (˚A)
+5.43
+4.36
+5.45
+Plane wave kinetic energy cutoff (eV)
+544
+1088 1088
+Qκ1
+11.83
+7.41 13.72
+Qκ2
+-11.83
+-2.63 -6.92
+Coarse k and q grids
+123
+123
+123
+Fine k and q electron grid
+1003
+1803
+1003
+Fine k and q hole grid
+1003
+1003
+1003
+For this expression to be amenable to Matthiessen’s rule,
+the scattering rates would need to be independent of the
+electronic state, say τ ph
+nk = τ ph and τ imp
+nk
+= τ imp. This is
+typically not the case in most semiconductors. Another
+special case where Matthiessen’s formula is meaningful
+occurs when one scattering mechanism dominates over
+the others. For example, in Eq. (27), when τ ph
+nk ≫ τ imp
+nk ,
+the expression reduces to the phonon-limited mobility. In
+this sense, Matthiessen’s rule constitutes a simple inter-
+polation formula between the limiting cases of phonon-
+limited and impurity-limited mobilities. We will analyze
+these aspects quantitatively in Sec. IV.
+III.
+COMPUTATIONAL METHODS
+All calculations are performed using the Quantum
+ESPRESSO materials simulation suite [53], the EPW
+code [38], and the Wannier90 code [54]. We employ the
+PBE exchange and correlation functional [55] and op-
+timized norm-conserving Vanderbilt (ONCV) pseudopo-
+tentials from the PseudoDojo repository [56, 57].
+For
+consistency with previous work, we use the experimental
+lattice constant of Si, SiC, and GaP at room temperature,
+and the plane-wave kinetic energy cutoff and quadrupole
+tensors reported in Ref. [58]. We include spin-orbit cou-
+pling for the valence bands only, to capture the splitting
+of the valence band top. Key calculation parameters are
+summarized in Tab. I.
+We calculate effective mass tensors by finite differences,
+using a wavevector increment of 0.01 × 2π/a, where a
+is the lattice constant reported in Tab. I. The dynam-
+ical matrix, the variations of the self-consistent poten-
+tial, and the vibrational eigenfrequencies and eigenmodes
+are calculated using a square convergence threshold of
+10−16 Ry2. This threshold refers to the change of the
+potential variation between two successive iterations, av-
+eraged over the unit cell.
+Electron energies, phonon
+frequencies, and electron-phonon matrix elements are
+initially computed on a coarse wavevector mesh using
+the EPW code. The electron Hamiltonian, the dynam-
+ical matrix, and the electron-phonon matrix elements
+are then interpolated onto fine Brillouin zone grids us-
+ing Wannier-Fourier interpolation [33, 34]. Long-range
+dipole and quadrupole corrections are employed for im-
+proved interpolation of the electron-phonon matrix ele-
+ments [47–49, 58, 59].
+To compute carrier mobilities, only states within a nar-
+row energy window of the band extrema are necessary.
+We find that, for the range of temperatures considered in
+this work (up to 500 K), a window of 400 meV is sufficient
+to obtain converged electron mobilities, and a window
+of 300 meV is sufficient for hole mobilities. At 300 K,
+converged results can be obtained by using a 200 meV
+window for both electrons and holes.
+To evaluate the overlap matrices Bmn,G(k, q) required
+in Eq. (17) in the fine Brillouin zone grid, we follow the
+procedure of Ref. [47] and approximate them as:
+Bmn,G(k, q) ≈
+�
+U(k + q)U †(k)
+�
+mn ,
+(28)
+where the unitary matrix Umn(k) is the diagonalizer of
+the interpolated Hamiltonian into the wavevector k of
+the fine grid. This approximation is motivated by the
+fact that the carrier-impurity matrix element in Eq. (17)
+is strongly peaked at q + G = 0.
+The Dirac delta functions appearing in Eqs. (4) and
+(17) are computed using Gaussian functions with a small
+broadening parameter. The results are sensitive to the
+choice of this parameter, therefore we accelerate the con-
+vergence by employing adaptive smearing. The proce-
+dure for the adaptive smearing of the carrier-phonon scat-
+tering rate, which involves a so-called type-III integral,
+is discussed in Refs. [6, 58, 60]. The calculation of the
+carrier-impurity scattering rates involves instead a type-
+II integral of the form:
+III
+nk =
+�
+m
+�
+dq
+ΩBZ
+fmn(k, q) δ(ϵmk+q − ϵnk),
+(29)
+where ΩBZ is the volume of the Brillouin zone. In this
+case, adaptive broadening can be achieved by using a
+state-dependent width σmk+q. We follow the procedure
+by Ref. [60], which gives:
+σmk+q = α
+3
+3
+�
+i=1
+vmk+q · bi
+Ni
+,
+(30)
+where vmk+q is the band velocity, bi is a primitive vector
+of the reciprocal lattice, and Ni denotes the number of
+k-points along the direction of bi. The coefficient α is a
+tunable parameter.
+Previous work has used α = 0.29
+for electron-phonon scattering rates [6, 58].
+We have
+performed a detailed converged test by comparing fixed-
+smearing and variable-smearing calculations, and found
+that values α = 0.1-0.3 provide similar results. For sim-
+plicity, in this work we use α = 0.29 as in previous work.
+In principle we could perform calculations of carrier
+mobilities by setting the impurity concentration and the
+carrier concentration separately. This would be required,
+for example, for the investigation of compensation dop-
+ing of semiconductors. To keep our results are general
+
+7
+as possible, in this work we choose to focus on the sim-
+pler scenario where each impurity creates one free car-
+rier, therefore we set the carrier density to be equal to
+the impurity concentration. We do not consider carrier
+freeze-out at low temperature, since this would require
+the knowledge of defect energy levels.
+In our calcula-
+tions, the role of the carrier concentration is mainly to
+modulate the effective Thomas-Fermi screening wavevec-
+tor in Eq. (22).
+IV.
+RESULTS AND DISCUSSION
+A.
+Electronic structure
+Given the importance of effective masses in mobility
+calculations, in this section we review briefly the band
+structures and effective masses of Si, SiC, and GaP. Ta-
+ble II shows our calculated directional effective masses.
+Hole masses are given for the heavy-hole (hh) band, light
+hole (lh) band, and the spin-orbit split-off (so) band. The
+longitudinal (∥) and transverse (⊥) electron masses cor-
+respond to the principal axes of the ellipsoidal conduction
+band extrema.
+In Tab. II we see that the light hole and split-off
+hole masses are fairly isotropic for all compounds con-
+sidered in this work. For the heavy hole masses, the Γ-X
+direction ([100] crystallographic direction) exhibits the
+lightest masses, whereas considerably heavier masses are
+found along the Γ-K ([110]) and Γ-L ([111]) directions.
+Similarly, in all compounds considered here the longitu-
+dinal electron masses are considerably heavier than the
+corresponding transverse masses, as expected. SiC ex-
+hibits the heaviest hole masses among SiC, GaP, and Si;
+while GaP exhibits the heaviest electron masses.
+Our calculated effective masses are in good agreement
+with previous calculations at the DFT level [10] as well as
+previous calculations at the GW level [10]. When com-
+paring to experimental data, we see from Tab. II that
+our electron effective masses are within 10% of the corre-
+sponding experimental values, which is remarkable con-
+sidering that we are using DFT/PBE.
+In the case of the hole masses, our calculations are also
+in good agreement with experiments. Here we emphasize
+that the experimental values usually quoted are not the
+effective masses, but the cyclotron masses, which depend
+on the direction of the magnetic field and are reported
+in Tab. II. These cyclotron masses correspond to aver-
+ages of the directional masses and cannot be compared
+directly to DFT calculations. To extract the correct di-
+rectional effective masses, in the case of silicon we used
+the Dresselhaus k · p model which was fitted to experi-
+mental cyclotron data. In this model the heavy hole and
+TABLE II.
+Calculated band effective masses, band gaps,
+high-frequency and static dielectric constants of Si, 3C-SiC,
+and GaP. All calculations performed within DFT/PBE. Ex-
+perimental data are from (a) [61] and [62], (b) [63], (c) [64],
+(d) [65], (e) [66], (f) [67], (g) [68], (h) [69], (i) [70], (j) [71], (k)
+[72], (l) [73]. All masses are give in units of the electron mass.
+The band gaps are in eV. The lines tagged “Dresselhaus” refer
+to the effective masses obtained from the Dresselhaus model
+fitted to experimental cyclotron data, from Ref. [61].
+This work
+Si
+SiC
+GaP
+Γ-X
+0.260
+0.592 0.374
+m∗
+hh
+Γ-K
+0.550
+1.412 0.837
+Γ-L
+0.655
+1.646 1.091
+Γ-X
+0.189
+0.423 0.143
+m∗
+lh
+Γ-K
+0.143
+0.328 0.125
+Γ-L
+0.134
+0.309 0.117
+Γ-X
+0.225
+0.490 0.213
+m∗
+so
+Γ-K
+0.223
+0.472 0.217
+Γ-L
+0.214
+0.436 0.206
+m∗
+e,∥
+0.959
+0.672 1.069
+m∗
+e,⊥
+0.196
+0.230 0.232
+Eg
+0.554
+1.359 1.566
+ε∞
+13.00
+6.93 10.53
+ε0
+13.00
+10.23 12.57
+Experiment
+Si
+SiC
+GaP
+B along [001]
+0.46a
+m∗
+hh
+B along [110]
+0.53a
+B along [111]
+0.56a
+0.54c
+Dresselhaus Γ-X 0.40
+Dresselhaus Γ-K 0.56
+Dresselhaus Γ-L 0.62
+B along [001]
+0.171a 0.45b
+m∗
+lh
+B along [110]
+0.163a
+B along [111]
+0.160a
+0.16c
+Dresselhaus Γ-X 0.18
+Dresselhaus Γ-K 0.16
+Dresselhaus Γ-L 0.15
+m∗
+e,∥
+0.97a
+0.68d 1.15c, 2.0k
+m∗
+e,⊥
+0.19a
+0.25d 0.21c, 0.25k
+Eg
+1.13f
+2.42g 2.26h
+ε∞
+11.7i
+6.52j 9.11j
+ε0
+11.7i
+9.72j 11.1j
+light hole masses are parameterized as:
+ϵhh(k) =Ak2 + [B2k4 + C2(k2
+xk2
+y + k2
+yk2
+z + k2
+zk2
+x)]1/2,
+(31)
+ϵlh(k) =Ak2 − [B2k4 + C2(k2
+xk2
+y + k2
+yk2
+z + k2
+zk2
+x)]1/2,
+(32)
+where k
+=
+|k| and the coefficients A, B, and C
+are −4.1 ¯h2/2me, −1.6 ¯h2/2me, and 3.3 ¯h2/2me, respec-
+tively [61]. From this parameterization we obtained the
+effective masses reported in Tab. II under the keyword
+“Dresselhaus”. From this table we can see that, in the
+
+8
+case of silicon, the light hole and heavy hole masses are
+close to our calculated results, with the exception of the
+Γ − X heavy-hole effective mass which is 65% of the ex-
+perimental value[10].
+Our calculated dielectric constants overestimate the
+experimental values by 15% at most, as expected from
+the underestimation of the band gaps [70, 71].
+In
+Sec. IV E we discuss how one can improve the calculated
+mobilities by introducing a posteriori corrections to the
+theoretical effective masses and dielectric constants.
+B.
+Carrier mobilities
+1.
+Silicon
+Figure 1 shows a comparison between our calculated
+mobilities of silicon and available experimental data, as
+a function of temperature and impurity concentration.
+The mobilities without carrier-impurity scattering [black
+lines in panels (a) and (b)] decrease rapidly with tem-
+perature, as expected. We find temperature slopes (the
+β in µ ∼ T β) of −2.1 for electrons and −2.4 for holes,
+in agreement with previous work [10, 58]. As we include
+carrier-impurity scattering, the room-temperature elec-
+tron mobility of silicon reduces from 1381 cm2/Vs to
+1153 cm2/Vs at 1.75×1016 cm−3 [blue line in panel (a)]
+and to 812 cm2/Vs at 1.3×1017 cm−3 [red line in panel
+(a)].
+Similarly, the room-temperature hole mobility of
+silicon decreases from 600 cm2/Vs in the absence of im-
+purities to 517 cm2/Vs for an impurity concentration of
+2.4×1016 cm−3[blue line in panel (b)], and to 359 cm2/Vs
+at the impurity concentration of 2.0×1017 cm2/Vs [red
+line in panel (b)].
+Our calculations for the temperature-dependent elec-
+tron and hole mobilities show that a single power law be-
+comes inadequate in the presence of impurity scattering.
+This is also seen in the experimental data from Refs. [74–
+77], which are shown as open circles in Fig. 1. We note
+that our calculations are in good agreement with the ex-
+periments over a broad temperature range. The agree-
+ment worsens slightly at low temperature, where carrier-
+impurity scattering dominates. This effect likely relates
+to the fact that in our calculations all donors and ac-
+ceptors are assumed to be fully ionized at all tempera-
+tures; as a result of this approximation, we are neglecting
+carrier freeze-out and hence we are likely overestimating
+the impurity concentration at low temperature. In Ap-
+pendix A we show that, by taking into account the the
+effects of partial impurity ionization, the agreement with
+experiments improves at low temperature and high im-
+purity concentration.
+Panel (c) of Fig. 1 shows the room temperature elec-
+tron mobility of silicon, as a function of impurity con-
+centration. The electron mobility is relatively insensitive
+to the impurity concentration up to 1016 cm−3. A steep
+decrease in the electron mobility is seen as we approach
+a doping density of 1017 cm−3.
+Up to this concentra-
+tion, our calculations (blue line) are in excellent agree-
+ment with experimental data (open black circles). Above
+1018 cm−3, while the agreement with experiment is still
+good, we tend to slightly overestimate the measured elec-
+tron mobility. This is likely due to two effects: (i) our
+formalism does not take into account multiple scattering
+events that become important at high impurity concen-
+tration, and (ii) our calculations do not include scattering
+by free-carrier plasmons, which dominate the mobility at
+high carrier density, as shown in Refs. [12, 23]. A similar
+overestimation was observed in Ref. [14].
+Panel (d) of Fig. 1 shows the room temperature hole
+mobility of silicon as a function of impurity concentra-
+tion. As for the electrons, we find generally good agree-
+ment between calculations (blue line) and experiments
+(open black circles) throughout the doping range. We
+emphasize that the vertical scales in panels (c) and (d)
+are different, and that the theory/experiment deviation
+at high impurity concentration is similar in both panels
+in absolute terms. At low impurity concentration, our
+calculations slightly overestimate the experimental data.
+This effect can be ascribed to the fact that our light hole
+effective masses are smaller than in experiments.
+2.
+Silicon carbide
+In Fig. 2 we show our calculated mobilities of 3C-SiC
+as a function of temperature and impurity concentration,
+and we compare to experimental data from Refs. [24, 78–
+84]. In the case of 3C-SiC, the comparison with exper-
+iments is complicated by the high concentration of line
+defects that nucleate at lattice-mismatched growth sub-
+strates such as Si or 6H-SiC [83, 85], which makes it diffi-
+cult to obtain data for defect-free samples. Furthermore,
+most experimental data are for co-doped samples, for
+which the impurity and carrier concentrations are more
+difficult to estimate.
+In the absence of impurity scattering [black line in
+panel (a)], the low electron effective mass of SiC leads
+to very high theoretical mobilities, up to 33000 cm2/Vs
+at 100 K and up to 2000 cm2/Vs at room temperature.
+These high mobilities are in agreement with previous the-
+oretical results [58]. In this case, we calculate an electron
+temperature exponent β = −2.9.
+In panel (a) of Fig. 2 we compare our calculations (blue
+line) with the data reported in Ref. [78] (red open cir-
+cles).
+In that work, they synthesized 3C-SiC with n-
+type impurity density of 5.0×1016 cm−3, and obtained
+electron mobilities at 100 K and 300 K of 2040 cm2/Vs
+and 584 cm2/Vs, respectively. In our calculations, when
+we consider the same impurity concentration, we find
+2773 cm2/Vs and 1369 cm2/Vs at 100 K and 300 K,
+respectively; therefore we overestimate the experimental
+data by a factor of 30%-230%.
+In panel (b) of Fig. 2 we show our calculated hole mo-
+bility of 3C-SiC as a function of temperature.
+In the
+absence of impurities (black line), the mobility decreases
+
+9
+with a temperature exponent β = −2.1.
+In this case
+we could not find experimental data for uncompensated
+samples to compare with. Upon including impurity scat-
+tering with an impurity concentration of 1018 cm−3, we
+find a significant reduction of the mobility at low tem-
+perature (blue line), from 1373 cm2/Vs to 148 cm2/Vs.
+At 300 K, the mobility is reduced from 165 cm2/Vs with-
+out impurities to 81 cm2/Vs, in good agreement with the
+measured value of 50 cm2/Vs reported in Ref. [84].
+Panels (c) and (d) of Fig. 2 show the room temper-
+ature electron and hole mobilities as a function of im-
+purity concentration, respectively. The electron mobility
+calculated (blue line) at low ionized donor concentration
+(1014 cm−3) is 2048 cm2/Vs, and significantly overesti-
+mates the measured value 1000 cm2/Vs by Ref. [79] (open
+black symbols). However, our calculations get closer to
+experimental data in the range of concentrations above
+1018 cm−3 [24, 80, 86].
+The hole mobility of 3C-SiC is significantly lower than
+the electron mobility, as expected from much heavier hole
+masses shown in Tab II. Our calculations (blue line) at
+low doping yield a mobility of 164 cm2/Vs, to be com-
+pared to 220 cm2/Vs measured in p-channel 3C-SiC de-
+vices [82] (open black symbols). We note that the vertical
+scales in panels (c) and (d) differ, and that our calculated
+hole mobilities are in better agreement with experiment
+in relative terms. In particular, our data for the hole mo-
+bility fall right in the middle of the experimental trend
+shown in panel (d).
+3.
+Gallium phosphide
+Figure 3 shows our mobility calculations for GaP and a
+comparison with experimental data. In panel (a) we have
+the calculated electron mobilities as a function of temper-
+ature. In the absence of impurities, the calculated elec-
+tron mobility (black line) decreases with a temperature
+exponent β = −2.2; the calculated mobilities at 100 K
+and 300 K are 4293 cm2/Vs and 328 cm2/Vs, respec-
+tively. Upon including the effect of impurity scattering
+(blue line), the mobility decreases significantly, reach-
+ing 157 cm2/Vs at room temperature for an impurity
+concentration of 2.5×1018 cm−3. This value is in good
+agreement with the measured mobility of 100 cm2/Vs by
+Ref. [87] (blue open circles). We note that the electron
+mobility of GaP is significantly lower than in silicon, de-
+spite the electron effective masses being comparable. In
+Sec. IV C we show that this effect arises from the addi-
+tional polar phonon scattering that electrons experience
+in GaP, which is absent in silicon.
+Panel (b) of Fig. 3 shows the calculated phonon-limited
+hole mobility (black line), the mobility calculated by in-
+cluding impurity scattering (blue line), and experimen-
+tal data (open red circles).
+The phonon-limited hole
+mobility decreases with temperature with an exponent
+β = −2.5. The calculated mobilities in the absence of
+impurities are 5096 cm2/Vs and 252 cm2/Vs at 100 K
+and 300 K, respectively. Upon including impurity scat-
+tering with a concentration of 2×1018 cm−3, the mobil-
+ity at room temperature decreases to 124 cm2/Vs, in
+good agreement with the measured value of 90 cm2/Vs
+by Ref. [87].
+Panel (c) of Fig. 3 shows the room temperature elec-
+tron mobility of GaP as a function of impurity concentra-
+tion. In the absence of impurity scattering, we calculate
+a mobility of 328 cm2/Vs(blue line), which compares well
+with the maximum value 258 cm2/Vs measured in ultra-
+pure samples in Ref. [88] (open black symbols). In the
+intermediate doping regime, our calculated electron mo-
+bilities overestimate the experimental data by a factor of
+two [87–90], but the agreement improves at high doping
+levels.
+Figure 3(d) shows the room temperature hole mobil-
+ity of GaP as a function of impurity concentration. The
+calculated hole mobility is 269 cm2/Vs at low impurity
+concentration, and decreases to 94 cm2/Vs at a concen-
+tration of 1019 cm−3 (blue line).
+Our calculations are
+within a factor of two from the highest measured hole
+mobilities across the same doping range [87, 91, 92] (open
+black symbols). We note that electron and hole mobilities
+in GaP are very similar across a wide range of tempera-
+tures and impurity concentrations (both in experiments
+and in our calculations), therefore GaP is an ambipo-
+lar semiconductor with well-balanced electron and hole
+transport.
+C.
+Carrier scattering rates
+In this section we analyze and compare the scattering
+rates resulting from carrier-phonon and carrier-impurity
+processes in Si, SiC, and GaP. The Brooks-Herring model
+for carrier-impurity scattering [21], which is based on the
+parabolic band approximation, predicts a scattering rate
+that scales as ϵ−3/2, where ϵ is the electron eigenvalue
+referred to the band extremum. This trend is a result of
+two competing effects: as the energy of the initial state
+increases above the band bottom, the scattering phase
+space increases as ϵ1/2, while at the same time the square
+modulus of the carrier-impurity matrix element given in
+Eq. (17) decreases as 1/q4, which is of the order of ϵ−2.
+This simple trend is opposite to what is expected from
+non-polar optical scattering and acoustic phonon scatter-
+ing, which tend to increase with energy.
+Figure 4 shows the scattering rates τ −1
+nk of holes and
+electrons in Si [panels (a) and (b)], SiC [panels (c) and
+(d)], and GaP [panels (e) and (f)]. For consistency, we
+set the impurity concentration to 1017 cm−3 in all cases,
+which is in the middle of the range considered in Figs. 1-3,
+and the temperature to 300 K. In line with the above dis-
+cussion, the carrier-impurity scattering rates decrease as
+we move away from the band extrema, while the carrier-
+phonon scattering rates increase. In the two polar semi-
+conductors that we are considering, SiC and GaP, we
+also see a sudden jump in the carrier-phonon scatter-
+
+10
+ing rates. This effect happens when the carrier energy
+reaches the threshold for the emission of a longitudinal
+optical phonon, thereby activating polar phonon scatter-
+ing [47].
+Panels (a) and (b) of Fig. 4 show that, in the case of
+silicon, the carrier-ionized-impurity scattering rates near
+the band edges are an order of magnitude higher than
+carrier-phonon rates (for an impurity concentration of
+1017 cm−3). The additional scattering by carriers causes
+a reduction of the mobility by ∼ 30% for both electrons
+and holes, indicating that impurity scattering is a sig-
+nificant effect at this impurity concentration. The rise
+of the carrier-electron scattering rates at energies around
+150 meV that can be seen in panel (b) correspond to
+interband scattering between the two lowest conduction
+bands.
+Panels (c) and (d) of Fig. 4 show the scattering rates
+in SiC. Unlike in silicon, here the electron and hole scat-
+tering rates differ considerably.
+In the case of holes,
+the carrier-phonon and carrier-impurity scattering rates
+are comparable in magnitude near the band edge, while
+in the case of electrons the carrier-impurity scattering
+dominates.
+This difference is reflected in the calcu-
+lated mobilities, where carrier-impurity scattering re-
+duces the phonon-limited mobility of holes by ∼ 20%
+and of electrons by ∼ 50% (for the impurity concentra-
+tion 1017 cm−3).
+Data for GaP are shown in panels (e) and (f) of Fig. 4.
+In this case the carrier-phonon scattering rates are com-
+parable to the carrier-impurity scattering rates. Accord-
+ingly, the mobilities are reduced by ∼10% from their val-
+ues without impurity scattering.
+D.
+Deviations from Matthiessen’s Rule
+In Sec. IV D we discussed how Matthiessen’s rule is for-
+mally justified only when the scattering rates are state-
+independent constants, or when one scattering mecha-
+nism dominates over all other mechanisms. To place that
+reasoning on a quantitative footing, in Fig. 5 we explicitly
+assess the predictive accuracy of the Matthiessen rule.
+For this test, we compute the mobilities of Si, SiC,
+and GaP by considering the following four scenarios:
+(i) phonon-limited mobility µph (i.e., without including
+carrier-impurity scattering); (ii) impurity-limited mobil-
+ity µimp (i.e., without including carrier-phonon scatter-
+ing); (iii) the mobility according to Matthiessen’s rule,
+as obtained by combining (i) and (ii) using 1/µM =
+1/µph + 1/µimp; (iv) the mobility µ calculated by includ-
+ing both carrier-phonon scattering and carrier-impurity
+scattering using the aiBTE.
+In panels (a), (c), and (e) we see this comparison for
+Si, SiC, and GaP, respectively, as a function of temper-
+ature. As expected, in all cases the phonon-limited mo-
+bilities (black lines) decrease with temperature while the
+impurity-limited mobilities (red lines) do increase. Their
+combination results into the characteristic smooth peak
+which is best seen in the cases of Si and SiC. In these pan-
+els, the dashed blue lines are from Matthiessen’s rule,
+and the solid blue lines are the complete aiBTE solu-
+tions. We see that the Matthiessen rule tends to overes-
+timate the aiBTE mobility, and the deviation is partic-
+ularly pronounced when the phonon and impurity con-
+tributions to the mobility reduction are comparable. To
+quantify the deviation between aiBTE calculations and
+the Matthiessen results, in panels (b), (d), and (f) of
+Fig. 5 we show the ratio between the two values, as a
+function of temperature.
+In all cases we see that the
+use of Matthiessen’s rule leads to an overestimation of
+the mobilities by up to 50%, which is significant in the
+context of predictive calculations of transport properties.
+More importantly, for the compounds considered in this
+work (Si, SiC, and GaP), the use of Matthiessen’s rule
+would worsen the agreement between calculated mobili-
+ties and experimental data.
+Based on these findings, we caution against the use
+of Matthiessen’s rule in future ab initio calculations of
+carrier mobilities.
+E.
+Improving the predictive power of the aiBTE
+In this section we investigate simple approaches to im-
+prove the predictive accuracy of the aiBTE by overcom-
+ing two standard limitations of DFT.
+The first limitation is that the DFT band gap prob-
+lem typically leads to an overestimation of the dielectric
+screening. As a result, both carrier-phonon and carrier-
+impurity matrix elements tend to be underestimated in
+DFT [35, 93], and mobilities tend to be overestimated. In
+Ref. [58] it was shown that, for a set of ten semiconduc-
+tors, this effect leads to mobilities which can overestimate
+experimental data by as much as a factor of two. To miti-
+gate this effect, we investigate a simple scaling correction
+to the matrix elements as follow:
+gcorr
+mnν(k, q) = εDFT
+εexp
+gDFT
+mnν (k, q),
+(33)
+where ϵDFT is our calculated value, and ϵexp is the exper-
+imental value. We use the high-frequency dielectric con-
+stant for the carrier-phonon matrix elements, as it was
+done in Ref. [10], and the static dielectric constants for
+the carrier-impurity matrix elements [see Eq. (9)]. This
+approach is meaningful for the systems considered in this
+work, because the majority of scattering processes oc-
+cur near the band extrema, and therefore involve small
+scattering wavevectors q, thus justifying the re-scaling of
+screening at long wavelength only.
+The second limitation of DFT calculations lies in the
+inaccurate curvature of the bands, which is also linked
+to the band gap problem, leading to slightly inaccurate
+carrier effective masses. This limitation could be over-
+come by performing GW calculations, but in this work
+we investigate a simpler mass scaling.
+According to Drude’s formula, carrier mobilities are
+inversely proportional to the effective masses. Based on
+
+11
+this observation, we consider the following scaling correc-
+tion, which is directly applied to the calculated mobility:
+µcorr = m∗
+DFT
+m∗exp
+µDFT,
+(34)
+where all masses are isotropic averages.
+The three compounds considered in this work all have
+ellipsoidal conduction band extrema, therefore we can
+evaluate the average isotropic mass as follows:
+m∗ = 3(1/m∗
+∥ + 2/m∗
+⊥)−1.
+(35)
+Evaluating the average hole mass is more complicated
+owing to the band degeneracy at Γ and the fact that ex-
+perimental data usually are reported for a given magnetic
+field direction as opposed to a crystallographic direction
+(see Sec. IV A). In the case of silicon, we evaluate the av-
+erage mass using the values extracted from Dresselhaus’
+model (see Sec. IV A). After this averaging procedure,
+the hole mass is calculated following Ref. [58]:
+m∗ = m∗,5/2
+hh
++ m∗,5/2
+lh
+m∗,3/2
+hh
++ m∗,3/2
+lh
+,
+(36)
+where all quantities on the r.h.s. are spherical averages in
+k-space. In the case of SiC and GaP we are not aware of
+a parametrization similar to Dresselhaus’, therefore we
+do not investigate mass corrections in these cases.
+The carrier mobilities obtained by applying the above
+corrections are shown in Fig. 6. In all cases we use the
+experimental dielectric constants reported in Tab. II.
+Panels (a) and (b) show our results for silicon. The
+screening correction to the electron mobilities of Si re-
+duces the calculated value at low impurity concentra-
+tion from 1381 cm2/Vs to 1133 cm2/Vs. This reduction
+causes an underestimation of the experimental value by
+approximately 20%. At higher impurity concentration,
+the corrected mobility agrees again well with experimen-
+tal results. The corrections to the electron effective mass
+of Si are minor and do not affect the mobility. In the
+case of holes, the screening and mass corrections improve
+considerably the agreement between theory and experi-
+ment (our calculated average hole mass is 0.43 me while
+the experimental value is 0.48 me). In fact, we obtain a
+hole mobility of 463 cm2/Vs at low impurity concentra-
+tion, which is within the measured value between 450 and
+500 cm2/Vs[76, 77]. The improvement is also noticeable
+at higher impurity concentration.
+Results for SiC are shown in panels (c) and (d) of
+Fig. 6.
+In this case, we find that screening and mass
+corrections do not significantly improve the agreement
+with experiments at low impurity concentration. In par-
+ticular, the screening correction reduces the electron mo-
+bility from 2047 cm2/Vs to 1815 cm2/Vs, and the mass
+correction further reduces this value to 1688 cm2/Vs. De-
+spite these corrections, the calculated electron mobility
+remains too high by about a factor of two. It is possible
+that additional scattering mechanisms such as disloca-
+tions could contribute to reduce this difference. In the
+case of the hole mobility, the screening correction reduces
+the calculated value at low impurity concentration from
+164 cm2/Vs to 148 cm2/Vs, which is not significant when
+compared to the large spread of experimental values [81–
+84].
+The screening correction appears to be successful in
+the case of GaP, as seen in panels (e) and (f) of Fig. 6.
+The electron mobility at low impurity concentration re-
+duces from 326 cm2/Vs to 243 cm2/Vs upon applying
+the screening correction. This value is in better agree-
+ment with the experimental data. Improved agreement
+with experiments is also found at higher impurity con-
+centration. The correction to the electron effective mass
+of GaP is small, and as a result the change in mobility is
+not significant. The screening correction for holes brings
+the calculated data closer to the experiments. In partic-
+ular, at low impurity concentration the hole mobility is
+reduced from 269 cm2/Vs to 226 cm2/Vs.
+The key takeaway from this analysis is that the screen-
+ing correction to the scattering matrix elements improves
+the agreement between theory and experiment for the
+compounds considered in this work. Based on the above
+observations, we suggest that screening and mass cor-
+rections could be used for the purpose of uncertainty
+quantification in future ab initio calculations of trans-
+port properties.
+V.
+CONCLUSIONS
+In this work we have demonstrated non-empirical cal-
+culations of carrier mobilities in semiconductors using the
+ab initio Boltzmann transport equations, including car-
+rier scattering by phonons and by ionized impurities. To
+this end, we developed an ab initio formalism to incor-
+porate ionized-impurity scattering within the transport
+workflow based on Wannier-Fourier interpolation and im-
+plemented in the EPW code.
+We described ionized impurities by randomly dis-
+tributed Coulomb scatters, and we obtained the carrier
+relaxation time by using the Kohn-Luttinger ensemble
+averaging procedure. We also incorporated the screen-
+ing of the impurity potential by free-carriers, within a
+parameter-free effective Thomas-Fermi model.
+We validated our approach by performing an extensive
+set of calculations of the electron and hole mobilities of
+three common semiconductors, namely Si, 3C-SiC, and
+GaP. In all cases we find a reasonably good agreement
+with experimental data, except possibly for the electron
+mobility in SiC which is probably reduced by additional
+scattering at line defects in real samples. Our calcula-
+tions follow closely the experimental data both as a func-
+tion of temperature (at fixed impurity concentration) and
+as a function of impurity concentration (at fixed temper-
+ature).
+Impurity scattering is found to dominate over phonon
+scattering at high impurity concentration and at low tem-
+perature. In the former case, the thermal distribution
+
+12
+function of the carrier is peaked near the band edges,
+therefore small-q elastic scattering by impurities dom-
+inates.
+In the latter case, the phonon population be-
+comes negligible at low temperature, therefore impurities
+remain the only active scattering channel. These trends
+are fully consistent with the general understanding of car-
+rier transport in semiconductors [94]. We also found that
+the energy-dependent carrier scattering rates are strongly
+dependent on the detailed mechanisms at play in each
+compound, and vary significantly over the energy range
+of relevance for transport phenomena. This finding un-
+derlines the importance of detailed ab initio calculations
+to achieve predictive accuracy in the description of trans-
+port phenomena of real materials.
+In the presence of multiple scattering channels, it
+is common to analyze mobility data using the clas-
+sic Matthiessen rule.
+However, by directly comparing
+aiBTE calculations including both phonon and impurity
+scattering with estimates based on Matthiessen’s rule, we
+found that the latter lead to inaccurate results, with de-
+viations of up to 50% with respect to aiBTE calculations.
+This finding indicates that Matthiessen’s rule should not
+be employed in predictive calculations of transport prop-
+erties.
+Lastly, we investigated simple corrections to DFT cal-
+culations of carrier mobilities, by scaling the calculated
+dielectric screening and the effective masses via their
+corresponding experimental values. We found that the
+screening correction generally improves agreement with
+experiments.
+Overall, our present approach offers a powerful tool for
+calculating transport properties in a variety of semicon-
+ducting materials of immediate interest, as well as for
+screening new putative semiconductors in the context of
+materials discovery.
+Several improvements upon this work are possible. For
+one, we do not account for neutral impurity scatter-
+ing.
+This additional channel could be added by gen-
+eralizing our monopole model to account for dipoles
+and quadrupoles, following similar work performed in
+the context of electron-phonon interactions [47–49, 59].
+Generalizations to the case of two-dimensional materi-
+als should also be possible, for example by following the
+related generalization of the Fr¨ohlich matrix element to
+two-dimensional systems [95, 96]. At high impurity con-
+centration one should also account for carrier-plasmon
+scattering, for example as discussed in Ref. [12]. And of
+course, any improvement in the DFT band structures and
+electron-phonon matrix elements would be highly bene-
+ficial to further enhance the predictive power of these
+calculations [93]. We hope that this study will stimulate
+further work along these and other promising directions.
+ACKNOWLEDGMENTS
+This research is primarily supported by the Compu-
+tational Materials Sciences Program funded by the U.S.
+Department of Energy, Office of Science, Basic Energy
+Sciences, under Award No.
+DE-SC0020129.
+This re-
+search used resources of the National Energy Research
+Scientific Computing Center, a DOE Office of Science
+User Facility supported by the Office of Science of the
+U.S. Department of Energy under Contract No.
+DE-
+AC02-05CH11231. The authors acknowledge the Texas
+Advanced Computing Center (TACC) at The University
+of Texas at Austin for providing HPC resources that have
+contributed to the research results reported within this
+paper: https://www.tacc.utexas.edu.
+Appendix A: Incomplete ionization of dopant
+In all calculations presented in this work, we have con-
+sidered that the carrier density coincides with the impu-
+rity concentration. The implicit assumption underlying
+this choice is that all impurities are ionized at all temper-
+atures. This is obviously a simplification, since the frac-
+tion of ionized impurities depends on the defect energy,
+the quasi Fermi level of the system, and the temperature.
+These aspects have already been discussed in the case of
+silicon in Ref. [14].
+In this Appendix we analyze the effect of incomplete
+ionization for the case of silicon. To estimate the fraction
+f of ionized impurities at a given temperature, we use the
+Fermi-Dirac distribution evaluated at the defect level of
+the impurity atom, ϵd [52]:
+f =
+1
+N uc
+imp
+�
+N uc
+imp
+�
+n,k
+1
+e(ϵnk−ϵd)/kBT + 1.
+(A1)
+Here, N uc
+imp is the number of impurities per unit cell. This
+fraction vanishes when the temperature goes to zero, and
+approaches unity at high temperature.
+In Fig. 7 we show the influence of incomplete impu-
+rity ionization on the electron mobility of Si. In these
+calculations, we used ϵd =45 meV as measured from the
+conduction band bottom [97]. By comparing these curves
+with Fig. 1(a), we see that the effect of incomplete ion-
+ization improves the agreement with experiments at low
+temperature and high doping (red curves in both figures).
+This is precisely the range where carrier-impurity scatter-
+ing tends to dominate over phonon scattering, therefore
+it is important to have a precise determination of the
+impurity concentration in this range. A more systematic
+assessment of these effects will require a broader database
+of experimental mobilities to compare with.
+
+13
+FIG. 1. Comparison between our calculated carrier mobilities in Si with experimental data. (a) Electron mobility of Si as a
+function of temperature. The black line and symbols are for low impurity concentration (no impurities in the calculations;
+< 1012 cm−3 impurities in the experiment); the blue line and symbols are for an impurity concentration of 1.75×1016 cm−3; the
+red line and symbols are for a concentration of 1.3×1017 cm−3. Filled disks are calculated values, open circles are experimental
+data from Ref. [75] (black) and [74] (blue and red). (b) Hole mobility of Si as a function of temperature. The black line and
+symbols are for low impurity concentration (no impurities in the calculations; 1012 cm−3 impurities in the experiment); the
+blue line and symbols are for an impurity concentration of 2.4×1016 cm−3; the red line and symbols are for a concentration
+of 2.0·1017 cm−3. Filled disks are calculated values, open circles are experimental data from Ref. [98] (black) and [74] (blue
+and red). (c) Room temperature electron mobility of Si as a function of impurity concentration. Blue line and filled disks are
+calculated data, open black circles are experimental data from Ref. [76].
+
+104
+electron mobility (cm?Ns)
+1500
+1250
+1000
+750
+103
+500
+250
+(a)
+(c)
+0
+104
+600
+500
+400
+103
+300
+O
+200
+100
+000
+(b)
+(d)
+102,
+0
+15
+100
+200
+300
+400
+500
+4
+16
+17
+18
+19
+10
+10
+10
+10
+10
+10
+temperature (K)
+ionized impurity density (cm-3)14
+FIG. 2. Comparison between our calculated carrier mobilities in 3C-SiC with experimental data. (a) Electron mobility of Si
+as a function of temperature. The black line and symbols are phonon-limited mobilities (no impurities in the calculations); the
+blue line and symbols are for an impurity concentration of 5×1016 cm−3. Filled disks are calculated values, open circles are
+experimental data from Ref. [78]. (b) Hole mobility of 3C-SiC as a function of temperature. The black line and symbols are
+phonon-limited mobilities (no impurities); the blue line and symbols are for an impurity concentration of 1018 cm−3. All data
+are calculated values. (c) Room temperature electron mobility of 3C-SiC as a function of impurity concentration. Blue line
+and filled disks are calculated data, open symbols are experimental data from Ref. [24], Ref. [79], Ref. [86], and Ref. [80]. (d)
+Room temperature hole mobility of 3C-SiC as a function of impurity concentration. Blue line and filled disks are calculated
+data, open symbols are experimental data from Ref. [81], Ref. [82], Ref. [83], and Ref. [84].
+
+2000
+(a)
+electron mobility (cm?Ns)
+104
+1500
+1000
+103.
+8
+500
+(c)
+0
+102
+250
+O
+(b)
+103
+hole mobility (cm?/Ns)
+200
+150
+102
+100
+50
+00
+(d)
+101
+0
+200
+300
+100
+400
+500
+14
+15
+16
+17
+18
+19
+10
+10
+10
+10
+10
+10
+temperature (K)
+lonized Dopant Density (cm-3)15
+FIG. 3. Comparison between our calculated carrier mobilities in GaP with experimental data. (a) Electron mobility of GaP
+as a function of temperature. The black line and symbols are phonon-limited mobilities (no impurities in the calculations);
+the blue line and symbols are for an impurity concentration of 2.5×1018 cm−3. Filled disks are calculated values, open circles
+are experimental data from Ref. [87]. (b) Hole mobility of GaP as a function of temperature. The black line and symbols
+are phonon-limited mobilities (no impurities); the blue line and symbols are for an impurity concentration of 2 × 1018 cm−3.
+Filled disks are calculated values, open circles are experimental data from Ref. [87]. (c) Room temperature electron mobility
+of GaP as a function of impurity concentration. Blue line and filled disks are calculated data, open symbols are experimental
+data from Ref. [87], Ref. [88], Ref. [90], and Ref. [89]. (d) Room temperature hole mobility of GaP as a function of impurity
+concentration. Blue line and filled disks are calculated data, open symbols are experimental data from Ref. [87], Ref. [91], and
+Ref. [92].
+
+300
+500
+250
+400
+electron mobility
+200
+300
+150-
+200
+100
+100
+50
+(a)
+(c)
+0
+0
+500
+250
+400
+200
+300
+150
+200
+100
+100
+000000
+50
+(b)
+(d)
+0
+0
+200
+400
+100
+300
+500
+14
+15
+16
+17
+18
+19
+10
+10
+10
+10
+10
+10
+temperature (K)
+ionized impurity density (cm-3)16
+FIG. 4.
+Calculated carrier scattering rates at 300 K, for an impurity concentration of 1017 cm−3. (a) Hole scattering rates
+in Si: carrier-phonon scattering rates (black disks) and carrier-impurity scattering rates (blue disks), as a function of energy
+referred to the valence band maximum (VBM). The dashed line and the shaded area represents the thermal distribution of
+carriers. (b) Electron scattering rates in Si: carrier-phonon scattering rates (black disks) and carrier-impurity scattering rates
+(blue disks), as a function of energy referred to the conduction band minimum (CBM). (c) and (d): same as (a) and (b), but
+for 3C-SiC. (e) and (f): same as (a) and (b), but for GaP.
+
+60
+60
+(a)
+(b)
+carrier-ph
+50
+50
+carrier-i Ni=1017 cm-3
+electrons
+40
+40
+les
+hol
+30
+30
+S
+20
+S 20
+10
+10
+60
+60
+(d)
+(C)
+50
+50
+(zHI)
+electrons
+es
+40
+40
+rate
+loy
+BC-Sic
+30
+30
+scattering
+-SiC
+20
+20
+3
+10
+10
+60
+60
+(e)
+(f)
+50
+50
+lectrons
+40
+40
+holes
+30
+30
+e
+R
+20
+20
+10
+10
+-0.20
+-0.05
+-0.15
+-0.10
+0.00
+0.00
+0.05
+0.10
+0.15
+0.20
+energy below VBM (eV)
+energy above CBM (eV)17
+FIG. 5.
+Comparison between mobility calculations performed using the aiBTE by including both carrier-phonon and carrier-
+impurity scattering, and mobilities obtained by using the Matthiessen’s rule. (a) Temperature-dependent electron mobility
+of Si. The black line and symbols indicate the phonon-limited mobility; the red line is the impurity-limited mobility, for an
+impurity concentration of 1.3 × 1017 cm−3; the dashed blue line is the mobility obtained from Matthiessen’s rule; the solid blue
+line is the aiBTE calculation including both phonons and impurities. (b) Ratio between the electron mobility of Si calculated
+using Matthiessen’s rule and the result of the aiBTE calculation with phonon and impurities, as a function of temperature. (c)
+and (d): Same as in (a), for for 3C-SiC with an impurity concentration of 2.5 × 1018 cm−3. (e) and (f): Same as in (a), for for
+3C-SiC with an impurity concentration of 5 × 1016 cm−3.
+
+(a)
+(b)
+1.5
+104
+1.4
+μaiBT
+S
+1.3
+103
+phonon only
+impurity only
+1.1
+Matthiessen
+phonon+impurity
+1.0
+(d)
+(c)
+1.5
+104
+μaiBTE
+1.4
+1.3
+latt.
+3
+103
+1.2
+3
+phonon only
+impurity only
+1.1
+Matthiessen
+phonon+impurity
+102
+1.0
+(e)
+(f)
+1.5
+μaiBTE
+103
+1.4
+P
+Gal
+1.3
+μMatt.
+1.2
+O- phonon only
+102
+impurity only
+1.1
+Matthiessen
+phonon+impurity
+1.0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+temperature (K)
+temperature (K)18
+FIG. 6.
+Comparison of correction schemes for improving the predictive accuracy of aiBTE calculations of mobilities. (a)
+Room-temperature electron mobility of Si, as a function of impurity concentration. Blue lines and disks indicate the uncorrected
+aiBTE results; green lines and disks indicate calculations with matrix elements corrected for screening; purple lines and disks
+are calculations corrected for the effective masses; yellow lines and disks include corrections for both the screening and the
+effective masses. Open black circles are experimental data. (b) Room temperature hole mobility of Si as a function of impurity
+concentration: uncorrected (blue); with screening correction (green); with effective mass correction (purple); and with both
+screening and mass correction (yellow). (c) and (d): Same as in (a) and (b) but for 3C-SiC. (e) and (f): Same as in (a) and
+(b) but for GaP. The experimental data are the same as those reported in Figs. 1, 2, and 3 [24, 76, 79–84, 86–92].
+
+1500
+exp
+exp.
+600
+calculated
+calculated
+scr. scaled
+scr. scaled
+1250
+500
+mass scaled
+mass scaled
+ (cm²Ns)
+2Ns)
+scr. + mass scaled
+scr. + mass scaled
+1000
+400
+o)
+750
+300
+an!
+500
+S 200
+S
+250
+100
+(b)
+(a)
+250
+2000
+exp.
+exp.
+calculated
+calculated
+ (cm²/Ns)
+scr. scaled
+scr. scaled
+mass scaled
+1500
+scr. + mass scaled
+150
+100
+500
+50
+(c)
+(d)
+0
+0
+exp.
+250
+300
+calculated
+scr. scaled
+250
+200
+200
+.150
+urd
+GaP
+100
+des
+exp.
+100
+calculated
+scr. scaled
+50
+50
+mass scaled
+(f)
+scr. + mass scaled
+(e)
+0-
+0
+14
+17
+15
+19
+15
+16
+18
+19
+14
+16
+17
+18
+10
+10
+10
+10
+10
+10
+10
+10
+10
+10
+10
+10
+ionized impurity density (cm-3)
+ionized impurity density (cm-3)19
+FIG. 7. Electron mobility in Si as a function of temperature, including the effect of incomplete ionization of the dopants. The
+black line and disks are the calculated phonon-limited mobilities. These data are compared to measurements for pristine silicon
+(impurity concentration < 1012cm−3), from Ref. [75]. The blue disks and line are calculations for an impurity concentration of
+1.75 × 1016 cm−3, taking into account incomplete dopant ionization as described in Appendix A. Experimental data are from
+Ref. [74]. The red disks and line are for an impurity concentration of 1.3 × 1017 cm−3, taking into account incomplete dopant
+ionization. Experimental data are from Ref. [74].
+
+104
+Si μe (cm?Ns)
+103
+100
+200
+300
+400
+500
+temperature (K)20
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diff --git a/n9E0T4oBgHgl3EQfZgDd/content/tmp_files/load_file.txt b/n9E0T4oBgHgl3EQfZgDd/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf,len=1775
+page_content='Ab initio calculation of carrier mobility in semiconductors  including ionized-impurity scattering  Joshua Leveillee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 2 Xiao Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3 Emmanouil Kioupakis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3 and Feliciano Giustino1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 2  1Oden Institute for Computational Engineering and Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The University of Texas at Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Texas 78712,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' USA  2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The University of Texas at Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Texas 78712,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' USA∗  3Department of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 48109,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' USA  (Dated: January 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 2023)  The past decade has seen the emergence of ab initio computational methods for calculating  phonon-limited carrier mobilities in semiconductors with predictive accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' More realistic calcu-  lations ought to take into account additional scattering mechanisms such as, for example, impurity  and grain-boundary scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In this work, we investigate the effect of ionized-impurity scattering  on the carrier mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We model the impurity potential by a collection of randomly distributed  Coulomb scattering centers, and we include this relaxation channel into the ab initio Boltzmann  transport equation, as implemented in the EPW code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We demonstrate this methodology by con-  sidering silicon, silicon carbide, and gallium phosphide, for which detailed experimental data are  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Our calculations agree reasonably well with experiments over a broad range of tempera-  tures and impurity concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' For each compound investigated here, we compare the relative  importance of electron-phonon scattering and ionized-impurity scattering, and we critically assess  the reliability of Matthiessen’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We also show that an accurate description of dielectric screening  and carrier effective masses cam improve quantitative agreement with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  INTRODUCTION  The ability to predict the charge transport properties  of semiconductors using non-empirical ab initio meth-  ods is of paramount importance for the design of next-  generation electronics, neuromorphic computing, energy-  efficient lighting, and energy conversion and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' For  example, as beyond-silicon materials for next-generation  field-effect transistors are being explored, such as wide-  gap semiconductors like GaN [1], SiC [2], and Ga2O3 [3],  or high-mobility materials such as GaAs [4], ab initio  methods for calculating transport properties with pre-  dictive accuracy are acquiring an increasingly important  role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The past decade has seen numerous developments  in first-principles calculations of phonon-limited charge  transport coefficients such as the electrical conductivity  in metals, and the drift and Hall mobilities in semicon-  ductors [5–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' More recently, several groups turned their  attention to ab initio calculations of additional scattering  mechanisms [5, 12–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Among the various mechanisms,  impurity scattering is of particular interest since ionized  donors and acceptors are ubiquitous in high-purity doped  semiconductors, and intrinsic point defects are unavoid-  able in all other materials [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In this work we fo-  cus on ionized-impurity scattering, which is expected to  provide the most significant contribution to the carrier  relaxation rates beyond phonons, given the long-ranged  nature of the Coulomb potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Ionized-impurity scattering in semiconductors has first  been studied via the Conwell-Weisskopf model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In this  ∗ fgiustino@oden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='edu  model, the scattering potential of the impurity is de-  scribed using a Coulomb monopole immersed in the di-  electric background of the semiconductor [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The long-  range nature of this potential makes it ill-behaved at  long-wavelength, and the singularity at long wavelengths  is removed using an ad hoc infrared cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  A better  handling of this singularity is achieved in the Brooks-  Herring model by considering free-carrier screening [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  This latter model proved very successful [22], and is still  widely used owing to its simplicity as it only requires  the electronic density of states, the carrier effective mass,  the high-frequency dielectric constant, and the impurity  concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Further improvements upon these models  were subsequently introduced, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=', carrier statistics, dis-  persive electronic screening, two-impurity scattering, and  atomic form factors [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' While this class of models en-  joyed considerable success with calculations of the carrier  mobility of silicon, they do not perform as well with other  semiconductors [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' These and similar other empir-  ical adjustments make it harder to quantify the role of  each scattering channels, and most importantly decrease  the transferability of the models and ultimately their use-  fulness in materials design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  During the past decade, considerable progress has been  achieved in ab initio calculations of charge carrier mo-  bilities [5–8, 14, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  These approaches are based on  the use of electronic band structures from density func-  tional theory (DFT) [27, 28], as well as phonon disper-  sion relations and electron-phonon matrix elements from  supercell calculations or from density-functional pertur-  bation theory (DFPT) [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' To achieve a numerically  converged sampling of the Brillouin zone, most calcula-  tions by now employ Wannier-Fourier interpolation [33–  35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Mobilities are then obtained by solving the ab initio  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='02323v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='mtrl-sci]  5 Jan 2023  2  Boltzmann transport equation (aiBTE) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The first  study of ionized-impurity scattering from first principles  was reported by Restrepo and Pantelides [5], and more  recent, state-of-the-art calculations have been reported  by Lu and coworkers [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In this latter work, the au-  thors find good agreement between calculated mobilities  and experimental data for silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Additional work using  a semi-empirical approach combining DFT calculations  and models was also reported recently [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In this work, we investigate from first principles the  effect of ionized-impurity scattering on the carrier mobil-  ity of semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' To this aim, we take into account  both carrier-phonon and carrier-impurity scattering on  the same footing, within the aiBTE formalism as imple-  mented in the EPW code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [38] Given that the shape of  the impurity potential depends on the details of the crys-  tal structure and its evaluation would require thermody-  namic calculations of defects and defect levels [39], we  limit ourselves to consider the monopole term of the scat-  tering potential and a random distribution of impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  This simplification allows us to achieve an elegant and  compact formalism, and to compute carrier mobilities by  using solely the concentration of ionized impurities as  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' To validate our methodology, we perform calcu-  lations for three test systems: Si, 3C-SiC, and GaP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' For  Si there is an abundance of experimental data and previ-  ous calculations to compare with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 3C-SiC, which is also  referred to as cubic SiC or β-SiC in the literature, is con-  sidered a promising candidate for next-generation power  electronics [40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Several experimental data sets are  available for carrier mobility in 3C-SiC, especially for n-  type (N) doping and less so for p-type doping (Al).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' GaP  is a standard optoelectronic semiconductor which is of in-  terest in non-linear optical switching [43–45];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' experimen-  tal mobility data for GaP are available both for n-type  doping (Sn) and p-type doping (Zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' For each of these  compounds we calculate the temperature-dependent car-  rier mobility at variable impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We in-  vestigate the relative importance of carrier-phonon and  carrier-impurity scattering, and we examine the validity  of the classic Matthiessen’s rule [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The manuscript is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' II we  briefly summarize the aiBTE formalism, we provide a  detailed derivation of the matrix elements for carrier-  impurity scattering, and we discuss the key approxima-  tions involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In this section we also discuss free-carrier  screening, and we examine under which conditions the  Matthiessen rule can reliably be used in transport cal-  culations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Section III is devoted to the implementa-  tion details and the calculation parameters used in this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IV we discuss our results for Si, SiC,  and GaP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In particular, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IV B we present our cal-  culated temperature- and concentration-dependent mo-  bilities and compare our data with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IV C we analyze the relative importance of phonon-  and impurity-mediated scattering processes in the carrier  relaxation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IV D we test Matthiessen’s rule  by comparing full aiBTE calculations with the results  of separate calculations including only phonon-limited or  impurity-limited mobilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IV E we investigate  how the DFT dielectric screening and carrier effective  masses influence calculated mobilities, and we test sim-  ple correction schemes along the lines of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' V we summarize our findings and offer our conclu-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Additional details on the calculation procedure are  discussed in the Appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  THEORETICAL APPROACH  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Carrier mobility from the ab initio Boltzmann  transport equation  A detailed derivation of the aiBTE formalism is given  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Here we limit ourselves to summarize the key  equations in order to keep this manuscript self-contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Within the linearized Boltzmann transport equation, the  carrier mobility tensor is obtained as:  µαβ = −  2  Ωucnc  1  Nuc  �  nk  vα  nk∂Eβfnk,  (1)  where the factor of 2 is for the spin degeneracy, Greek  indices indicate Cartesian directions, Eβ indicate the  Cartesian components of the electric field, and ∂Eβfnk  is the linear variation of the electronic occupation of the  state with band index n and wavevector k in response  to the applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' vα  nk represents the expectation value  of the velocity operator along the direction α, for the  Kohn-Sham state nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' e, nc, Ωuc, and Nuc indicate the  electron charge, the carrier density, the volume of the  unit cell, and the number of unit cells in the Born-von  K´arm´an (BvK) supercell, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The n-summation  extends over all Kohn-Sham states, although in practice  only those states near the chemical potential contribute  to the mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The k-summation is over a uniform Bril-  louin zone grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The variation ∂Eβfnk is obtained from the self-  consistent solution of the equation:  −evβ  nk  ∂f 0  nk  ∂ϵnk  =  �  mq  �  τ −1  mk+q→nk ∂Eβfmk+q  −τ −1  nk→mk+q ∂Eβfnk  �  ,  (2)  where f 0  nk denotes the Fermi-Dirac occupation of the  state nk in the absence of electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The quantity  τ −1  nk→mk+q is the partial scattering rate from the Kohn-  Sham state nk to the state mk + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In many-body per-  turbation theory, this rate is derived from the imaginary  parts of the electron self-energy, therefore different scat-  tering mechanisms simply add up to the lowest order in  perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In this work, we write the scattering  rate as the sum of the rates of carrier-phonon scattering  (ph) and carrier-impurity (imp) scattering:  1  τnk→mk+q  =  1  τ ph  nk→mk+q  +  1  τ imp  nk→mk+q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  (3)  3  The partial carrier-phonon scattering rate is given  by [26]:  1  τ ph  nk→mk+q  =  1  Nuc  �  ν  2π  ¯h |gmnν(k, q)|2  ×  �  (nqν + 1 − f 0  mk+q)δ(ϵnk−ϵmk+q − ¯hωqν)  +(nqν + f 0  mk+q)δ(ϵnk−ϵmk+q + ¯hωqν)  �  ,  (4)  where ϵnk denote Kohn-Sham eigenstates,  and ωqν  stands for the frequency of a phonon with branch in-  dex ν, wavevector q, and Bose-Einstein occupation nqν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The matrix elements gmnν(k, q) indicate the probability  amplitude for the scattering of an electron from state nk  to state mk+q via a phonon qν [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The partial rate in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (4) can be obtained either from Fermi’s golden rule or  from many-body perturbation theory [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The carrier-  impurity scattering rate required in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (3) is derived in  the next section and is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Together, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (1)-(4) and (17) define the aiBTE  framework employed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This approach consis-  tently captures back-scattering and Umklapp processes,  with a computational cost that is similar to more ap-  proximate approaches based on various relaxation-time  approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [26] for a  comprehensive review of common approximations to the  Boltzmann transport equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Scattering of Carriers by ionized impurities in  the monopole approximation  To  obtain  the  carrier-impurity  scattering  rate  1/τ imp  nk→mk+q we proceed as follows: (i) We derive the  matrix element of the scattering potential for a single  impurity in a periodic BvK supercell of the crystal unit  cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (ii) We generalize the matrix element to consider  a number Nimp of impurities in the BvK supercell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (iii)  From this matrix element, we obtain the scattering rate  corresponding to the Nimp impurities by using the first  Born approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (iv) We average the resulting rate  over a random uniform distribution of impurity positions  using a method due to Kohn and Luttinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Scattering potential and matrix element for single  impurity  We employ the monopole approximation to describe  the potential of an impurity of charge Ze located at the  position r0 in the BvK supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' A more refined choice  would entail explicitly calculating the impurity poten-  tial in DFT and its matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This approach was  pursued in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [5] and [14], but it carries the disadvan-  tage that one needs to compute defect energetics prior  to mobility calculations, and then perform rotational av-  erages to account for the randomness of the impurity  orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Our simpler approach is useful for system-  atic transport calculations when detailed knowledge of  the atomic-scale structure of impurities is lacking, and  can be made more accurate by incorporating dipole and  quadrupole terms along the lines of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  By solving the Poisson equation in the BvK supercell  and considering a background anisotropic static dielec-  tric constant tensor ε0 = ε0  αβ, the potential of this point  charge is found to be [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (S3) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [47]]:  φ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' r0) = 4π  Ωsc  Ze  4πε0  �  q  �  G̸=−q  ei(q+G)·(r−r0)  (q + G)·ε0· (q + G),  (5)  modulo an inessential constant that reflects the compen-  sating background charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In this expression, ε0 is the  vacuum permittivity, G is a reciprocal lattice vector,  and the wavevector q belongs to a uniform Brillouin-  zone grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Here an in the following, we consider that the  BvK cell consists of Nuc unit cells, so that its volume is  Ωsc = NucΩuc, and that the Brillouin zone is discretized  in a uniform grid of Nuc points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The potential φ(r, r0) is  periodic over the BvK supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The perturbation potential resulting from this impu-  rity is V = ∓eφ for electrons and holes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  For definiteness, we consider electrons in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The matrix elements of the perturbation V between the  Kohn-Sham states ψnk and ψmk+q is given by:  gimp  mn (k, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' r0) = ⟨ψmk+q|V (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' r0)|ψnk⟩sc,  (6)  where the integral is over the supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The states can  be written as ψnk = N −1/2  uc  eik·runk, where unk is the  Bloch-periodic part and is normalized in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The combination of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (5) and (6) yields:  gimp  mn (k, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' r0) = −e2  4πε0  4πZ  Ωsc  �  G̸=−q  e−i(q+G)·r0Bmn,G(k, q)  (q + G)·ε0· (q + G) ,  (7)  having defined the overlap integral:  Bmn,G(k, q) = ⟨umk+q|eiG·r|unk⟩uc,  (8)  which is evaluated over the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Scattering rate from multiple impurities within the first  Born approximation  We now consider N sc  imp impurities located at the po-  sitions r1, r2, · · · , rNimp in the BvK supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The  corresponding perturbation potential is the sum of  the potentials obtained in the previous section, V  =  �N sc  imp  I=1 V (r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' rI), therefore the generalization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (7)  to the case of multiple identical impurities reads:  gimp  mn (k, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' {rI}) = −e2  4πε0  4πZ  Ωsc  �  G̸=−q  Bmn,G(k, q)  (q + G)·ε0· (q + G)  ×  �N sc  imp  I=1 e−i(q+G)·rI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  (9)  4  The total scattering rate out of state nk associated  with this matrix element can be written using the first  Born approximation for the scattering matrix [50] [Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='16) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='32)]:  1  τ imp  nk  =  �  mq  2π  ¯h |gimp  mn (k, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' {rI})|2δ(ϵnk − ϵmk+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  (10)  We note that this expression is an intensive quantity,  as expected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' it does not scale with the size of the  BvK supercell [see discussion after Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (17)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The partial  scattering rate needed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (3) is then defined as:  1  τ imp  nk→mk+q  = 2π  ¯h |gimp  mn (k, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' {rI})|2δ(ϵnk−ϵmk+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (11)  Unlike Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (4), in this expressions we do not have the  Fermi-Dirac occupations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' These occupations drop out in  the linearized Boltzmann transport equation, as it can  be verified, for example, by setting nqν = 0 and ωqν = 0  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (11) the Dirac delta function ensures  energy conservation, consistent with the fact that we are  considering the scattering by a fixed potential, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' we  are neglecting the recoil of the impurity upon collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  By combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (9) and (11) we find:  1  τ imp  nk→mk+q  ({ri}) = 2π  ¯h  � e2  4πε0  4πZ  Ωsc  �2  δ(ϵnk − ϵmk+q)  ×  �  G,G′̸=−q  Bmn,G(k, q)B∗  mn,G′(k, q)  (Q ·ε0· Q)(Q′ ·ε0· Q′)  �N sc  imp  I,J=1ei(Q′·rJ−Q·rI),  (12)  having defined Q = q + G and Q′ = q + G′ for conve-  nience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Kohn-Luttinger ensamble averaging of the scattering rate  In order to account for the randomness in the dis-  tribution of impurities, we perform a configuration av-  erage of the scattering rate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (12) by considering  a uniform probability distribution, following the Kohn-  Luttinger approach [51]:  1  τ imp,ave  nk→mk+q  =  �  sc  dr1 · · · drN sc  imp  Ω  N sc  imp  sc  1  τ imp  nk→mk+q  ({ri}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (13)  The only term that depends on the impurity positions in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (12) is the sum over I, J on the second line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Below we  evaluate the ensemble average of this sum by separating  the I = J and I ̸= J terms:  �  sc  dr1 · · · drN sc  imp  ΩNimp  sc  �N sc  imp  I,J=1ei(Q′·rJ−Q·rI)  = N sc  imp  Ωsc  �  sc  dr ei(Q′−Q)·r  +N sc  imp(N sc  imp − 1)  Ω2sc  ��  sc  dr eiQ′·r  ���  sc  dr e−iQ·r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (14)  Both terms on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' require the evaluation of an in-  tegral of the type:  �  sc  dr eiQ·r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  (15)  This integral equals Ωsc for Q = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' for finite Q, we note  that the integral becomes the Fourier representation of  the Dirac delta when Nuc → ∞, therefore it vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In  this limit, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (14) reduces to:  �  sc  dr1 · · · drNimp  Ω  N sc  imp  sc  �N sc  imp  I,J=1ei(Q′·rJ−Q·rI)  = N sc  imp δG,G′ + N sc  imp(N sc  imp − 1) δG,−qδG′,−q,  (16)  Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (16) and (12) inside Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (13), we obtain:  1  τ imp,ave  nk→mk+q  =  1  Nuc  N uc  imp  2π  ¯h  � e2  4πε0  4πZ  Ωuc  �2  ×  �  G̸=−q  |Bmn,G(k, q)|2  |(q + G) ·ε0· (q + G)|2 δ(ϵnk − ϵmk+q), (17)  where we use N uc  imp = N sc  imp/Nuc to denote the number  of impurities per unit cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' N uc  imp is a dimensionless quan-  tity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We note that, in practical calculations, the prefac-  tor 1/Nuc in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (17), which also appears in the partial  carrier-phonon scattering rate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (4), is included as  a k-point weight in Brillouin zone summations, so that  the sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (2) becomes N −1  uc  �  q and is independent  of the size of the BvK supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The scattering rate given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (17) is similar but not  identical to alternative forms used in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' For  example, it differs from classic approaches such as the  Conwell-Weisskopf formula [20] and the Brooks-Herring  formula [21] in that here the details of band structures,  Kohn-Sham orbital overlaps, and anisotropic dielectric  screening are fully taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Furthermore, it  differs from more recent ab initio approaches such as  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [5] in that the long-range nature of the Coulomb in-  teraction is taken into account from the start, as opposed  to being included as an ad hoc correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Our expression  is similar to the formula provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [14], except that  here we take into account the periodicity of the impurity  potential over the BvK supercell and the anisotropy of  the dielectric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The fact that we reached a similar  expression as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [14] starting from a rather different  viewpoint involving the Kohn-Luttinger ensemble aver-  age lends support to both approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Free-carrier screening of the impurity potential  The carrier-impurity scattering rate given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (17)  contains a singular q−4 term that is not integrable (with  q = |q|), and leads to incorrect results when used in the  aiBTE of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This problem was already identified by  5  Conwell and Weisskopf [20], who introduced an infrared  cutoff to suppress the Coulomb singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The formal way to overcome this difficulty is to observe  that ionized impurities are accompanied by free-carriers,  which introduce metallic-like screening of the impurity  potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the Thomas-Fermi model, free-carriers in-  troduce an additional screening  εTF(q) = 1 + q2  TF  q2 ,  (18)  where qTF is the Thomas-Fermi wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' When used  in combination with the impurity potential appearing in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (17), this additional screening lifts the Coulomb sin-  gularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In fact, by temporarily ignoring the G vectors  and the anisotropy of the dielectric tensor, free-carrier  screening modifies the denominator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (17) as fol-  lows:  1  (ε0q2)2  −−→  1  [εTF(q)ε0q2]2 =  1  [ε0(q2 + q2  TF)]2 , (19)  which tends to the finite value 1/(ε0q2  TF)2 at long wave-  length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  To incorporate free-carrier screening in our calcula-  tions, while taking into account all details of band struc-  tures and effective masses, we employ the Lindhard di-  electric function instead of the Thomas-Fermi model, fol-  lowing Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The same approach was employed in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The Lindhard dielectric function is given by:  εL(q) = 1 −  e2  4πε0  4π  q2  2  NucΩuc  �  nk  f 0  nk+q − f 0  nk  ϵnk+q − ϵnk  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  (20)  Since the density of free-carriers is typically low in doped  semiconductors, we only need the long wavelength limit  of this expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In this limit, (f 0  nk+q − f 0  nk)/(ϵnk+q −  ϵnk) = ∂f 0  nk/∂ϵnk, therefore we can write:  εL(q) = 1 + q2  TF  q2 ,  (21)  having introduced the effective Thomas-Fermi vector:  qTF =  e2  4πε0  2 · 4π  NucΩuc  �  nk  ����  ∂f 0  nk  ∂ϵnk  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  (22)  For parabolic bands, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (21) reduces to the Thomas-  Fermi or Debye model in the respective temperature lim-  its.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The free-carrier screening provides an additional  screening mechanism to the dielectric screening of the  insulating semiconductors, and is included in our calcu-  lations by replacing ε0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (17) by the total dielectric  function:  ε0  −−→  ε0 + 1q2  TF  q2 ,  (23)  where 1 denotes the 3 × 3 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We note that  this improved description of the screening includes tem-  perature effects via the Fermi-Dirac occupations entering  the definition of the effective Thomas-Fermi wavevector,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Matthiessen’s Rule  Matthiessen’s rule [46] is widely employed to interpret  transport measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the context of carrier trans-  port in semiconductors, this rule can be stated as follows:  the contributions of different scattering channels to the  mobility can be obtained by adding the reciprocals of the  individual mobilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the case of carrier-phonon and  impurity-phonon scattering, we would have:  1  µ =  1  µph  +  1  µimp  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  (24)  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IV we proceed to quantify the reliability of this  approximation by comparing mobility data calculated us-  ing the complete aiBTE including both phonons and im-  purities with the prediction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (24) obtained by cal-  culating the mobility with these two scattering channels  taken individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We will show that this rule does not  carry predictive power for the examples considered in this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  From a formal standpoint, the rule expressed by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (24) is obviously related to the choice of expressing  the total scattering rates as the sum or the individual  rates, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  That choice was motivated by the  observation that, to first order in perturbation theory,  different scattering channels do not mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' However, it is  easy to see that, even when Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (3) is a good approx-  imation, the additivity of the rates does not imply the  Matthiessen rule as expressed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' To appreciate  this point, we observe that the aiBTE in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (2) can be  recast as a linear system of the type:  A × {∂Eβfnk} = b,  (25)  where the matrix A contains the partial scattering rates  τ −1  nk→n′k′, the vector b contains the drift term on the left  hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (2), and {∂Eβfnk} denotes the vector of  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' If we break down the matrix A into its contri-  butions from carrier-phonon and carrier-impurity scat-  tering, Aph and Aimp respectively, we see immediately  that  {∂Eβfnk} = (Aph + Aimp)−1b ̸= A−1  ph b + A−1  impb,  (26)  therefore the additivity of the scattering rates does not  imply the Matthiessen rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  This point can be made  even more explicit by considering the self-energy relax-  ation time approximation to the aiBTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The approxi-  mation consists of neglecting the first term on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (2), and yields the following expression for the  mobility:  µαβ = −  e  Ωucnc  2  Nuc  �  nk  ∂f 0  nk  ∂ϵnk  vα  nkvβ  nk  ×  1  1  τ ph  nk  +  1  τ imp  nk  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  (27)  6  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Calculation parameters used in this work: Experi-  mental lattice constant, plane wave kinetic energy cutoff, and  non-vanishing elements of the quadrupole tensor are chosen  to be consistent with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Si 3C-SiC GaP  Lattice constant (˚A)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='43  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='36  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='45  Plane wave kinetic energy cutoff (eV)  544  1088 1088  Qκ1  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='83  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='41 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='72  Qκ2  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='83  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='63 -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='92  Coarse k and q grids  123  123  123  Fine k and q electron grid  1003  1803  1003  Fine k and q hole grid  1003  1003  1003  For this expression to be amenable to Matthiessen’s rule,  the scattering rates would need to be independent of the  electronic state, say τ ph  nk = τ ph and τ imp  nk  = τ imp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This is  typically not the case in most semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Another  special case where Matthiessen’s formula is meaningful  occurs when one scattering mechanism dominates over  the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' For example, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (27), when τ ph  nk ≫ τ imp  nk ,  the expression reduces to the phonon-limited mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In  this sense, Matthiessen’s rule constitutes a simple inter-  polation formula between the limiting cases of phonon-  limited and impurity-limited mobilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We will analyze  these aspects quantitatively in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  COMPUTATIONAL METHODS  All calculations are performed using the Quantum  ESPRESSO materials simulation suite [53], the EPW  code [38], and the Wannier90 code [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We employ the  PBE exchange and correlation functional [55] and op-  timized norm-conserving Vanderbilt (ONCV) pseudopo-  tentials from the PseudoDojo repository [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  For  consistency with previous work, we use the experimental  lattice constant of Si, SiC, and GaP at room temperature,  and the plane-wave kinetic energy cutoff and quadrupole  tensors reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We include spin-orbit cou-  pling for the valence bands only, to capture the splitting  of the valence band top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Key calculation parameters are  summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  We calculate effective mass tensors by finite differences,  using a wavevector increment of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='01 × 2π/a, where a  is the lattice constant reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The dynam-  ical matrix, the variations of the self-consistent poten-  tial, and the vibrational eigenfrequencies and eigenmodes  are calculated using a square convergence threshold of  10−16 Ry2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This threshold refers to the change of the  potential variation between two successive iterations, av-  eraged over the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Electron energies, phonon  frequencies, and electron-phonon matrix elements are  initially computed on a coarse wavevector mesh using  the EPW code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The electron Hamiltonian, the dynam-  ical matrix, and the electron-phonon matrix elements  are then interpolated onto fine Brillouin zone grids us-  ing Wannier-Fourier interpolation [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Long-range  dipole and quadrupole corrections are employed for im-  proved interpolation of the electron-phonon matrix ele-  ments [47–49, 58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  To compute carrier mobilities, only states within a nar-  row energy window of the band extrema are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  We find that, for the range of temperatures considered in  this work (up to 500 K), a window of 400 meV is sufficient  to obtain converged electron mobilities, and a window  of 300 meV is sufficient for hole mobilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' At 300 K,  converged results can be obtained by using a 200 meV  window for both electrons and holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  To evaluate the overlap matrices Bmn,G(k, q) required  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (17) in the fine Brillouin zone grid, we follow the  procedure of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [47] and approximate them as:  Bmn,G(k, q) ≈  �  U(k + q)U †(k)  �  mn ,  (28)  where the unitary matrix Umn(k) is the diagonalizer of  the interpolated Hamiltonian into the wavevector k of  the fine grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This approximation is motivated by the  fact that the carrier-impurity matrix element in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (17)  is strongly peaked at q + G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The Dirac delta functions appearing in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (4) and  (17) are computed using Gaussian functions with a small  broadening parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The results are sensitive to the  choice of this parameter, therefore we accelerate the con-  vergence by employing adaptive smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The proce-  dure for the adaptive smearing of the carrier-phonon scat-  tering rate, which involves a so-called type-III integral,  is discussed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [6, 58, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The calculation of the  carrier-impurity scattering rates involves instead a type-  II integral of the form:  III  nk =  �  m  �  dq  ΩBZ  fmn(k, q) δ(ϵmk+q − ϵnk),  (29)  where ΩBZ is the volume of the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In this  case, adaptive broadening can be achieved by using a  state-dependent width σmk+q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We follow the procedure  by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [60], which gives:  σmk+q = α  3  3  �  i=1  vmk+q · bi  Ni  ,  (30)  where vmk+q is the band velocity, bi is a primitive vector  of the reciprocal lattice, and Ni denotes the number of  k-points along the direction of bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The coefficient α is a  tunable parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Previous work has used α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='29  for electron-phonon scattering rates [6, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  We have  performed a detailed converged test by comparing fixed-  smearing and variable-smearing calculations, and found  that values α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3 provide similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' For sim-  plicity, in this work we use α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='29 as in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In principle we could perform calculations of carrier  mobilities by setting the impurity concentration and the  carrier concentration separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This would be required,  for example, for the investigation of compensation dop-  ing of semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' To keep our results are general  7  as possible, in this work we choose to focus on the sim-  pler scenario where each impurity creates one free car-  rier, therefore we set the carrier density to be equal to  the impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We do not consider carrier  freeze-out at low temperature, since this would require  the knowledge of defect energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In our calcula-  tions, the role of the carrier concentration is mainly to  modulate the effective Thomas-Fermi screening wavevec-  tor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  RESULTS AND DISCUSSION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Electronic structure  Given the importance of effective masses in mobility  calculations, in this section we review briefly the band  structures and effective masses of Si, SiC, and GaP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Ta-  ble II shows our calculated directional effective masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Hole masses are given for the heavy-hole (hh) band, light  hole (lh) band, and the spin-orbit split-off (so) band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The  longitudinal (∥) and transverse (⊥) electron masses cor-  respond to the principal axes of the ellipsoidal conduction  band extrema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' II we see that the light hole and split-off  hole masses are fairly isotropic for all compounds con-  sidered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' For the heavy hole masses, the Γ-X  direction ([100] crystallographic direction) exhibits the  lightest masses, whereas considerably heavier masses are  found along the Γ-K ([110]) and Γ-L ([111]) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Similarly, in all compounds considered here the longitu-  dinal electron masses are considerably heavier than the  corresponding transverse masses, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' SiC ex-  hibits the heaviest hole masses among SiC, GaP, and Si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  while GaP exhibits the heaviest electron masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Our calculated effective masses are in good agreement  with previous calculations at the DFT level [10] as well as  previous calculations at the GW level [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' When com-  paring to experimental data, we see from Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' II that  our electron effective masses are within 10% of the corre-  sponding experimental values, which is remarkable con-  sidering that we are using DFT/PBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In the case of the hole masses, our calculations are also  in good agreement with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Here we emphasize  that the experimental values usually quoted are not the  effective masses, but the cyclotron masses, which depend  on the direction of the magnetic field and are reported  in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' These cyclotron masses correspond to aver-  ages of the directional masses and cannot be compared  directly to DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' To extract the correct di-  rectional effective masses, in the case of silicon we used  the Dresselhaus k · p model which was fitted to experi-  mental cyclotron data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In this model the heavy hole and  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Calculated band effective masses, band gaps,  high-frequency and static dielectric constants of Si, 3C-SiC,  and GaP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' All calculations performed within DFT/PBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Ex-  perimental data are from (a) [61] and [62], (b) [63], (c) [64],  (d) [65], (e) [66], (f) [67], (g) [68], (h) [69], (i) [70], (j) [71], (k)  [72], (l) [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' All masses are give in units of the electron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The band gaps are in eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The lines tagged “Dresselhaus” refer  to the effective masses obtained from the Dresselhaus model  fitted to experimental cyclotron data, from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  This work  Si  SiC  GaP  Γ-X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='260  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content='374  m∗  hh  Γ-K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content='837  Γ-L  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='655  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='646 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='091  Γ-X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content='143  m∗  lh  Γ-K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content='117  Γ-X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content='213  m∗  so  Γ-K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content='214  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='206  m∗  e,∥  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='959  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='672 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='069  m∗  e,⊥  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='196  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='232  Eg  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='554  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='359 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='566  ε∞  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='00  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='93 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='53  ε0  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='00  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='23 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='57  Experiment  Si  SiC  GaP  B along [001]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='46a  m∗  hh  B along [110]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='53a  B along [111]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='56a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='54c  Dresselhaus Γ-X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='40  Dresselhaus Γ-K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='56  Dresselhaus Γ-L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='62  B along [001]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='171a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='45b  m∗  lh  B along [110]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='163a  B along [111]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='160a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='16c  Dresselhaus Γ-X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='18  Dresselhaus Γ-K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='16  Dresselhaus Γ-L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='15  m∗  e,∥  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='97a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='68d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='15c, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='0k  m∗  e,⊥  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='19a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='25d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='21c, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='25k  Eg  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='13f  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='42g 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='26h  ε∞  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='7i  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='52j 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='11j  ε0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='7i  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='72j 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='1j  light hole masses are parameterized as:  ϵhh(k) =Ak2 + [B2k4 + C2(k2  xk2  y + k2  yk2  z + k2  zk2  x)]1/2,  (31)  ϵlh(k) =Ak2 − [B2k4 + C2(k2  xk2  y + k2  yk2  z + k2  zk2  x)]1/2,  (32)  where k  =  |k| and the coefficients A, B, and C  are −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='1 ¯h2/2me, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='6 ¯h2/2me, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3 ¯h2/2me, respec-  tively [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' From this parameterization we obtained the  effective masses reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' II under the keyword  “Dresselhaus”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' From this table we can see that, in the  8  case of silicon, the light hole and heavy hole masses are  close to our calculated results, with the exception of the  Γ − X heavy-hole effective mass which is 65% of the ex-  perimental value[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Our calculated dielectric constants overestimate the  experimental values by 15% at most, as expected from  the underestimation of the band gaps [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IV E we discuss how one can improve the calculated  mobilities by introducing a posteriori corrections to the  theoretical effective masses and dielectric constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Carrier mobilities  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Silicon  Figure 1 shows a comparison between our calculated  mobilities of silicon and available experimental data, as  a function of temperature and impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The mobilities without carrier-impurity scattering [black  lines in panels (a) and (b)] decrease rapidly with tem-  perature, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We find temperature slopes (the  β in µ ∼ T β) of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='1 for electrons and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='4 for holes,  in agreement with previous work [10, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' As we include  carrier-impurity scattering, the room-temperature elec-  tron mobility of silicon reduces from 1381 cm2/Vs to  1153 cm2/Vs at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='75×1016 cm−3 [blue line in panel (a)]  and to 812 cm2/Vs at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3×1017 cm−3 [red line in panel  (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Similarly, the room-temperature hole mobility of  silicon decreases from 600 cm2/Vs in the absence of im-  purities to 517 cm2/Vs for an impurity concentration of  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='4×1016 cm−3[blue line in panel (b)], and to 359 cm2/Vs  at the impurity concentration of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='0×1017 cm2/Vs [red  line in panel (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Our calculations for the temperature-dependent elec-  tron and hole mobilities show that a single power law be-  comes inadequate in the presence of impurity scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  This is also seen in the experimental data from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [74–  77], which are shown as open circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We note  that our calculations are in good agreement with the ex-  periments over a broad temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The agree-  ment worsens slightly at low temperature, where carrier-  impurity scattering dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This effect likely relates  to the fact that in our calculations all donors and ac-  ceptors are assumed to be fully ionized at all tempera-  tures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' as a result of this approximation, we are neglecting  carrier freeze-out and hence we are likely overestimating  the impurity concentration at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In Ap-  pendix A we show that, by taking into account the the  effects of partial impurity ionization, the agreement with  experiments improves at low temperature and high im-  purity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Panel (c) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 1 shows the room temperature elec-  tron mobility of silicon, as a function of impurity con-  centration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The electron mobility is relatively insensitive  to the impurity concentration up to 1016 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' A steep  decrease in the electron mobility is seen as we approach  a doping density of 1017 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Up to this concentra-  tion, our calculations (blue line) are in excellent agree-  ment with experimental data (open black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Above  1018 cm−3, while the agreement with experiment is still  good, we tend to slightly overestimate the measured elec-  tron mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This is likely due to two effects: (i) our  formalism does not take into account multiple scattering  events that become important at high impurity concen-  tration, and (ii) our calculations do not include scattering  by free-carrier plasmons, which dominate the mobility at  high carrier density, as shown in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [12, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' A similar  overestimation was observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Panel (d) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 1 shows the room temperature hole  mobility of silicon as a function of impurity concentra-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' As for the electrons, we find generally good agree-  ment between calculations (blue line) and experiments  (open black circles) throughout the doping range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We  emphasize that the vertical scales in panels (c) and (d)  are different, and that the theory/experiment deviation  at high impurity concentration is similar in both panels  in absolute terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' At low impurity concentration, our  calculations slightly overestimate the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  This effect can be ascribed to the fact that our light hole  effective masses are smaller than in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Silicon carbide  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 2 we show our calculated mobilities of 3C-SiC  as a function of temperature and impurity concentration,  and we compare to experimental data from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [24, 78–  84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the case of 3C-SiC, the comparison with exper-  iments is complicated by the high concentration of line  defects that nucleate at lattice-mismatched growth sub-  strates such as Si or 6H-SiC [83, 85], which makes it diffi-  cult to obtain data for defect-free samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Furthermore,  most experimental data are for co-doped samples, for  which the impurity and carrier concentrations are more  difficult to estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In the absence of impurity scattering [black line in  panel (a)], the low electron effective mass of SiC leads  to very high theoretical mobilities, up to 33000 cm2/Vs  at 100 K and up to 2000 cm2/Vs at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  These high mobilities are in agreement with previous the-  oretical results [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In this case, we calculate an electron  temperature exponent β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 2 we compare our calculations (blue  line) with the data reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [78] (red open cir-  cles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In that work, they synthesized 3C-SiC with n-  type impurity density of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='0×1016 cm−3, and obtained  electron mobilities at 100 K and 300 K of 2040 cm2/Vs  and 584 cm2/Vs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In our calculations, when  we consider the same impurity concentration, we find  2773 cm2/Vs and 1369 cm2/Vs at 100 K and 300 K,  respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' therefore we overestimate the experimental  data by a factor of 30%-230%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In panel (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 2 we show our calculated hole mo-  bility of 3C-SiC as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In the  absence of impurities (black line), the mobility decreases  9  with a temperature exponent β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In this case  we could not find experimental data for uncompensated  samples to compare with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Upon including impurity scat-  tering with an impurity concentration of 1018 cm−3, we  find a significant reduction of the mobility at low tem-  perature (blue line), from 1373 cm2/Vs to 148 cm2/Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  At 300 K, the mobility is reduced from 165 cm2/Vs with-  out impurities to 81 cm2/Vs, in good agreement with the  measured value of 50 cm2/Vs reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Panels (c) and (d) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 2 show the room temper-  ature electron and hole mobilities as a function of im-  purity concentration, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The electron mobility  calculated (blue line) at low ionized donor concentration  (1014 cm−3) is 2048 cm2/Vs, and significantly overesti-  mates the measured value 1000 cm2/Vs by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [79] (open  black symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' However, our calculations get closer to  experimental data in the range of concentrations above  1018 cm−3 [24, 80, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The hole mobility of 3C-SiC is significantly lower than  the electron mobility, as expected from much heavier hole  masses shown in Tab II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Our calculations (blue line) at  low doping yield a mobility of 164 cm2/Vs, to be com-  pared to 220 cm2/Vs measured in p-channel 3C-SiC de-  vices [82] (open black symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We note that the vertical  scales in panels (c) and (d) differ, and that our calculated  hole mobilities are in better agreement with experiment  in relative terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In particular, our data for the hole mo-  bility fall right in the middle of the experimental trend  shown in panel (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Gallium phosphide  Figure 3 shows our mobility calculations for GaP and a  comparison with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In panel (a) we have  the calculated electron mobilities as a function of temper-  ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the absence of impurities, the calculated elec-  tron mobility (black line) decreases with a temperature  exponent β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' the calculated mobilities at 100 K  and 300 K are 4293 cm2/Vs and 328 cm2/Vs, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Upon including the effect of impurity scattering  (blue line), the mobility decreases significantly, reach-  ing 157 cm2/Vs at room temperature for an impurity  concentration of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='5×1018 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This value is in good  agreement with the measured mobility of 100 cm2/Vs by  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [87] (blue open circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We note that the electron  mobility of GaP is significantly lower than in silicon, de-  spite the electron effective masses being comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IV C we show that this effect arises from the addi-  tional polar phonon scattering that electrons experience  in GaP, which is absent in silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Panel (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 3 shows the calculated phonon-limited  hole mobility (black line), the mobility calculated by in-  cluding impurity scattering (blue line), and experimen-  tal data (open red circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The phonon-limited hole  mobility decreases with temperature with an exponent  β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The calculated mobilities in the absence of  impurities are 5096 cm2/Vs and 252 cm2/Vs at 100 K  and 300 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Upon including impurity scat-  tering with a concentration of 2×1018 cm−3, the mobil-  ity at room temperature decreases to 124 cm2/Vs, in  good agreement with the measured value of 90 cm2/Vs  by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Panel (c) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 3 shows the room temperature elec-  tron mobility of GaP as a function of impurity concentra-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the absence of impurity scattering, we calculate  a mobility of 328 cm2/Vs(blue line), which compares well  with the maximum value 258 cm2/Vs measured in ultra-  pure samples in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [88] (open black symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the  intermediate doping regime, our calculated electron mo-  bilities overestimate the experimental data by a factor of  two [87–90], but the agreement improves at high doping  levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Figure 3(d) shows the room temperature hole mobil-  ity of GaP as a function of impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The  calculated hole mobility is 269 cm2/Vs at low impurity  concentration, and decreases to 94 cm2/Vs at a concen-  tration of 1019 cm−3 (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Our calculations are  within a factor of two from the highest measured hole  mobilities across the same doping range [87, 91, 92] (open  black symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We note that electron and hole mobilities  in GaP are very similar across a wide range of tempera-  tures and impurity concentrations (both in experiments  and in our calculations), therefore GaP is an ambipo-  lar semiconductor with well-balanced electron and hole  transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Carrier scattering rates  In this section we analyze and compare the scattering  rates resulting from carrier-phonon and carrier-impurity  processes in Si, SiC, and GaP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The Brooks-Herring model  for carrier-impurity scattering [21], which is based on the  parabolic band approximation, predicts a scattering rate  that scales as ϵ−3/2, where ϵ is the electron eigenvalue  referred to the band extremum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This trend is a result of  two competing effects: as the energy of the initial state  increases above the band bottom, the scattering phase  space increases as ϵ1/2, while at the same time the square  modulus of the carrier-impurity matrix element given in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (17) decreases as 1/q4, which is of the order of ϵ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  This simple trend is opposite to what is expected from  non-polar optical scattering and acoustic phonon scatter-  ing, which tend to increase with energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Figure 4 shows the scattering rates τ −1  nk of holes and  electrons in Si [panels (a) and (b)], SiC [panels (c) and  (d)], and GaP [panels (e) and (f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' For consistency, we  set the impurity concentration to 1017 cm−3 in all cases,  which is in the middle of the range considered in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 1-3,  and the temperature to 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In line with the above dis-  cussion, the carrier-impurity scattering rates decrease as  we move away from the band extrema, while the carrier-  phonon scattering rates increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the two polar semi-  conductors that we are considering, SiC and GaP, we  also see a sudden jump in the carrier-phonon scatter-  10  ing rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This effect happens when the carrier energy  reaches the threshold for the emission of a longitudinal  optical phonon, thereby activating polar phonon scatter-  ing [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Panels (a) and (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 4 show that, in the case of  silicon, the carrier-ionized-impurity scattering rates near  the band edges are an order of magnitude higher than  carrier-phonon rates (for an impurity concentration of  1017 cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The additional scattering by carriers causes  a reduction of the mobility by ∼ 30% for both electrons  and holes, indicating that impurity scattering is a sig-  nificant effect at this impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The rise  of the carrier-electron scattering rates at energies around  150 meV that can be seen in panel (b) correspond to  interband scattering between the two lowest conduction  bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Panels (c) and (d) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 4 show the scattering rates  in SiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Unlike in silicon, here the electron and hole scat-  tering rates differ considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In the case of holes,  the carrier-phonon and carrier-impurity scattering rates  are comparable in magnitude near the band edge, while  in the case of electrons the carrier-impurity scattering  dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  This difference is reflected in the calcu-  lated mobilities, where carrier-impurity scattering re-  duces the phonon-limited mobility of holes by ∼ 20%  and of electrons by ∼ 50% (for the impurity concentra-  tion 1017 cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Data for GaP are shown in panels (e) and (f) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In this case the carrier-phonon scattering rates are com-  parable to the carrier-impurity scattering rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Accord-  ingly, the mobilities are reduced by ∼10% from their val-  ues without impurity scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Deviations from Matthiessen’s Rule  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IV D we discussed how Matthiessen’s rule is for-  mally justified only when the scattering rates are state-  independent constants, or when one scattering mecha-  nism dominates over all other mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' To place that  reasoning on a quantitative footing, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 5 we explicitly  assess the predictive accuracy of the Matthiessen rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  For this test, we compute the mobilities of Si, SiC,  and GaP by considering the following four scenarios:  (i) phonon-limited mobility µph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=', without including  carrier-impurity scattering);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (ii) impurity-limited mobil-  ity µimp (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=', without including carrier-phonon scatter-  ing);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (iii) the mobility according to Matthiessen’s rule,  as obtained by combining (i) and (ii) using 1/µM =  1/µph + 1/µimp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (iv) the mobility µ calculated by includ-  ing both carrier-phonon scattering and carrier-impurity  scattering using the aiBTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In panels (a), (c), and (e) we see this comparison for  Si, SiC, and GaP, respectively, as a function of temper-  ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' As expected, in all cases the phonon-limited mo-  bilities (black lines) decrease with temperature while the  impurity-limited mobilities (red lines) do increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Their  combination results into the characteristic smooth peak  which is best seen in the cases of Si and SiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In these pan-  els, the dashed blue lines are from Matthiessen’s rule,  and the solid blue lines are the complete aiBTE solu-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We see that the Matthiessen rule tends to overes-  timate the aiBTE mobility, and the deviation is partic-  ularly pronounced when the phonon and impurity con-  tributions to the mobility reduction are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' To  quantify the deviation between aiBTE calculations and  the Matthiessen results, in panels (b), (d), and (f) of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 5 we show the ratio between the two values, as a  function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In all cases we see that the  use of Matthiessen’s rule leads to an overestimation of  the mobilities by up to 50%, which is significant in the  context of predictive calculations of transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  More importantly, for the compounds considered in this  work (Si, SiC, and GaP), the use of Matthiessen’s rule  would worsen the agreement between calculated mobili-  ties and experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Based on these findings, we caution against the use  of Matthiessen’s rule in future ab initio calculations of  carrier mobilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Improving the predictive power of the aiBTE  In this section we investigate simple approaches to im-  prove the predictive accuracy of the aiBTE by overcom-  ing two standard limitations of DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The first limitation is that the DFT band gap prob-  lem typically leads to an overestimation of the dielectric  screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' As a result, both carrier-phonon and carrier-  impurity matrix elements tend to be underestimated in  DFT [35, 93], and mobilities tend to be overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [58] it was shown that, for a set of ten semiconduc-  tors, this effect leads to mobilities which can overestimate  experimental data by as much as a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' To miti-  gate this effect, we investigate a simple scaling correction  to the matrix elements as follow:  gcorr  mnν(k, q) = εDFT  εexp  gDFT  mnν (k, q),  (33)  where ϵDFT is our calculated value, and ϵexp is the exper-  imental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We use the high-frequency dielectric con-  stant for the carrier-phonon matrix elements, as it was  done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [10], and the static dielectric constants for  the carrier-impurity matrix elements [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (9)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This  approach is meaningful for the systems considered in this  work, because the majority of scattering processes oc-  cur near the band extrema, and therefore involve small  scattering wavevectors q, thus justifying the re-scaling of  screening at long wavelength only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The second limitation of DFT calculations lies in the  inaccurate curvature of the bands, which is also linked  to the band gap problem, leading to slightly inaccurate  carrier effective masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This limitation could be over-  come by performing GW calculations, but in this work  we investigate a simpler mass scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  According to Drude’s formula, carrier mobilities are  inversely proportional to the effective masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Based on  11  this observation, we consider the following scaling correc-  tion, which is directly applied to the calculated mobility:  µcorr = m∗  DFT  m∗exp  µDFT,  (34)  where all masses are isotropic averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The three compounds considered in this work all have  ellipsoidal conduction band extrema, therefore we can  evaluate the average isotropic mass as follows:  m∗ = 3(1/m∗  ∥ + 2/m∗  ⊥)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  (35)  Evaluating the average hole mass is more complicated  owing to the band degeneracy at Γ and the fact that ex-  perimental data usually are reported for a given magnetic  field direction as opposed to a crystallographic direction  (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IV A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the case of silicon, we evaluate the av-  erage mass using the values extracted from Dresselhaus’  model (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IV A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' After this averaging procedure,  the hole mass is calculated following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [58]:  m∗ = m∗,5/2  hh  + m∗,5/2  lh  m∗,3/2  hh  + m∗,3/2  lh  ,  (36)  where all quantities on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' are spherical averages in  k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the case of SiC and GaP we are not aware of  a parametrization similar to Dresselhaus’, therefore we  do not investigate mass corrections in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The carrier mobilities obtained by applying the above  corrections are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In all cases we use the  experimental dielectric constants reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Panels (a) and (b) show our results for silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The  screening correction to the electron mobilities of Si re-  duces the calculated value at low impurity concentra-  tion from 1381 cm2/Vs to 1133 cm2/Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This reduction  causes an underestimation of the experimental value by  approximately 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' At higher impurity concentration,  the corrected mobility agrees again well with experimen-  tal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The corrections to the electron effective mass  of Si are minor and do not affect the mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the  case of holes, the screening and mass corrections improve  considerably the agreement between theory and experi-  ment (our calculated average hole mass is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='43 me while  the experimental value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='48 me).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In fact, we obtain a  hole mobility of 463 cm2/Vs at low impurity concentra-  tion, which is within the measured value between 450 and  500 cm2/Vs[76, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The improvement is also noticeable  at higher impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Results for SiC are shown in panels (c) and (d) of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In this case, we find that screening and mass  corrections do not significantly improve the agreement  with experiments at low impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In par-  ticular, the screening correction reduces the electron mo-  bility from 2047 cm2/Vs to 1815 cm2/Vs, and the mass  correction further reduces this value to 1688 cm2/Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' De-  spite these corrections, the calculated electron mobility  remains too high by about a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' It is possible  that additional scattering mechanisms such as disloca-  tions could contribute to reduce this difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the  case of the hole mobility, the screening correction reduces  the calculated value at low impurity concentration from  164 cm2/Vs to 148 cm2/Vs, which is not significant when  compared to the large spread of experimental values [81–  84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The screening correction appears to be successful in  the case of GaP, as seen in panels (e) and (f) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The electron mobility at low impurity concentration re-  duces from 326 cm2/Vs to 243 cm2/Vs upon applying  the screening correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This value is in better agree-  ment with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Improved agreement  with experiments is also found at higher impurity con-  centration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The correction to the electron effective mass  of GaP is small, and as a result the change in mobility is  not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The screening correction for holes brings  the calculated data closer to the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In partic-  ular, at low impurity concentration the hole mobility is  reduced from 269 cm2/Vs to 226 cm2/Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  The key takeaway from this analysis is that the screen-  ing correction to the scattering matrix elements improves  the agreement between theory and experiment for the  compounds considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Based on the above  observations, we suggest that screening and mass cor-  rections could be used for the purpose of uncertainty  quantification in future ab initio calculations of trans-  port properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  CONCLUSIONS  In this work we have demonstrated non-empirical cal-  culations of carrier mobilities in semiconductors using the  ab initio Boltzmann transport equations, including car-  rier scattering by phonons and by ionized impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' To  this end, we developed an ab initio formalism to incor-  porate ionized-impurity scattering within the transport  workflow based on Wannier-Fourier interpolation and im-  plemented in the EPW code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  We described ionized impurities by randomly dis-  tributed Coulomb scatters, and we obtained the carrier  relaxation time by using the Kohn-Luttinger ensemble  averaging procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We also incorporated the screen-  ing of the impurity potential by free-carriers, within a  parameter-free effective Thomas-Fermi model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  We validated our approach by performing an extensive  set of calculations of the electron and hole mobilities of  three common semiconductors, namely Si, 3C-SiC, and  GaP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In all cases we find a reasonably good agreement  with experimental data, except possibly for the electron  mobility in SiC which is probably reduced by additional  scattering at line defects in real samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Our calcula-  tions follow closely the experimental data both as a func-  tion of temperature (at fixed impurity concentration) and  as a function of impurity concentration (at fixed temper-  ature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Impurity scattering is found to dominate over phonon  scattering at high impurity concentration and at low tem-  perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In the former case, the thermal distribution  12  function of the carrier is peaked near the band edges,  therefore small-q elastic scattering by impurities dom-  inates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In the latter case, the phonon population be-  comes negligible at low temperature, therefore impurities  remain the only active scattering channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' These trends  are fully consistent with the general understanding of car-  rier transport in semiconductors [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We also found that  the energy-dependent carrier scattering rates are strongly  dependent on the detailed mechanisms at play in each  compound, and vary significantly over the energy range  of relevance for transport phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This finding un-  derlines the importance of detailed ab initio calculations  to achieve predictive accuracy in the description of trans-  port phenomena of real materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In the presence of multiple scattering channels, it  is common to analyze mobility data using the clas-  sic Matthiessen rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  However, by directly comparing  aiBTE calculations including both phonon and impurity  scattering with estimates based on Matthiessen’s rule, we  found that the latter lead to inaccurate results, with de-  viations of up to 50% with respect to aiBTE calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  This finding indicates that Matthiessen’s rule should not  be employed in predictive calculations of transport prop-  erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Lastly, we investigated simple corrections to DFT cal-  culations of carrier mobilities, by scaling the calculated  dielectric screening and the effective masses via their  corresponding experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We found that the  screening correction generally improves agreement with  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Overall, our present approach offers a powerful tool for  calculating transport properties in a variety of semicon-  ducting materials of immediate interest, as well as for  screening new putative semiconductors in the context of  materials discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Several improvements upon this work are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' For  one, we do not account for neutral impurity scatter-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  This additional channel could be added by gen-  eralizing our monopole model to account for dipoles  and quadrupoles, following similar work performed in  the context of electron-phonon interactions [47–49, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Generalizations to the case of two-dimensional materi-  als should also be possible, for example by following the  related generalization of the Fr¨ohlich matrix element to  two-dimensional systems [95, 96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' At high impurity con-  centration one should also account for carrier-plasmon  scattering, for example as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' And of  course, any improvement in the DFT band structures and  electron-phonon matrix elements would be highly bene-  ficial to further enhance the predictive power of these  calculations [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' We hope that this study will stimulate  further work along these and other promising directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This research is primarily supported by the Compu-  tational Materials Sciences Program funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Department of Energy, Office of Science, Basic Energy  Sciences, under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  DE-SC0020129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  This re-  search used resources of the National Energy Research  Scientific Computing Center, a DOE Office of Science  User Facility supported by the Office of Science of the  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Department of Energy under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  DE-  AC02-05CH11231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The authors acknowledge the Texas  Advanced Computing Center (TACC) at The University  of Texas at Austin for providing HPC resources that have  contributed to the research results reported within this  paper: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='tacc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Appendix A: Incomplete ionization of dopant  In all calculations presented in this work, we have con-  sidered that the carrier density coincides with the impu-  rity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The implicit assumption underlying  this choice is that all impurities are ionized at all temper-  atures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This is obviously a simplification, since the frac-  tion of ionized impurities depends on the defect energy,  the quasi Fermi level of the system, and the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  These aspects have already been discussed in the case of  silicon in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In this Appendix we analyze the effect of incomplete  ionization for the case of silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' To estimate the fraction  f of ionized impurities at a given temperature, we use the  Fermi-Dirac distribution evaluated at the defect level of  the impurity atom, ϵd [52]:  f =  1  N uc  imp  �  N uc  imp  �  n,k  1  e(ϵnk−ϵd)/kBT + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  (A1)  Here, N uc  imp is the number of impurities per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' This  fraction vanishes when the temperature goes to zero, and  approaches unity at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 7 we show the influence of incomplete impu-  rity ionization on the electron mobility of Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' In these  calculations, we used ϵd =45 meV as measured from the  conduction band bottom [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' By comparing these curves  with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 1(a), we see that the effect of incomplete ion-  ization improves the agreement with experiments at low  temperature and high doping (red curves in both figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  This is precisely the range where carrier-impurity scatter-  ing tends to dominate over phonon scattering, therefore  it is important to have a precise determination of the  impurity concentration in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' A more systematic  assessment of these effects will require a broader database  of experimental mobilities to compare with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  13  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Comparison between our calculated carrier mobilities in Si with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (a) Electron mobility of Si as a  function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The black line and symbols are for low impurity concentration (no impurities in the calculations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  < 1012 cm−3 impurities in the experiment);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' the blue line and symbols are for an impurity concentration of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='75×1016 cm−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' the  red line and symbols are for a concentration of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3×1017 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Filled disks are calculated values, open circles are experimental  data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [75] (black) and [74] (blue and red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (b) Hole mobility of Si as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The black line and  symbols are for low impurity concentration (no impurities in the calculations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 1012 cm−3 impurities in the experiment);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' the  blue line and symbols are for an impurity concentration of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='4×1016 cm−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' the red line and symbols are for a concentration  of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='0·1017 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Filled disks are calculated values, open circles are experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [98] (black) and [74] (blue  and red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (c) Room temperature electron mobility of Si as a function of impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Blue line and filled disks are  calculated data, open black circles are experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  104  electron mobility (cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='Ns)  1500  1250  1000  750  103  500  250  (a)  (c)  0  104  600  500  400  103  300  O  200  100  000  (b)  (d)  102,  0  15  100  200  300  400  500  4  16  17  18  19  10  10  10  10  10  10  temperature (K)  ionized impurity density (cm-3)14  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Comparison between our calculated carrier mobilities in 3C-SiC with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (a) Electron mobility of Si  as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The black line and symbols are phonon-limited mobilities (no impurities in the calculations);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' the  blue line and symbols are for an impurity concentration of 5×1016 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Filled disks are calculated values, open circles are  experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (b) Hole mobility of 3C-SiC as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The black line and symbols are  phonon-limited mobilities (no impurities);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' the blue line and symbols are for an impurity concentration of 1018 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' All data  are calculated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (c) Room temperature electron mobility of 3C-SiC as a function of impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Blue line  and filled disks are calculated data, open symbols are experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [24], Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [79], Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [86], and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (d)  Room temperature hole mobility of 3C-SiC as a function of impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Blue line and filled disks are calculated  data, open symbols are experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [81], Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [82], Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [83], and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  2000  (a)  electron mobility (cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='Ns)  104  1500  1000  103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  8  500  (c)  0  102  250  O  (b)  103  hole mobility (cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='/Ns)  200  150  102  100  50  00  (d)  101  0  200  300  100  400  500  14  15  16  17  18  19  10  10  10  10  10  10  temperature (K)  lonized Dopant Density (cm-3)15  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Comparison between our calculated carrier mobilities in GaP with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (a) Electron mobility of GaP  as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The black line and symbols are phonon-limited mobilities (no impurities in the calculations);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  the blue line and symbols are for an impurity concentration of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='5×1018 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Filled disks are calculated values, open circles  are experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (b) Hole mobility of GaP as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The black line and symbols  are phonon-limited mobilities (no impurities);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' the blue line and symbols are for an impurity concentration of 2 × 1018 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Filled disks are calculated values, open circles are experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (c) Room temperature electron mobility  of GaP as a function of impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Blue line and filled disks are calculated data, open symbols are experimental  data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [87], Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [88], Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [90], and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (d) Room temperature hole mobility of GaP as a function of impurity  concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Blue line and filled disks are calculated data, open symbols are experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [87], Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [91], and  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  300  500  250  400  electron mobility  200  300  150-  200  100  100  50  (a)  (c)  0  0  500  250  400  200  300  150  200  100  100  000000  50  (b)  (d)  0  0  200  400  100  300  500  14  15  16  17  18  19  10  10  10  10  10  10  temperature (K)  ionized impurity density (cm-3)16  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Calculated carrier scattering rates at 300 K, for an impurity concentration of 1017 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (a) Hole scattering rates  in Si: carrier-phonon scattering rates (black disks) and carrier-impurity scattering rates (blue disks), as a function of energy  referred to the valence band maximum (VBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The dashed line and the shaded area represents the thermal distribution of  carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (b) Electron scattering rates in Si: carrier-phonon scattering rates (black disks) and carrier-impurity scattering rates  (blue disks), as a function of energy referred to the conduction band minimum (CBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (c) and (d): same as (a) and (b), but  for 3C-SiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (e) and (f): same as (a) and (b), but for GaP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  60  60  (a)  (b)  carrier-ph  50  50  carrier-i Ni=1017 cm-3  electrons  40  40  les  hol  30  30  S  20  S 20  10  10  60  60  (d)  (C)  50  50  (zHI)  electrons  es  40  40  rate  loy  BC-Sic  30  30  scattering  SiC  20  20  3  10  10  60  60  (e)  (f)  50  50  lectrons  40  40  holes  30  30  e  R  20  20  10  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='20  energy below VBM (eV)  energy above CBM (eV)17  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Comparison between mobility calculations performed using the aiBTE by including both carrier-phonon and carrier-  impurity scattering, and mobilities obtained by using the Matthiessen’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (a) Temperature-dependent electron mobility  of Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The black line and symbols indicate the phonon-limited mobility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' the red line is the impurity-limited mobility, for an  impurity concentration of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3 × 1017 cm−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' the dashed blue line is the mobility obtained from Matthiessen’s rule;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' the solid blue  line is the aiBTE calculation including both phonons and impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (b) Ratio between the electron mobility of Si calculated  using Matthiessen’s rule and the result of the aiBTE calculation with phonon and impurities, as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (c)  and (d): Same as in (a), for for 3C-SiC with an impurity concentration of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='5 × 1018 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (e) and (f): Same as in (a), for for  3C-SiC with an impurity concentration of 5 × 1016 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  (a)  (b)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='5  104  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='4  μaiBT  S  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3  103  phonon only  impurity only  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='1  Matthiessen  phonon+impurity  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='0  (d)  (c)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='5  104  μaiBTE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3  latt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  3  103  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='2  3  phonon only  impurity only  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='1  Matthiessen  phonon+impurity  102  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='0  (e)  (f)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='5  μaiBTE  103  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='4  P  Gal  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3  μMatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='2  O- phonon only  102  impurity only  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='1  Matthiessen  phonon+impurity  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='0  100  200  300  400  500  100  200  300  400  500  temperature (K)  temperature (K)18  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Comparison of correction schemes for improving the predictive accuracy of aiBTE calculations of mobilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (a)  Room-temperature electron mobility of Si, as a function of impurity concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Blue lines and disks indicate the uncorrected  aiBTE results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' green lines and disks indicate calculations with matrix elements corrected for screening;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' purple lines and disks  are calculations corrected for the effective masses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' yellow lines and disks include corrections for both the screening and the  effective masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Open black circles are experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (b) Room temperature hole mobility of Si as a function of impurity  concentration: uncorrected (blue);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' with screening correction (green);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' with effective mass correction (purple);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' and with both  screening and mass correction (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (c) and (d): Same as in (a) and (b) but for 3C-SiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' (e) and (f): Same as in (a) and  (b) but for GaP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The experimental data are the same as those reported in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 1, 2, and 3 [24, 76, 79–84, 86–92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  1500  exp  exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  600  calculated  calculated  scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' scaled  scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' scaled  1250  500  mass scaled  mass scaled  (cm²Ns)  2Ns)  scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' + mass scaled  scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' + mass scaled  1000  400  o)  750  300  an!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  500  S 200  S  250  100  (b)  (a)  250  2000  exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  calculated  calculated  (cm²/Ns)  scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' scaled  scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' scaled  mass scaled  1500  scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' + mass scaled  150  100  500  50  (c)  (d)  0  0  exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  250  300  calculated  scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' scaled  250  200  200  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='150  urd  GaP  100  des  exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  100  calculated  scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' scaled  50  50  mass scaled  (f)  scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' + mass scaled  (e)  0-  0  14  17  15  19  15  16  18  19  14  16  17  18  10  10  10  10  10  10  10  10  10  10  10  10  ionized impurity density (cm-3)  ionized impurity density (cm-3)19  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Electron mobility in Si as a function of temperature, including the effect of incomplete ionization of the dopants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The  black line and disks are the calculated phonon-limited mobilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' These data are compared to measurements for pristine silicon  (impurity concentration < 1012cm−3), from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The blue disks and line are calculations for an impurity concentration of  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='75 × 1016 cm−3, taking into account incomplete dopant ionization as described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Experimental data are from  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' The red disks and line are for an impurity concentration of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3 × 1017 cm−3, taking into account incomplete dopant  ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Experimental data are from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  104  Si μe (cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='Ns)  103  100  200  300  400  500  temperature (K)20  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Pushpakaran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Subburaj, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Bayne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 49, 6247 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Ramkumar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Priya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Rajakumari, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Val-  salan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Chakravarthi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Latha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Math-  upriya, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Rajan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Nanomater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 2022, 8648284  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Green, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Speck, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Xing, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Moens, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Allerstam,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Gumaelius, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Neyer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Arias-Purdue, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Mehro-  tra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Kuramata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Sasaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Watanabe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Koshi,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Blevins, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Bierwagen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Krishnamoorthy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Leedy,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Arehart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Neal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Mou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Ringel,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Sharma, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Ghosh, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Singisetti, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Li,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Chabak, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Liddy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Islam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Rajan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Graham,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Choi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Cheng, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Higashiwaki, APL Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 10,  029201 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  [4] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Papez, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Dallaev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' T¸alu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Kastyl, Materials  14, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3390/ma14113075 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  [5] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Restrepo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Varga, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Pantelides, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Carrete, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Katcho, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Mingo, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 185, 1747 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Fiorentini and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Bonini, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' B 94, 085204  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  [8] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Kim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Park, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Marzari, Nano Letters  16, 2439 (2016), pMID: 26907524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Mustafa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Bernardi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Neaton, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Louie,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Margine, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Giustino, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' B  97, 121201 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Protik and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Kozinsky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' B 102, 245202  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Giustino, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Bernardi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Park, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Bernardi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Materials 6, L010801 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Wright,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Crothers, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Giustino, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Herz, and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Johnston, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Kioupakis, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Davies, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content='  [36] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Graziosi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Kumarasinghe, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Neophytou, ACS  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Energy Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 3, 5913 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Ganose, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Park, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Faghaninia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Woods-  Robinson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Persson, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Jain, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  12, 2222 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Ponc´e, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Margine, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Verdi, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Giustino, Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 209, 116 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  [39] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Freysoldt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Grabowski, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Hickel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Neugebauer,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Kresse, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Janotti, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Van de Walle, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 86, 253 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Bhatnagar and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Baliga, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Electron De-  vices 40, 645 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Roccaforte, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Greco, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Fiorenza, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' La Via,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' P´erez-Tomas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Evans, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Fisher, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Mon-  aghan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Mawby, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Jennings, Materials 14,  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='3390/ma14195831 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  [42] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Renz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' P´erez-Tom´as, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Shah, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Gammon,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Via, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Jennings, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Mawby, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content='  [43] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Zipperian, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Chaffin, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Dawson, IEEE  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' IE-29, 129 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Hughes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Zipperian, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content='  Biefeld, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Dvorack, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Guo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Gao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Dong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Zhao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Leng,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Ma, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Liang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Zhang,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Etheridge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Yan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Gonze, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Rignanese,  and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Hautier, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Napolitano, Modern Quantum Mechan-  ics (2nd Edition) (Pearson;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 2nd edition, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Mermin, Solid State Physics (Saun-  21  ders College Publishing: Fort Worth, TX, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=', 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Brumme, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Bunau,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Cococcioni, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Colonna, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Kawa-  mura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Vitale, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Arita, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Bl¨ugel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Freimuth,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' G´eranton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Gibertini, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Gresch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Johnson,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Iba˜nez-Azpiroz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Lihm,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Marchand, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Marrazzo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Mokrousov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Nohara, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Nomura, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Paulatto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Ponc´e, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Pon-  weiser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Wierzbowska,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Marzari, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Vanderbilt, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' 48, 197 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Kamiya,  IEEE Electron Device Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Krieger, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Naga-  sawa, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Hatta, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Chelnokov, Semicond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Tech-  nol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Sonomura, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Yamamoto, Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E0T4oBgHgl3EQfZgDd/content/2301.02323v1.pdf'}
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+Violation-Aware Contextual Bayesian Optimization for
+Controller Performance Optimization with Unmodeled
+Constraints
+Wenjie Xu∗†, Colin N Jones∗, Bratislav Svetozarevic†,
+Christopher R. Laughman‡, Ankush Chakrabarty‡§
+January 31, 2023
+Abstract
+We study the problem of performance optimization of closed-loop control systems with unmodeled dynamics.
+Bayesian optimization (BO) has been demonstrated to be effective for improving closed-loop performance by
+automatically tuning controller gains or reference setpoints in a model-free manner. However, BO methods have
+rarely been tested on dynamical systems with unmodeled constraints and time-varying ambient conditions. In
+this paper, we propose a violation-aware contextual BO algorithm (VACBO) that optimizes closed-loop perfor-
+mance while simultaneously learning constraint-feasible solutions under time-varying ambient conditions. Unlike
+classical constrained BO methods which allow unlimited constraint violations, or ‘safe’ BO algorithms that are
+conservative and try to operate with near-zero violations, we allow budgeted constraint violations to improve con-
+straint learning and accelerate optimization. We demonstrate the effectiveness of our proposed VACBO method
+for energy minimization of industrial vapor compression systems under time-varying ambient temperature and
+humidity.
+1
+INTRODUCTION
+Closed-loop systems can often be optimized after deployment by altering controller gains or reference inputs guided
+by the performance observed through operational data. Manually tuning these control parameters often requires care
+and effort along with considerable task-specific expertise. Algorithms that can automatically adjust these control
+parameters to achieve optimal performance are therefore invaluable for saving manual effort, time, and expenditure.
+The optimal performance of a control system is generally defined via domain-specific performance functions whose
+arguments are outputs measured from the closed-loop system. While the map from measurements to performance may
+be clearly defined, the map from control parameters (that can actually be tuned) to performance is often unmodeled
+or unknown, since closed-form system dynamics may not be available during tuning [8]. It is thus common to treat
+the control parameters-to-performance map as a black-box, and design a data-driven tuning algorithm, where data is
+collected by experiments or simulations. However, since both experimentation and high-fidelity software simulations
+are expensive, tuning algorithms must be designed to assign a near-optimal set of control parameters with as few
+experiments/simulations (equivalently, performance function evaluations) as possible. Therefore, existing data-driven
+methods that need a large number of samples, such as genetic algorithms [9], can be impractical.
+It is precisely for this reason that Bayesian optimization (BO)1 has received widespread attention in the context
+of closed-loop performance optimization. BO is a sample-efficient derivative-free global optimization method [17,35]
+that utilizes probabilistic machine learning to intelligently search through parameter spaces. [12] gives a detailed
+survey of Bayesian Optimization. In recent work, BO has demonstrated potential in controller gain tuning. For
+example, BO has been applied to the tuning of the PI controller of a heat pump [18] and the tuning of PID cascade
+controller gains [19]. BO has also been applied to the performance optimization of model predictive control. For
+∗Laboratoire
+d’Automatique,
+´Ecole
+polytechnique
+f´ed´erale
+de
+Lausanne,
+Lausanne,
+Switzerland.
+�
+{wenjie.xu,
+colin.jones}@epfl.ch.
+†Swiss Federal Laboratories for Materials Science and Technology, Switzerland. � bratislav.svetozarevic@empa.ch
+‡Mitsubishi Electric Research Laboratories, Cambridge, MA, USA. � laughman@merl.com
+§Corresponding author. � achakrabarty@ieee.org. � +1 (617) 758-6175. � 201 Broadway, 8th Floor, Cambridge, MA 02139, USA.
+1Also known as efficient global optimization (e.g., in [17]) or kriging (e.g., in [16]) in optimization and engineering literature.
+1
+arXiv:2301.12099v1  [cs.LG]  28 Jan 2023
+
+−10.0
+−7.5
+−5.0
+−2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+x
+−4
+−2
+0
+2
+4
+y
+predicted mean
+ f (x) = g(x) 
+zero constraint 
+OPT
+Local minima
+(a) Safe BO [32]
+−10.0
+−7.5
+−5.0
+−2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+x
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+y
+predicted mean
+ f (x) = g(x) 
+zero constraint
+OPT
+Intolerable
+(b) Constrained BO [14]
+−10.0
+−7.5
+−5.0
+−2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+x
+−4
+−2
+0
+2
+4
+y
+predicted mean
+ f (x) = g(x) 
+zero constraint
+OPT
+(c) Our violation-aware BO
+Figure 1: A motivating example comparing our violation-aware BO to existing state-of-the-art methods. Safe BO
+gets stuck in a local minimum and fails to identify the global optimum, while generic constrained BO identifies the
+global optimum but can incur large constraint violations during the sampling process. In contrast, our method can
+simultaneously identify the global minimum and manages the constraint violation well.
+example, BO was applied to optimize the nominal linear model of a predictive controller [3], to tune the parameters
+of MPC to optimize the closed-loop performance [29], and to generate candidate parameters for data-driven scenario
+optimization [28]. BO has also been proposed to select closed loop kernel based model [4]. BO was used in many other
+various real-world control applications, such as wind energy systems parameter tuning [1,2], engine calibration [26],
+and space cooling system optimization [8].
+A challenge that has garnered recent interest is that of safe Bayesian optimization; that is, BO in the presence
+of safety-critical system constraints. These constraints may also be unmodeled (‘black-box’), as a mathematical
+representation of the constraint with respect to the control parameters is not always known or straightforward to
+represent. To handle these constraints, safe BO methods have been recently proposed in [32], improved in [31],
+and extended to a more general setting in [33].
+These methods either operate on the principle of not allowing
+any constraint violations during optimization, or leverage partial model knowledge to ensure safety via Lyapunov
+arguments [7].
+In either case, safe BO learns feasible optima without violating unmodeled constraints, or risks
+their violation with a predefined small probability. Often, this conservativeness results in obtaining local minima,
+slow convergence speeds, and reduced data efficiency. Conversely, generic constrained optimization with BO learns
+constraints without paying heed to the amount of constraint violation during the exploration phase [13, 14]. More
+recently, a group of works [34, 36] propose an optimistic constrained optimization approach, with applications to
+control system tuning. These methods are mostly agnostic to the deleterious consequences of constraint violation,
+such as long-term damage to expensive hardware caused by large violations, rendering them impractical for many
+industrial applications. Another direction of BO research proposes the use of budgets on the cost of samples (e.g.,
+neural network training time [30], wall clock time [22], sample number [21] and system failures [23]). However,
+in these existing BO settings, the budget considered is usually related to the effort or failure risk for performance
+function evaluation, and does not provide a way to manage the magnitude of constraint violations.
+For many industrial systems, small constraint violations over a short period are often acceptable if that exploration
+improves the convergence rate of optimization, but large violations are strongly discouraged. For example, in vapor
+compression systems (VCSs), it is imperative that constraint violation on variables such as compressor discharge
+temperature are limited to short time periods. We aim to find a set of near-optimal parameters within as few samples
+as possible since performance evaluation is time-consuming and available tuning time can be limited. Therefore, it
+may be desirable to systematically trade tolerable constraint violations for faster convergence and potentially skipping
+local minima. In other words, for VCSs, the benefits of accelerated global convergence outweigh the cost of short-term
+constraint violations.
+The performance of control systems are also influenced by exogenous signals such as variations in the environment.
+We refer to such signals as context variables. It is often necessary to adapt control policies to maintain the performance
+and feasibility of the control system despite changes in the context variables. For instance, a controller designed
+for both performance and safety may adapt to prioritize safety over performance if the context variable predicts
+an unsafe event about to occur. In order to systematically incorporate contextual information, contextual Bayesian
+optimization approaches have been proposed in [20], where the inputs to the learner include the context variables
+augmented with the optimization variables. Contextual Bayesian optimization approaches have recently been tested
+on controller design applications: for example, safe Bayesian optimization has been extended to the contextual
+case in [11] to tune a room temperature controller via PID gain tuning. Furthermore, in [27], contextual Bayesian
+optimization is applied to cooperative wind farm control to maximize the efficiency of generated power. However,
+2
+
+the incorporation of context in our setting, that is: how to simultaneously tune the controlled system efficiently and
+manage the constraint violations under a tolerable level with time-varying contexts, has not been studied previously,
+to the best of our knowledge.
+In this paper, we propose a novel violation-aware contextual Bayesian optimization approach (VACBO 2) that
+exhibits accelerated convergence compared to safe BO, while ensuring the violation cost is within a prescribed budget
+under time-varying contextual variables. We demonstrate that our VACBO algorithm is less conservative than ‘safe
+BO’ algorithms that tend to be sample-inefficient and can get stuck in a local minimum because they cannot allow
+any constraint violation. The VACBO algorithm is also more cautious than constrained Bayesian optimization, which
+is agnostic to constraint violations and thus, is likely to incur large violation costs. Our VACBO algorithm is based
+on the principle of encouraging performance function evaluation at combinations of control parameters that greatly
+assist the optimization process, as long as it does not incur high constraint violations likely to result in system failure
+or irreversible damage.
+Our VACBO algorithm is an extension of the VABO algorithm [37] to the contextual case. We augment the
+input space with contextual variables and design a tractable auxiliary acquisition optimization problem specific to
+the contextual setting. More specifically, the VABO algorithm can not directly incorporate the impact of contextual
+variables, which can be significant in many applications. For example, the impact of ambient temperature and ambient
+humidity can be significant in vapor compression system set-point tuning. To incorporate the contextual variables, we
+augment the input variables, which we can control, with contextual variables that are measured from the environment.
+With this augmentation, our method can learn the joint impact of the input variables and contextual variables based
+on Gaussian process learning. Furthermore, the existing constrained expected improvement acquisition function
+used in VABO [37] and other constrained BO papers [13, 14] are not readily applicable to the contextual case. To
+address this issue, we extend the commonly used Constrained Expected Improvement (CEI) acquisition function to
+the contextual case and propose the Constrained Proxy Expected Improvement (CPEI) acquisition function.
+Our contributions include:
+1. We propose a new variant of constrained BO methods for control parameter tuning that improves global
+convergence rates within a prescribed amount of constraint violation with guaranteed high probability under a
+time-varying contextual setting;
+2. We propose a simple and tractable constrained auxiliary acquisition function optimization problem for trading
+off performance improvement and constraint violation;
+3. To incorporate the environmental conditions that impact the objective and constraints, we augment the in-
+put space by contextual variables and propose a new acquisition function by extending the commonly used
+Constrained Expected Improvement (CEI) [13,14] to the contextual setting; and,
+4. We validate our algorithms on a set-point optimization problem using a high-fidelity VCS that has been
+calibrated on an industrial HVAC system, with ambient temperature and ambient humidity as two context
+variables. Simulation results with real-world weather signals as context variables demonstrate that our method
+efficiently minimizes the power consumption while simultaneously managing constraint violations within a
+tolerable level.
+We now state the organization of our paper. In Sec. 2, we present the statement of our problem and our proposed
+solution concept. Then in Sec. 3, we present our VACBO algorithm. After that, we give the application result of a
+case study on vapor compression system in Sec. 4. Finally, we conclude the paper in Sec. 5.
+2
+Preliminaries
+2.1
+Problem Statement
+We consider closed-loop systems of the form
+ξ+ = F(ξ, θ, z),
+(1)
+where ξ, ξ+ ∈ Rnξ denote the system state and the successor state (respectively), θ ∈ Θ ⊂ Rnθ the control parameters
+(e.g., set-points) to be tuned, z ∈ Z ⊂ Rnz the context variables (e.g., ambient temperature) that affect the dynamics,
+and F(·, ·, ·) the closed-loop dynamics with initial condition ξ0. We assume that the closed-loop system (1) is designed
+such that it is exponentially stable to a control parameter and context dependent equilibrium state ξ∞(θ, z) for every
+θ ∈ Θ. We further assume that ξ∞(·) is a continuous map on Θ.
+2Code is available at https://github.com/PREDICT-EPFL/VACBO.
+3
+
+To determine the system performance, we define a continuous cost function ℓ(θ, z) : Rnθ+nz → R to be minimized,
+which is an unknown/unmodeled function of the parameters θ and the context variables z. This is not unusual: while
+ℓ may be well-defined in terms of system outputs, it is often the case that the map from control parameters and
+context variables to cost remains unmodeled; in fact, ℓ may not even admit a closed-form representation on Θ and
+Z, c.f. [6,8].
+We also define N unmodeled constraints on the system outputs that require caution during tuning. The i-th
+such constraint is given by gi(θ, z) : Rnθ+nz → R, i ∈ [N], where the notation [N]
+def
+= {i ∈ N, 1 ≤ i ≤ N}; we assume
+each gi(·, ·) is continuous on Θ × Z. We assume that the cost function ℓ(θ, z) and every constraint gi(θ, z), i ∈ [N]
+can be ascertained, either by measurement or estimation, during the hardware/simulation experiment. We introduce
+the brief notation g(·, ·) ≤ 0 ≜ gi(·, ·) ≤ 0, i ∈ [N] and assume that an initial feasible set of solutions is available at
+design time.
+Assumption 1. The designer has access to a non-empty safe set S0 ⊂ Θ × Z such that for any (θ, z) ∈ S0, all
+constraints are satisfied; that is, g(θ, z) ≤ 0 for every (θ, z) ∈ S0.
+While such an initial set S0 can be derived using domain expertise, it is likely that S0 contains only a few
+feasible (θ, z), and at worst could even be a singleton set. Assumption 1 is a common assumption in the literature
+of safe optimization (e.g., [32]). Without this assumption, it is possible that in the initial steps, we may already
+sample some points with large violations, which makes the problem unlikely to be tractable at all. Furthermore,
+we notice that in many applications (e.g., vapor compression system set-points tuning), an initial set of feasible
+(maybe suboptimal) set-points are known based on domain knowledge. Therefore, Assumption 1 is a necessary but
+not restrictive assumption for our problem.
+We cast the control parameter tuning problem as a black-box constrained optimization problem, formally de-
+scribed by
+min
+θ∈Θ
+ℓ(θ, z),
+(2a)
+subject to:
+gi(θ, z) ≤ 0,
+∀i ∈ [N].
+(2b)
+The contextual variable z represents some quantities reflecting the environmental conditions, which we can measure
+but not directly control at each step. We give an example in the following.
+Example 1. Ambient temperature and ambient humidity are two important contextual variables that impact the
+operation of the vapor compression system. Both of them can be measured but not directly controlled during the
+operations of the vapor compression systems.
+Furthermore, z can be time-varying with its own dynamics. We use zt to denote the value of z at time step t.
+Our objective is to solve the constrained optimization problem (2) with limited constraint violations during the
+optimization process. Since the constraints are assumed to be unmodeled and a limited set of feasible solutions is
+known at design time, we do not expect a guarantee of zero constraint violation. The tolerable amount and duration
+of constraint violations are problem-dependent. In some applications, such as vapor compression systems, small
+constraint violations over a short-term are acceptable, while large constraint violations are strongly discouraged.
+In such cases, instead of being overly cautious and ending up with suboptimal solutions, we allow small constraint
+violations as long as the resulting knowledge gathered by evaluating an infeasible (in terms of constraint violation) θ
+accelerates the optimization process or helps avoid local minima.
+Remark. Our formulation (2) can also optimize batch processes over finite-time horizons, say Th.
+This would
+involve defining the objective and constraints over a batch trajectory with stage loss ℓ(θ, z) :=
+1
+Th
+� Th
+0
+l(τ, θ, z) dτ.
+2.2
+Proposed Solution
+We propose a modified Bayesian optimization framework to solve the problem (2) that is violation-aware: the
+algorithm automatically updates the degree of risk-taking in the current iteration based on the severity of constraint
+violations in prior iterations. Concretely, for an infeasible θ, the constraint violation cost is given by
+¯ci(θ, z) ≜ ci
+�
+[gi(θ, z)]+�
+,
+i ∈ [N]
+(3)
+where [gi(·, ·)]+ := max{gi(·, ·), 0} and ci : R≥0 → R≥0. Note that gi corresponds to physically meaningful system
+outputs that we can measure, e.g., temperature.
+This violation cost function ci is user-defined as a means to
+explicitly weigh the severity of ‘small’ versus ‘large’ constraint violations. While the function ci is at the discretion
+of the designer, it needs to satisfy the following mild assumptions in order to achieve desirable theoretical properties;
+see §3.
+4
+
+Assumption 2. The violation cost function ci satisfies:
+(A1) ci(0) = 0,
+(A2) ci(s1) ≥ ci(s2), if s1 > s2 ≥ 0,
+(A3) ci is left continuous on R≥0.
+Assumption 2 captures some intuitive properties required of the violation cost function.
+According to (A1),
+there is no cost associated with no violation. From (A2), we ensure that the violation cost is monotonically non-
+decreasing with increased violations. Finally (A3) ensures that this monotonic increase is smooth and does not
+exhibit discontinuous jumps from the left.
+To adapt the degree of risk-taking based on prior data obtained, we define a violation budget over a horizon of
+T ∈ N optimization iterations. Our goal is to sequentially search over T iterations {θt}T
+t=1 while using a prescribed
+budget of constraint violations in order to obtain a feasible and optimal set of parameters
+min
+θt∈Θ;
+g(θt,zt)≤0
+ℓ(θt, zt)
+subject to:
+T
+�
+t=1
+¯ci(θt, zt) ≤ Bi ,
+i ∈ [N]
+(4)
+where Bi denotes a budget allowed for the i-th violation cost. Note that this formulation is a generalization of
+well-known constrained/safe Bayesian optimization formulations proposed in the literature. If we set all Bi ≡ 0,
+then our formulation is closely related to safe BO [31, 32]. Alternatively, setting Bi ≡ ∞ reduces our problem to
+constrained BO agnostic to violation cost [13,14].
+3
+Violation-Aware Contextual Bayesian Optimization
+3.1
+Bayesian Optimization Preliminaries
+For Bayesian optimization, one models ℓ(x) and g(x) as functions sampled from independent Gaussian processes.
+In our case, the input x to the Gaussian process consists of control parameters θ and the context variables z. At
+iteration t, conditioned on previous input and function evaluation data D := {(θ, z)1:t, ℓ((θ, z)1:t)}, the posterior
+mean and standard deviation of ℓ is given by
+µℓ(x|D) = k⊤
+ℓ (x, xD)K−1
+ℓ
+∆yℓ + µℓ,0(x)
+and
+σ2
+ℓ(x|D) = kℓ(x, x) − k⊤
+ℓ (x, xD)K−1
+ℓ
+kℓ(xD, x),
+where xD = (θ, z)1:t is the set of control parameters and context variables with which previous experiments/simulations
+have been performed. Here,
+kℓ(x, xD) ≜ [kℓ(x, xi)]xi∈xD,
+kℓ(xD, x) ≜ [kℓ(xi, x)]xi∈xD,
+Kℓ ≜ (kℓ(xi, xj))xi,xj∈xD ,
+∆yℓ ≜ [ℓ(xi) − µℓ,0(xi)]xi∈xD,
+and kℓ(·, ·) is a user-defined kernel function and µℓ,0 is the prior mean function, both associated with ℓ; see [12] for
+more details on kernel and prior mean selection. The above quantities are all column vectors, except Kℓ, which is
+a positive-definite matrix. For the constraint functions g, similar expressions for the posterior mean µgi(x|D) and
+standard deviation σgi(x|D) can be obtained.
+The kernelized functions above provide tractable approximations of the cost of the closed-loop system, along with
+the constraint functions, both of which were hitherto unmodeled/unknown. Classical BO methods use the statistical
+information embedded within these approximations to intelligently explore the search space Θ via acquisition func-
+tions. A specific instance of an acquisition function commonly used in constrained BO is the constrained expected
+improved (CEI) function [13]. It is defined as the expectation of the multiplication of improvement as compared
+to the incumbent best objective sampled so far and the feasibility indicator. However, in the contextual case, the
+5
+
+objective may heavily rely on context. The incumbent best objective value under a favorable context may mislead
+the parameter search under the adversarial context.
+Therefore, instead of using the incumbent best objective, we propose a two-step approach. In the first step, we
+use the minimum value of the posterior mean of ℓ to construct a proxy of the best objective sampled so far under a
+different context as in (5).
+ˆℓmin
+t
+(z) = min
+θ∈Θ µℓ((θ, z)|D)
+(5)
+In the second step, we use the minimum value of posterior mean ˆℓmin
+t
+to construct a new acquisition function, the
+Constrained Proxy Expected Improvement (CPEI), given by
+CPEI((θ, z)|D) = E
+�
+� �
+i∈[N]
+1gi(θ,z)≤0 I(θ, z)|D
+�
+� ,
+(6)
+where 1 denotes the indicator function, E denotes the expectation operator, and I(θ, z) = max{0, ˆℓmin
+t
+(z) − ℓ(θ, z)}
+is the improvement of (θ, z) over the proxy ˆℓmin
+t
+(z) for the best incumbent solution over t iterations.
+As gi(θ, z), ∀i ∈ [N] and ℓ(θ, z) are independent, we deduce
+CPEI(θ, z|D) =
+�
+i∈[N]
+P(gi(θ, z) ≤ 0|D)E (I(θ, z)|D) .
+(7)
+We have P(gi(θ, z) ≤ 0|D) = Φ
+� −µgi(θ,z|D)
+σgi(θ,z|D)
+�
+, and the closed-form expression of expected improvement [17],
+E (I(θ, z)|D) = ∆l(θ, z|D)Φ (w) + σℓ(θ, z|D)φ (w) ,
+(8)
+where ∆l(θ, z|D) = ˆℓmin
+t
+(z) − µℓ(θ, z|D), w =
+ˆℓmin
+t
+(z)−µℓ(θ,z|D)
+σℓ(θ,z|D)
+, Φ(·) and φ(·) are the standard normal cumulative
+distribution and probability density functions, respectively.
+3.2
+VACBO Algorithm
+Our VACBO algorithm proposes an auxiliary optimization problem that leverages the constrained proxy expected
+improvement acquisition function to guide the search of feasible points with potentially lower objective to evaluate
+while ensuring (with high probability) that the violation cost will remain within a prescribed budget.
+Given the total violation cost budget, a question is how to allocate the budget across different samples. Intuitively,
+it may be beneficial to dynamically adjust the violation cost budget allocated to a single step. For example, if we
+find that we incur no violation cost for several steps, it is possible that we are overly cautious in those steps and
+may get stuck in a local minimum. So we can then increase the violation cost budget allocated for the next step. It
+is also possible that we incur significant violation in some step due to the over-confidence in the constraint function
+prediction by Gaussian process regression. In this case, we need to decrease the violation cost budget allocated to
+one single step. To capture this intuition in our algorithm, we design a violation cost budget allocation scheme to
+dynamically adjust the violation cost allocated to one single step. We use Bi,t to denote the violation cost budget
+allocated to the step t for the i-th constraint. Our violation cost budget allocation scheme is given as,
+Bi,t ≜ min
+�
+max
+�
+BiSi,t −
+t−1
+�
+τ=1
+¯ci(θτ, zτ), 0
+�
+, Bmax
+i
+�
+,
+(9)
+where Si,t is a non-negative and non-decreasing sequence that satisfies Si,T = 1 and Bmax
+i
+is the maximum violation
+cost tolerable for the i-th constraint, which is a user-provided parameter that captures the user’s maximum tolerance
+for constraint violation in one single step.
+At this iteration, after observing the context variables, we solve the following auxiliary problem
+θ⋆
+t := arg max
+θ∈Θ CPEI(θ, z|D),
+(10a)
+subject to:
+�
+i∈[N]
+P(¯ci(θ, z) ≤ Bi,t) ≥ 1 − ϵt,
+(10b)
+6
+
+to compute the next control parameter candidate θ⋆
+t , where 0 < ϵt ≪ 1 determines the probability of large constraint
+violation.
+Note that (10a) involves maximizing the constrained expected improvement type objective, which is
+common to cBO algorithms; c.f. [13]. Our modification using the budget, as written in (10b), enforces that the next
+sampled point will not use up more than the violation cost budget Bi,t for all constraints with a probability of at
+least 1 − ϵt, conditioned on the data seen so far. This modification allows us to trade a prescribed level of violation
+risk for more aggressive exploration, leading to faster convergence. Although we use the constrained proxy expected
+improvement acquisition function here, we can easily generalize to other acquisition functions by simply replacing
+CPEI in the objective (10a).
+We now discuss how to efficiently solve the auxiliary problem (10). Recall from Assumption 2 that ci is non-
+decreasing on R≥0 for every i ∈ [N]. Therefore, we can define an inverse violation function c−1
+i (s) = sup{r ∈ R≥0 |
+ci(r) ≤ s} for any s ∈ R≥0. Therefore, we can write P(¯ci(θ, z) ≤ Bi,t|D) = P([gi(θ, z)]+ ≤ c−1
+i (Bi,t)|D) = P(gi(θ, z) ≤
+c−1
+i (Bi,t)|D). Since gi(θ, z) follows a Gaussian distribution with mean µgi(θ, z|D) and variance σ2
+gi(θ, z|D), we get
+P(¯ci(θ, z) ≤ Bi,t|D) = Φ
+�c−1
+i (Bi,t) − µgi(θ|D)
+σgi(θ|D)
+�
+.
+When the number of control parameters nθ is small (e.g., < 6), we can place a grid on Θ and evaluate the cost
+and constraints of (10) at all the grid nodes. The maximum feasible solution can then be used as the solution to
+the auxiliary problem. When the number of control parameters is large (e.g., nθ > 6), we can use gradient-based
+methods with multiple starting points to solve problem (10), since evaluating the learned GPs approximating ℓ and
+g require very little computational time or effort when T is not large (empirically, < 2000). We provide pseudocode
+for implementation in Algorithm 1.
+The following proposition provides a probabilistic guarantee of keeping the violation cost below the given budget.
+It highlights the “violation-awareness” property exhibited by VACBO.
+Proposition 1. Fix δ ∈ (0, 1) and T ∈ N. If ϵt, t ∈ [T] are chosen such that δ = 1 − �T
+t=1(1 − ϵt), then the VACBO
+algorithm satisfies the probability that
+�
+max
+t∈[T ] ¯ci(θt, zt) ≤ Bmax
+i
+and
+T
+�
+t=1
+¯ci(θt, zt) ≤ Bi, ∀i ∈ [N]
+�
+is at least 1 − δ.
+Proof. Let
+E1
+t :=
+�
+max
+τ∈[t] ¯ci(θτ, zτ) ≤ Bmax
+i
+, ∀i ∈ [N]
+�
+E2
+t :=
+�
+t
+�
+τ=1
+¯ci(θτ, zτ) ≤ BiSi
+t, ∀i ∈ [N]
+�
+Et := E1
+t ∩ E2
+t ,
+where t ∈ [T]. Furthermore, we let
+∆E1
+t := {¯ci(θt, zt) ≤ Bmax
+i
+, ∀i ∈ [N]}
+Notice that E1
+t+1 = E1
+t ∩ ∆E1
+t+1. We have
+P (ET ) ≥ P (ET −1) P (ET | ET −1)
+= P (ET −1) P
+�
+E1
+T −1 ∩ ∆E1
+T ∩ E2
+T |ET −1
+�
+= P (ET −1) P
+�
+∆E1
+T ∩ E2
+T |ET −1
+�
+= P (ET −1) P (¯ci(θT , zT ) ≤ Bi,T , ∀i ∈ [N]|ET −1)
+≥ P(ET −1)(1 − ϵT ),
+(11)
+where the last equality follows by that
+Bi,T ≜ min
+�
+BiSi,T −
+T −1
+�
+τ=1
+¯ci(θτ, zτ), Bmax
+i
+�
+7
+
+conditioned on ET −1. By recursion, we have
+P (ET ) ≥ P (E1)
+T
+�
+t=2
+(1 − ϵt) ≥
+T
+�
+t=1
+(1 − ϵt) = 1 − δ,
+which concludes the proof since Si
+T = 1.
+Algorithm 1 Violation-Aware Contextual Bayesian Optimization
+1: Require: VACBO horizons T, violation total budget Bi, ∀i ∈ [N], maximum allowed violation cost for a single
+step Bmax
+i
+, ∀i ∈ [N], and an initial safe set of points X0
+2: Evaluate ℓ(θ, z), gi(θ, z), ∀i ∈ [N] for (θ, z) ∈ X0 by performing experiments or simulation, or using historical
+data
+3: Initialize dataset
+D = {(θt, zt), ℓ(θt, zt), g(θt, zt) ∀(θt, zt) ∈ X0}
+4: for t ∈ [T] do
+5:
+Observe the context variables zt for step t
+6:
+Bi,t ≜ min
+�
+max
+�
+BiSi,t − �t−1
+τ=1 ¯ci(θτ, zτ), 0
+�
+, Bmax
+i
+�
+7:
+Θt =
+��
+i∈[N] P(¯ci(θ, zt) ≤ Bi,t|D) ≥ 1 − ϵt|θ ∈ Θ
+�
+8:
+θ⋆
+t = arg maxθ∈Θt CPEI(θ, zt|D)
+▷ Solving (10)
+9:
+ℓ(θ⋆
+t , zt), g(θ⋆
+t , zt) ← perform experiment with θ⋆
+t under the context zt
+10:
+Update dataset,
+D ← D ∪ {(θ⋆
+t , zt), ℓ(θ⋆
+t , zt), g(θ⋆
+t , zt)}
+11:
+Update Gaussian process posterior
+12: end for
+4
+Case Study: Constrained VCS Optimization
+In this section, we apply the violation-aware contextual Bayesian optimization framework to safely tune the set-points
+of a vapor compression system (VCS). As shown in Fig. 2, a VCS typically consists of a compressor, a condenser, an
+expansion valve, and an evaporator. While physics-based models of these systems can be formulated as large sets of
+nonlinear differential algebraic equations to predict electrical power consumption, there are a variety of challenges in
+developing and calibrating these models. This motivates interest in directly using measurements of the power under
+different operating conditions to search for optimal set-points to the VCS actuators using data-driven, black-box
+optimization methods such as BO, to minimize the power consumption.
+During the tuning process, one constraint, considered here, is on the temperature of the refrigerant leaving the
+compressor, also referred to as the ‘discharge temperature’ (see Fig. 2). The discharge temperature must be managed
+because compressors are designed to operate within specific temperature ranges; excessively high temperatures can
+result in the breakdown of refrigerant oils and increase wear and tear. In addition, high temperatures are often
+correlated with high pressures, which can cause metal fatigue and compromise the integrity of the pressurized
+refrigerant pipes in extreme cases. While managing the constraints mentioned above in the long run is critical, we
+also observe that small violations over short periods of time have limited harmful effects. Indeed, it may be beneficial
+to take the risk of short-period limited violation to accelerate the tuning process.
+Meanwhile, another challenge of tuning comes from the time-varying ambient conditions. Two major ambient
+conditions that have significant impact on the performance of vapor compression systems are the ambient temperature
+and the ambient humidity. The feasibility and optimality of VCS setpoints may be strongly correlated with these
+ambient conditions. For example, one set of fixed setpoints may lead to a feasible discharge temperature with low
+ambient temperature, but result in discharge temperature violation when the ambient temperature becomes high. It
+is necessary to adapt the set-points to the ambient conditions.
+We cast the VCS optimization problem in the same form as (2), with ℓ((θ, z)) denoting the steady-state power
+of the VCS with set points θ and contextual variables z. The constraint g ≤ 0 is given by Td((θ, z)) − ˆTd ≤ 0,
+where Td((θ, z)) is the steady-state discharge temperature with set points θ and context variables z, and ˆTd is a
+safe upper bound. We close a feedback loop from compressor frequency to room temperature, leaving the set of
+8
+
+Figure 2: Schematic diagram of vapor compression system with our proposed VACBO controlling the EEV (electronic
+expansion valve) and the two fans’ speed. For simplicity, we do not show other measurements and controls.
+3 tunable set points θ as the expansion valve position and the indoor/outdoor fan speeds. We set the contextual
+variables to be ambient temperature and the ambient humidity, which are the two major ambient factors influencing
+the performance of vapor compression systems. The effects of these set points and contextual variables on power
+and discharge temperature are not easy to model, and no simple closed-form representation exists. In practice, we
+assign a setpoint θ under contexts z, wait for an adequate amount of time until the power signal resides within a
+95% settling zone, and use that power value as ℓ((θ, z)) and the corresponding discharge temperature as Td((θ, z)).
+Implementation Details
+We use a high-fidelity model of the dynamics of a prototype VCS3 written in the Modelica language [24] to collect
+data and optimize the set-points on-the-fly. A complete model description is available in [8]. The model was first
+developed in the Dymola [10] environment, and then exported as a functional mockup unit (FMU) [25]. Its current
+version comprises 12,114 differential equations.
+We sample the system state each second. To leave enough time
+for the system state to converge to the steady state, we update the set-points every 180s. Bayesian optimization is
+implemented in GPy [15].
+We define our set-point search space Θ := [200, 350]×[300, 450]×[500, 850], in expansion valve counts, indoor fan
+rpm (revolutions per minute), and outdoor fan rpm, respectively. We aim to keep the discharge temperature below
+ˆTd = 333 K; these constraints are set according to domain knowledge [6]. We initialize the simulator at an expansion
+valve position of 340 counts, an indoor fan speed of 440 rpm, and an outdoor fan speed of 840 rpm, which is known
+to be a feasible set-point based on experience.
+Constraint violations are penalized with the function ci(s) = s2, s ∈ R+. The quadratic nature of the violation
+cost implies that minor violations are not as heavily penalized as larger ones. The reason for this is that small
+violations over a small period of time are unlikely to prove deleterious to the long-term health of the VCS, whereas
+large violations could have more significant effects, even over short periods of time; for instance, damage to motor
+winding insulation or exceeding mechanical limits on the pressure vessel of the compressor. These constraints have
+been incorporated into the selection of the thresholds ˆTd.
+Of course, the threshold values and a parameterized
+violation cost could be considered to be hyperparameters, and could be optimized via further experimentation. We
+3Note that while the behavior of this model has been validated against a real VCS, the numerical values and/or performance presented
+in this work are not representative of any product.
+9
+
+VAPOR COMPRESSION SYSTEM (VCS)
+comp freq, CF
+discharge temp
+LNTERNAL VARIABLES TO BE
+②
+zone temp, T
+compressor
+ASSIGNED:
+evaporator
+EXPANSION VALVE POSITION
+condenser
+2.
+INDOOR FAN SPEED
+indoor fan
+3.
+OUTDOOR FAN SPEED
+speed, IFS
+outdoor fan I
+speed, OFS
+heat load
+EEV position
+. ambient air
+3
+temp.
+VACBO
+zone
+electronic
+expansion valv
+POWER (OBJECTIVE)
+DISCHARGE TEMPERATURE (CONSTRAINED VARIABLE)
+AMBIENT CONDITIONS (TEMPERATURE, HUMIDITY)choose the RBF kernel for our problem, which is commonly used in Bayesian optimization [12], and compare our
+method to two other state-of-the-art BO methods, namely, safe BO [5,11] and generic constrained BO (cBO) [14]. To
+ensure a fair comparison, we also extend the other two state-of-the-art BO methods to the contextual setting. More
+specifically, we augment the input space of Gaussian processes with contextual variables and inherit the algorithms
+of both safe BO and generic cBO. Based on our domain experience and prior knowledge, we set the kernel variance to
+be 15.0 (2.0, respectively), the kernel lengthscales in control parameters to be [50, 60, 70] ([20, 24, 28], respectively) for
+the objective (constraint) and the kernel lengthscales in contextual variables to be [1.0, 0.06] ([1.0, 0.06], respectively)
+for the objective (constraint).
+We showcase the effect of varying Bi, i = 1 by selecting B1 = 0, 10, and 20, and set S1,t = ai + bi t
+T , where
+ai + bi = 1 and ai ≥ 0 and bi ≥ 0.This choice allows the algorithm to use an increasing fraction of the total budget
+minus the violation cost incurred in the previous steps when approaching the sampling limit T. Intuitively, such a
+choice of budget sequence gradually increases the violation risk level without sampling too aggressively in the initial
+several steps and allows us to reduce the budget based on the violation cost in the previous steps. Proposition 1
+gives a conservative way of choosing ϵt. In this case study, setting ϵt to a small constant 0.01 works well. We use
+“VACBO B” to indicate violation aware contextual BO with budget B1 = B.
+Generation of Context Sequences. We consider both artificial recurring contexts and real-world contexts. In
+Sec. 4.1, we use artificially generated recurring contexts to showcase the effectiveness of our algorithm in optimizing
+the power consumption while managing the discharge temperature violation well. In Sec. 4.2, we apply our algorithm
+to more realistic setting with real-world historic context sequences measured in Zurich, Switzerland.
+4.1
+Artificial Recurring Contexts
+0
+100
+200
+300
+Time/min
+302.6
+302.8
+303.0
+303.2
+303.4
+303.6
+Ambient Temperature/K
+The First Contextual Variable
+0
+100
+200
+300
+Time/min
+0.600
+0.625
+0.650
+0.675
+0.700
+0.725
+Ambient Humidity
+The Second Contextual Variable
+Figure 3: Recurring contexts used in our experiment.
+To showcase the performance improvement brought by our method, we first apply an artificial sequence of
+recurring contexts. We periodically choose context zt from a pre-defined context list (zp
+i )i∈[n] with additive Gaussian
+noise. Fig. 3 shows the change of the two contextual variables with respect to time in our experiment.
+10
+
+0
+100
+200
+300
+Time/min
+400
+450
+500
+550
+600
+Power/W
+The Objective Value
+0
+100
+200
+300
+Time/min
+310
+315
+320
+325
+330
+335
+340
+Discharge Temperature/K
+The Constraint Variable
+Threshold
+Fixed
+CEI
+VACBO
+Figure 4: The evolution of power and discharge temperature with respect to time under recurring contexts.
+11
+
+0
+200
+400
+600
+Average Power/W
+0
+50
+100
+150
+200
+Maximum Violation/K
+Fixed Solution
+VACBO
+CEI
+Safe BO
+Figure 5: Comparison of average power and maximum constraint violation.
+Results and Discussion
+Fig. 4 illustrates the experimental results of violation-aware contextual Bayesian optimization compared to a fixed
+solution, which fixes the tuning parameters to default feasible values. The fixed solution can maintain the feasibility
+of discharge temperature constraint all the time.
+However, without any set-point exploration and optimization,
+the fixed solution maintains a relatively high operating power all the time.
+In contrast, our VACBO algorithm
+strategically explores the parameter space and optimizes the power function, with only small short-term tolerable
+violation. With more and more samples collected, the VACBO algorithm achieves lower and lower power, significantly
+reducing the energy consumption as compared to the fixed-parameter solution.
+40
+60
+80
+Time/min
+301.5
+302.0
+302.5
+303.0
+Ambient Temperature/K
+The First Contextual Variable
+40
+60
+80
+Time/min
+0.500
+0.525
+0.550
+0.575
+0.600
+Ambient Humidity
+The Second Contextual Variable
+Figure 6: The ambient temperature and the ambient humidity in Zurich, Switzerland, starting at 1:30 PM on 2nd,
+July, 2019.
+We then compare our VACBO algorithm to two state-of-the-art Bayesian optimization methods with constraints,
+safe Bayesian optimization [32] and generic constrained Bayesian optimization [13,14]. Fig. 5 gives the comparison of
+average power and the maximum discharge temperature violation. Our VACBO method reduces the average power
+12
+
+by about 12.2% as compared to fixing the setpoints, while managing the discharge temperature well as shown in
+Fig. 4. In comparison, generic constrained BO incurs discharge temperature that violates the constraint by as large
+as more than 150K, which is very dangerous for the system. As compared to the fixed solution, safe BO shows a
+very minor improvement in terms of the average power.
+4.2
+Real-world Contexts
+To further demonstrate the effectiveness of our method under real-world contexts, we use the ambient temperature
+and the ambient humidity in Zurich, Switzerland as the contexts.
+Fig. 6 shows the context data used for the real-world contexts. Fig. 7 shows the evolution of power and discharge
+temperature with respect to time under real-world contexts for a set of fixed set-points, the generic constrained
+Bayesian optimization method (CEI [13,14]), and VACBO. Similar to the results under recurring contexts, without
+exploration and optimization of the set-point parameter space, the fixed solution keeps operating with high power.
+We further show the average power and maximum discharge temperature violation in Fig. 8. Again, VACBO achieves
+significant power reduction compared to fixed set-point solution and other state-of-the-art method. Furthermore, our
+method incurs small and tolerable constraint violation (< 0.5K), in sharp contrast to the significant violation (≥ 2.0K)
+of generic contextual constrained BO (CEI method).
+2000
+3000
+4000
+5000
+Time/s
+400
+450
+500
+550
+Power/W
+The Objective Value
+2000
+3000
+4000
+5000
+Time/s
+326
+328
+330
+332
+334
+Discharge Temperature/K
+The Constraint Variable
+Fixed
+CEI
+VACBO
+Threshold
+Figure 7: The evolution of power and discharge temperature with respect to time under real-world contexts.
+5
+Conclusions
+In this paper, we design a sample-efficient and violation-aware contextual Bayesian optimization (VACBO) algorithm
+13
+
+0
+100
+200
+300
+400
+500
+Average Power/W
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Maximum Violation/K
+Fixed Solution
+VACBO
+CEI
+Safe BO
+Figure 8: The average power and maximum discharge temperature violation for different methods.
+to solve the closed-loop control performance optimization problem with unmodeled constraints and time-varying
+contextual factors, by leveraging the fact that small violations over a short period only incur limited costs during the
+optimization process in many applications such as for vapor-compression cycles. We strategically trade the budgeted
+violations for faster convergence by solving a tractable auxiliary problem with probabilistic budget constraints at
+each step. Our experiments on a VCS show that, as compared to existing safe BO and generic constrained BO, our
+method simultaneously exhibits improved optimization performance and manages the violation cost well.
+Acknowledgements
+This research was supported by the Swiss National Science Foundation under NCCR Automation, grant agreement
+51NF40 180545, and in part by the Swiss Data Science Center, grant agreement C20-13.
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diff --git a/oNFLT4oBgHgl3EQfgi-Y/content/tmp_files/load_file.txt b/oNFLT4oBgHgl3EQfgi-Y/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf,len=649
+page_content='Violation-Aware Contextual Bayesian Optimization for  Controller Performance Optimization with Unmodeled  Constraints  Wenjie Xu∗†, Colin N Jones∗, Bratislav Svetozarevic†,  Christopher R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Laughman‡, Ankush Chakrabarty‡§  January 31, 2023  Abstract  We study the problem of performance optimization of closed-loop control systems with unmodeled dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Bayesian optimization (BO) has been demonstrated to be effective for improving closed-loop performance by  automatically tuning controller gains or reference setpoints in a model-free manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' However, BO methods have  rarely been tested on dynamical systems with unmodeled constraints and time-varying ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In  this paper, we propose a violation-aware contextual BO algorithm (VACBO) that optimizes closed-loop perfor-  mance while simultaneously learning constraint-feasible solutions under time-varying ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Unlike  classical constrained BO methods which allow unlimited constraint violations, or ‘safe’ BO algorithms that are  conservative and try to operate with near-zero violations, we allow budgeted constraint violations to improve con-  straint learning and accelerate optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We demonstrate the effectiveness of our proposed VACBO method  for energy minimization of industrial vapor compression systems under time-varying ambient temperature and  humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  1  INTRODUCTION  Closed-loop systems can often be optimized after deployment by altering controller gains or reference inputs guided  by the performance observed through operational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Manually tuning these control parameters often requires care  and effort along with considerable task-specific expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Algorithms that can automatically adjust these control  parameters to achieve optimal performance are therefore invaluable for saving manual effort, time, and expenditure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  The optimal performance of a control system is generally defined via domain-specific performance functions whose  arguments are outputs measured from the closed-loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' While the map from measurements to performance may  be clearly defined, the map from control parameters (that can actually be tuned) to performance is often unmodeled  or unknown, since closed-form system dynamics may not be available during tuning [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' It is thus common to treat  the control parameters-to-performance map as a black-box, and design a data-driven tuning algorithm, where data is  collected by experiments or simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' However, since both experimentation and high-fidelity software simulations  are expensive, tuning algorithms must be designed to assign a near-optimal set of control parameters with as few  experiments/simulations (equivalently, performance function evaluations) as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Therefore, existing data-driven  methods that need a large number of samples, such as genetic algorithms [9], can be impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  It is precisely for this reason that Bayesian optimization (BO)1 has received widespread attention in the context  of closed-loop performance optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' BO is a sample-efficient derivative-free global optimization method [17,35]  that utilizes probabilistic machine learning to intelligently search through parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' [12] gives a detailed  survey of Bayesian Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In recent work, BO has demonstrated potential in controller gain tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' For  example, BO has been applied to the tuning of the PI controller of a heat pump [18] and the tuning of PID cascade  controller gains [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' BO has also been applied to the performance optimization of model predictive control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' For  ∗Laboratoire  d’Automatique,  ´Ecole  polytechnique  f´ed´erale  de  Lausanne,  Lausanne,  Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  �  {wenjie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='xu,  colin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='jones}@epfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  †Swiss Federal Laboratories for Materials Science and Technology, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' � bratislav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='svetozarevic@empa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='ch  ‡Mitsubishi Electric Research Laboratories, Cambridge, MA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' � laughman@merl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='com  §Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' � achakrabarty@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' � +1 (617) 758-6175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' � 201 Broadway, 8th Floor, Cambridge, MA 02139, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  1Also known as efficient global optimization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
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+page_content='0  x  −4  −2  0  2  4  y  predicted mean  f (x) = g(x)  zero constraint  OPT  (c) Our violation-aware BO  Figure 1: A motivating example comparing our violation-aware BO to existing state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Safe BO  gets stuck in a local minimum and fails to identify the global optimum, while generic constrained BO identifies the  global optimum but can incur large constraint violations during the sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In contrast, our method can  simultaneously identify the global minimum and manages the constraint violation well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  example, BO was applied to optimize the nominal linear model of a predictive controller [3], to tune the parameters  of MPC to optimize the closed-loop performance [29], and to generate candidate parameters for data-driven scenario  optimization [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' BO has also been proposed to select closed loop kernel based model [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' BO was used in many other  various real-world control applications, such as wind energy systems parameter tuning [1,2], engine calibration [26],  and space cooling system optimization [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  A challenge that has garnered recent interest is that of safe Bayesian optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' that is, BO in the presence  of safety-critical system constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' These constraints may also be unmodeled (‘black-box’), as a mathematical  representation of the constraint with respect to the control parameters is not always known or straightforward to  represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' To handle these constraints, safe BO methods have been recently proposed in [32], improved in [31],  and extended to a more general setting in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  These methods either operate on the principle of not allowing  any constraint violations during optimization, or leverage partial model knowledge to ensure safety via Lyapunov  arguments [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  In either case, safe BO learns feasible optima without violating unmodeled constraints, or risks  their violation with a predefined small probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Often, this conservativeness results in obtaining local minima,  slow convergence speeds, and reduced data efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Conversely, generic constrained optimization with BO learns  constraints without paying heed to the amount of constraint violation during the exploration phase [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' More  recently, a group of works [34, 36] propose an optimistic constrained optimization approach, with applications to  control system tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' These methods are mostly agnostic to the deleterious consequences of constraint violation,  such as long-term damage to expensive hardware caused by large violations, rendering them impractical for many  industrial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Another direction of BO research proposes the use of budgets on the cost of samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=',  neural network training time [30], wall clock time [22], sample number [21] and system failures [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' However,  in these existing BO settings, the budget considered is usually related to the effort or failure risk for performance  function evaluation, and does not provide a way to manage the magnitude of constraint violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  For many industrial systems, small constraint violations over a short period are often acceptable if that exploration  improves the convergence rate of optimization, but large violations are strongly discouraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' For example, in vapor  compression systems (VCSs), it is imperative that constraint violation on variables such as compressor discharge  temperature are limited to short time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We aim to find a set of near-optimal parameters within as few samples  as possible since performance evaluation is time-consuming and available tuning time can be limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Therefore, it  may be desirable to systematically trade tolerable constraint violations for faster convergence and potentially skipping  local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In other words, for VCSs, the benefits of accelerated global convergence outweigh the cost of short-term  constraint violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  The performance of control systems are also influenced by exogenous signals such as variations in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  We refer to such signals as context variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' It is often necessary to adapt control policies to maintain the performance  and feasibility of the control system despite changes in the context variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' For instance, a controller designed  for both performance and safety may adapt to prioritize safety over performance if the context variable predicts  an unsafe event about to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In order to systematically incorporate contextual information, contextual Bayesian  optimization approaches have been proposed in [20], where the inputs to the learner include the context variables  augmented with the optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Contextual Bayesian optimization approaches have recently been tested  on controller design applications: for example, safe Bayesian optimization has been extended to the contextual  case in [11] to tune a room temperature controller via PID gain tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Furthermore, in [27], contextual Bayesian  optimization is applied to cooperative wind farm control to maximize the efficiency of generated power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' However,  2  the incorporation of context in our setting, that is: how to simultaneously tune the controlled system efficiently and  manage the constraint violations under a tolerable level with time-varying contexts, has not been studied previously,  to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  In this paper, we propose a novel violation-aware contextual Bayesian optimization approach (VACBO 2) that  exhibits accelerated convergence compared to safe BO, while ensuring the violation cost is within a prescribed budget  under time-varying contextual variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We demonstrate that our VACBO algorithm is less conservative than ‘safe  BO’ algorithms that tend to be sample-inefficient and can get stuck in a local minimum because they cannot allow  any constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The VACBO algorithm is also more cautious than constrained Bayesian optimization, which  is agnostic to constraint violations and thus, is likely to incur large violation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Our VACBO algorithm is based  on the principle of encouraging performance function evaluation at combinations of control parameters that greatly  assist the optimization process, as long as it does not incur high constraint violations likely to result in system failure  or irreversible damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Our VACBO algorithm is an extension of the VABO algorithm [37] to the contextual case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We augment the  input space with contextual variables and design a tractable auxiliary acquisition optimization problem specific to  the contextual setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' More specifically, the VABO algorithm can not directly incorporate the impact of contextual  variables, which can be significant in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' For example, the impact of ambient temperature and ambient  humidity can be significant in vapor compression system set-point tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' To incorporate the contextual variables, we  augment the input variables, which we can control, with contextual variables that are measured from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  With this augmentation, our method can learn the joint impact of the input variables and contextual variables based  on Gaussian process learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Furthermore, the existing constrained expected improvement acquisition function  used in VABO [37] and other constrained BO papers [13, 14] are not readily applicable to the contextual case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' To  address this issue, we extend the commonly used Constrained Expected Improvement (CEI) acquisition function to  the contextual case and propose the Constrained Proxy Expected Improvement (CPEI) acquisition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Our contributions include:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We propose a new variant of constrained BO methods for control parameter tuning that improves global  convergence rates within a prescribed amount of constraint violation with guaranteed high probability under a  time-varying contextual setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We propose a simple and tractable constrained auxiliary acquisition function optimization problem for trading  off performance improvement and constraint violation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' To incorporate the environmental conditions that impact the objective and constraints, we augment the in-  put space by contextual variables and propose a new acquisition function by extending the commonly used  Constrained Expected Improvement (CEI) [13,14] to the contextual setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' and,  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We validate our algorithms on a set-point optimization problem using a high-fidelity VCS that has been  calibrated on an industrial HVAC system, with ambient temperature and ambient humidity as two context  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Simulation results with real-world weather signals as context variables demonstrate that our method  efficiently minimizes the power consumption while simultaneously managing constraint violations within a  tolerable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  We now state the organization of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 2, we present the statement of our problem and our proposed  solution concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Then in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 3, we present our VACBO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' After that, we give the application result of a  case study on vapor compression system in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Finally, we conclude the paper in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  2  Preliminaries  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='1  Problem Statement  We consider closed-loop systems of the form  ξ+ = F(ξ, θ, z),  (1)  where ξ, ξ+ ∈ Rnξ denote the system state and the successor state (respectively), θ ∈ Θ ⊂ Rnθ the control parameters  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=', set-points) to be tuned, z ∈ Z ⊂ Rnz the context variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=', ambient temperature) that affect the dynamics,  and F(·, ·, ·) the closed-loop dynamics with initial condition ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We assume that the closed-loop system (1) is designed  such that it is exponentially stable to a control parameter and context dependent equilibrium state ξ∞(θ, z) for every  θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We further assume that ξ∞(·) is a continuous map on Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  2Code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='com/PREDICT-EPFL/VACBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  3  To determine the system performance, we define a continuous cost function ℓ(θ, z) : Rnθ+nz → R to be minimized,  which is an unknown/unmodeled function of the parameters θ and the context variables z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' This is not unusual: while  ℓ may be well-defined in terms of system outputs, it is often the case that the map from control parameters and  context variables to cost remains unmodeled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' in fact, ℓ may not even admit a closed-form representation on Θ and  Z, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' [6,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  We also define N unmodeled constraints on the system outputs that require caution during tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The i-th  such constraint is given by gi(θ, z) : Rnθ+nz → R, i ∈ [N], where the notation [N]  def  = {i ∈ N, 1 ≤ i ≤ N};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' we assume  each gi(·, ·) is continuous on Θ × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We assume that the cost function ℓ(θ, z) and every constraint gi(θ, z), i ∈ [N]  can be ascertained, either by measurement or estimation, during the hardware/simulation experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We introduce  the brief notation g(·, ·) ≤ 0 ≜ gi(·, ·) ≤ 0, i ∈ [N] and assume that an initial feasible set of solutions is available at  design time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The designer has access to a non-empty safe set S0 ⊂ Θ × Z such that for any (θ, z) ∈ S0, all  constraints are satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' that is, g(θ, z) ≤ 0 for every (θ, z) ∈ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  While such an initial set S0 can be derived using domain expertise, it is likely that S0 contains only a few  feasible (θ, z), and at worst could even be a singleton set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Assumption 1 is a common assumption in the literature  of safe optimization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=', [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Without this assumption, it is possible that in the initial steps, we may already  sample some points with large violations, which makes the problem unlikely to be tractable at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Furthermore,  we notice that in many applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=', vapor compression system set-points tuning), an initial set of feasible  (maybe suboptimal) set-points are known based on domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Therefore, Assumption 1 is a necessary but  not restrictive assumption for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  We cast the control parameter tuning problem as a black-box constrained optimization problem, formally de-  scribed by  min  θ∈Θ  ℓ(θ, z),  (2a)  subject to:  gi(θ, z) ≤ 0,  ∀i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  (2b)  The contextual variable z represents some quantities reflecting the environmental conditions, which we can measure  but not directly control at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We give an example in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Ambient temperature and ambient humidity are two important contextual variables that impact the  operation of the vapor compression system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Both of them can be measured but not directly controlled during the  operations of the vapor compression systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Furthermore, z can be time-varying with its own dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We use zt to denote the value of z at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Our objective is to solve the constrained optimization problem (2) with limited constraint violations during the  optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Since the constraints are assumed to be unmodeled and a limited set of feasible solutions is  known at design time, we do not expect a guarantee of zero constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The tolerable amount and duration  of constraint violations are problem-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In some applications, such as vapor compression systems, small  constraint violations over a short-term are acceptable, while large constraint violations are strongly discouraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  In such cases, instead of being overly cautious and ending up with suboptimal solutions, we allow small constraint  violations as long as the resulting knowledge gathered by evaluating an infeasible (in terms of constraint violation) θ  accelerates the optimization process or helps avoid local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Our formulation (2) can also optimize batch processes over finite-time horizons, say Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  This would  involve defining the objective and constraints over a batch trajectory with stage loss ℓ(θ, z) :=  1  Th  � Th  0  l(τ, θ, z) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='2  Proposed Solution  We propose a modified Bayesian optimization framework to solve the problem (2) that is violation-aware: the  algorithm automatically updates the degree of risk-taking in the current iteration based on the severity of constraint  violations in prior iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Concretely, for an infeasible θ, the constraint violation cost is given by  ¯ci(θ, z) ≜ ci  �  [gi(θ, z)]+�  ,  i ∈ [N]  (3)  where [gi(·, ·)]+ := max{gi(·, ·), 0} and ci : R≥0 → R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Note that gi corresponds to physically meaningful system  outputs that we can measure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=', temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  This violation cost function ci is user-defined as a means to  explicitly weigh the severity of ‘small’ versus ‘large’ constraint violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' While the function ci is at the discretion  of the designer, it needs to satisfy the following mild assumptions in order to achieve desirable theoretical properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  4  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The violation cost function ci satisfies:  (A1) ci(0) = 0,  (A2) ci(s1) ≥ ci(s2), if s1 > s2 ≥ 0,  (A3) ci is left continuous on R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Assumption 2 captures some intuitive properties required of the violation cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  According to (A1),  there is no cost associated with no violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' From (A2), we ensure that the violation cost is monotonically non-  decreasing with increased violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Finally (A3) ensures that this monotonic increase is smooth and does not  exhibit discontinuous jumps from the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  To adapt the degree of risk-taking based on prior data obtained, we define a violation budget over a horizon of  T ∈ N optimization iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Our goal is to sequentially search over T iterations {θt}T  t=1 while using a prescribed  budget of constraint violations in order to obtain a feasible and optimal set of parameters  min  θt∈Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  g(θt,zt)≤0  ℓ(θt, zt)  subject to:  T  �  t=1  ¯ci(θt, zt) ≤ Bi ,  i ∈ [N]  (4)  where Bi denotes a budget allowed for the i-th violation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Note that this formulation is a generalization of  well-known constrained/safe Bayesian optimization formulations proposed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' If we set all Bi ≡ 0,  then our formulation is closely related to safe BO [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Alternatively, setting Bi ≡ ∞ reduces our problem to  constrained BO agnostic to violation cost [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  3  Violation-Aware Contextual Bayesian Optimization  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='1  Bayesian Optimization Preliminaries  For Bayesian optimization, one models ℓ(x) and g(x) as functions sampled from independent Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  In our case, the input x to the Gaussian process consists of control parameters θ and the context variables z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' At  iteration t, conditioned on previous input and function evaluation data D := {(θ, z)1:t, ℓ((θ, z)1:t)}, the posterior  mean and standard deviation of ℓ is given by  µℓ(x|D) = k⊤  ℓ (x, xD)K−1  ℓ  ∆yℓ + µℓ,0(x)  and  σ2  ℓ(x|D) = kℓ(x, x) − k⊤  ℓ (x, xD)K−1  ℓ  kℓ(xD, x),  where xD = (θ, z)1:t is the set of control parameters and context variables with which previous experiments/simulations  have been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Here,  kℓ(x, xD) ≜ [kℓ(x, xi)]xi∈xD,  kℓ(xD, x) ≜ [kℓ(xi, x)]xi∈xD,  Kℓ ≜ (kℓ(xi, xj))xi,xj∈xD ,  ∆yℓ ≜ [ℓ(xi) − µℓ,0(xi)]xi∈xD,  and kℓ(·, ·) is a user-defined kernel function and µℓ,0 is the prior mean function, both associated with ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' see [12] for  more details on kernel and prior mean selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The above quantities are all column vectors, except Kℓ, which is  a positive-definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' For the constraint functions g, similar expressions for the posterior mean µgi(x|D) and  standard deviation σgi(x|D) can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  The kernelized functions above provide tractable approximations of the cost of the closed-loop system, along with  the constraint functions, both of which were hitherto unmodeled/unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Classical BO methods use the statistical  information embedded within these approximations to intelligently explore the search space Θ via acquisition func-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' A specific instance of an acquisition function commonly used in constrained BO is the constrained expected  improved (CEI) function [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' It is defined as the expectation of the multiplication of improvement as compared  to the incumbent best objective sampled so far and the feasibility indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' However, in the contextual case, the  5  objective may heavily rely on context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The incumbent best objective value under a favorable context may mislead  the parameter search under the adversarial context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Therefore, instead of using the incumbent best objective, we propose a two-step approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In the first step, we  use the minimum value of the posterior mean of ℓ to construct a proxy of the best objective sampled so far under a  different context as in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  ˆℓmin  t  (z) = min  θ∈Θ µℓ((θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' z)|D)  (5)  In the second step,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' we use the minimum value of posterior mean ˆℓmin  t  to construct a new acquisition function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' the  Constrained Proxy Expected Improvement (CPEI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' given by  CPEI((θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' z)|D) = E  �  � �  i∈[N]  1gi(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='z)≤0 I(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' z)|D  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  (6)  where 1 denotes the indicator function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' E denotes the expectation operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' and I(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' z) = max{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' ˆℓmin  t  (z) − ℓ(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' z)}  is the improvement of (θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' z) over the proxy ˆℓmin  t  (z) for the best incumbent solution over t iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  As gi(θ, z), ∀i ∈ [N] and ℓ(θ, z) are independent, we deduce  CPEI(θ, z|D) =  �  i∈[N]  P(gi(θ, z) ≤ 0|D)E (I(θ, z)|D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  (7)  We have P(gi(θ, z) ≤ 0|D) = Φ  � −µgi(θ,z|D)  σgi(θ,z|D)  �  , and the closed-form expression of expected improvement [17],  E (I(θ, z)|D) = ∆l(θ, z|D)Φ (w) + σℓ(θ, z|D)φ (w) ,  (8)  where ∆l(θ, z|D) = ˆℓmin  t  (z) − µℓ(θ, z|D), w =  ˆℓmin  t  (z)−µℓ(θ,z|D)  σℓ(θ,z|D)  , Φ(·) and φ(·) are the standard normal cumulative  distribution and probability density functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='2  VACBO Algorithm  Our VACBO algorithm proposes an auxiliary optimization problem that leverages the constrained proxy expected  improvement acquisition function to guide the search of feasible points with potentially lower objective to evaluate  while ensuring (with high probability) that the violation cost will remain within a prescribed budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Given the total violation cost budget, a question is how to allocate the budget across different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Intuitively,  it may be beneficial to dynamically adjust the violation cost budget allocated to a single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' For example, if we  find that we incur no violation cost for several steps, it is possible that we are overly cautious in those steps and  may get stuck in a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' So we can then increase the violation cost budget allocated for the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' It  is also possible that we incur significant violation in some step due to the over-confidence in the constraint function  prediction by Gaussian process regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In this case, we need to decrease the violation cost budget allocated to  one single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' To capture this intuition in our algorithm, we design a violation cost budget allocation scheme to  dynamically adjust the violation cost allocated to one single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We use Bi,t to denote the violation cost budget  allocated to the step t for the i-th constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Our violation cost budget allocation scheme is given as,  Bi,t ≜ min  �  max  �  BiSi,t −  t−1  �  τ=1  ¯ci(θτ, zτ), 0  �  , Bmax  i  �  ,  (9)  where Si,t is a non-negative and non-decreasing sequence that satisfies Si,T = 1 and Bmax  i  is the maximum violation  cost tolerable for the i-th constraint, which is a user-provided parameter that captures the user’s maximum tolerance  for constraint violation in one single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  At this iteration, after observing the context variables, we solve the following auxiliary problem  θ⋆  t := arg max  θ∈Θ CPEI(θ, z|D),  (10a)  subject to:  �  i∈[N]  P(¯ci(θ, z) ≤ Bi,t) ≥ 1 − ϵt,  (10b)  6  to compute the next control parameter candidate θ⋆  t , where 0 < ϵt ≪ 1 determines the probability of large constraint  violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Note that (10a) involves maximizing the constrained expected improvement type objective, which is  common to cBO algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Our modification using the budget, as written in (10b), enforces that the next  sampled point will not use up more than the violation cost budget Bi,t for all constraints with a probability of at  least 1 − ϵt, conditioned on the data seen so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' This modification allows us to trade a prescribed level of violation  risk for more aggressive exploration, leading to faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Although we use the constrained proxy expected  improvement acquisition function here, we can easily generalize to other acquisition functions by simply replacing  CPEI in the objective (10a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  We now discuss how to efficiently solve the auxiliary problem (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Recall from Assumption 2 that ci is non-  decreasing on R≥0 for every i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Therefore, we can define an inverse violation function c−1  i (s) = sup{r ∈ R≥0 |  ci(r) ≤ s} for any s ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Therefore, we can write P(¯ci(θ, z) ≤ Bi,t|D) = P([gi(θ, z)]+ ≤ c−1  i (Bi,t)|D) = P(gi(θ, z) ≤  c−1  i (Bi,t)|D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Since gi(θ, z) follows a Gaussian distribution with mean µgi(θ, z|D) and variance σ2  gi(θ, z|D), we get  P(¯ci(θ, z) ≤ Bi,t|D) = Φ  �c−1  i (Bi,t) − µgi(θ|D)  σgi(θ|D)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  When the number of control parameters nθ is small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=', < 6), we can place a grid on Θ and evaluate the cost  and constraints of (10) at all the grid nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The maximum feasible solution can then be used as the solution to  the auxiliary problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' When the number of control parameters is large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=', nθ > 6), we can use gradient-based  methods with multiple starting points to solve problem (10), since evaluating the learned GPs approximating ℓ and  g require very little computational time or effort when T is not large (empirically, < 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We provide pseudocode  for implementation in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  The following proposition provides a probabilistic guarantee of keeping the violation cost below the given budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  It highlights the “violation-awareness” property exhibited by VACBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Fix δ ∈ (0, 1) and T ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' If ϵt, t ∈ [T] are chosen such that δ = 1 − �T  t=1(1 − ϵt), then the VACBO  algorithm satisfies the probability that  �  max  t∈[T ] ¯ci(θt, zt) ≤ Bmax  i  and  T  �  t=1  ¯ci(θt, zt) ≤ Bi, ∀i ∈ [N]  �  is at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Let  E1  t :=  �  max  τ∈[t] ¯ci(θτ, zτ) ≤ Bmax  i  , ∀i ∈ [N]  �  E2  t :=  �  t  �  τ=1  ¯ci(θτ, zτ) ≤ BiSi  t, ∀i ∈ [N]  �  Et := E1  t ∩ E2  t ,  where t ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Furthermore, we let  ∆E1  t := {¯ci(θt, zt) ≤ Bmax  i  , ∀i ∈ [N]}  Notice that E1  t+1 = E1  t ∩ ∆E1  t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We have  P (ET ) ≥ P (ET −1) P (ET | ET −1)  = P (ET −1) P  �  E1  T −1 ∩ ∆E1  T ∩ E2  T |ET −1  �  = P (ET −1) P  �  ∆E1  T ∩ E2  T |ET −1  �  = P (ET −1) P (¯ci(θT , zT ) ≤ Bi,T , ∀i ∈ [N]|ET −1)  ≥ P(ET −1)(1 − ϵT ),  (11)  where the last equality follows by that  Bi,T ≜ min  �  BiSi,T −  T −1  �  τ=1  ¯ci(θτ, zτ), Bmax  i  �  7  conditioned on ET −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' By recursion, we have  P (ET ) ≥ P (E1)  T  �  t=2  (1 − ϵt) ≥  T  �  t=1  (1 − ϵt) = 1 − δ,  which concludes the proof since Si  T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Algorithm 1 Violation-Aware Contextual Bayesian Optimization  1: Require: VACBO horizons T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' violation total budget Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' ∀i ∈ [N],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' maximum allowed violation cost for a single  step Bmax  i  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' ∀i ∈ [N],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' and an initial safe set of points X0  2: Evaluate ℓ(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' gi(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' ∀i ∈ [N] for (θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' z) ∈ X0 by performing experiments or simulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' or using historical  data  3: Initialize dataset  D = {(θt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' ℓ(θt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' g(θt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' zt) ∀(θt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' zt) ∈ X0}  4: for t ∈ [T] do  5:  Observe the context variables zt for step t  6:  Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='t ≜ min  �  max  �  BiSi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='t − �t−1  τ=1 ¯ci(θτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' zτ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 0  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Bmax  i  �  7:  Θt =  ��  i∈[N] P(¯ci(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' zt) ≤ Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='t|D) ≥ 1 − ϵt|θ ∈ Θ  �  8:  θ⋆  t = arg maxθ∈Θt CPEI(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' zt|D)  ▷ Solving (10)  9:  ℓ(θ⋆  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' g(θ⋆  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' zt) ← perform experiment with θ⋆  t under the context zt  10:  Update dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  D ← D ∪ {(θ⋆  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' ℓ(θ⋆  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' zt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' g(θ⋆  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' zt)}  11:  Update Gaussian process posterior  12: end for  4  Case Study: Constrained VCS Optimization  In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' we apply the violation-aware contextual Bayesian optimization framework to safely tune the set-points  of a vapor compression system (VCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 2, a VCS typically consists of a compressor, a condenser, an  expansion valve, and an evaporator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' While physics-based models of these systems can be formulated as large sets of  nonlinear differential algebraic equations to predict electrical power consumption, there are a variety of challenges in  developing and calibrating these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' This motivates interest in directly using measurements of the power under  different operating conditions to search for optimal set-points to the VCS actuators using data-driven, black-box  optimization methods such as BO, to minimize the power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  During the tuning process, one constraint, considered here, is on the temperature of the refrigerant leaving the  compressor, also referred to as the ‘discharge temperature’ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The discharge temperature must be managed  because compressors are designed to operate within specific temperature ranges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' excessively high temperatures can  result in the breakdown of refrigerant oils and increase wear and tear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In addition, high temperatures are often  correlated with high pressures, which can cause metal fatigue and compromise the integrity of the pressurized  refrigerant pipes in extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' While managing the constraints mentioned above in the long run is critical, we  also observe that small violations over short periods of time have limited harmful effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Indeed, it may be beneficial  to take the risk of short-period limited violation to accelerate the tuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Meanwhile, another challenge of tuning comes from the time-varying ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Two major ambient  conditions that have significant impact on the performance of vapor compression systems are the ambient temperature  and the ambient humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The feasibility and optimality of VCS setpoints may be strongly correlated with these  ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' For example, one set of fixed setpoints may lead to a feasible discharge temperature with low  ambient temperature, but result in discharge temperature violation when the ambient temperature becomes high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' It  is necessary to adapt the set-points to the ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  We cast the VCS optimization problem in the same form as (2), with ℓ((θ, z)) denoting the steady-state power  of the VCS with set points θ and contextual variables z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The constraint g ≤ 0 is given by Td((θ, z)) − ˆTd ≤ 0,  where Td((θ, z)) is the steady-state discharge temperature with set points θ and context variables z, and ˆTd is a  safe upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We close a feedback loop from compressor frequency to room temperature, leaving the set of  8  Figure 2: Schematic diagram of vapor compression system with our proposed VACBO controlling the EEV (electronic  expansion valve) and the two fans’ speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' For simplicity, we do not show other measurements and controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  3 tunable set points θ as the expansion valve position and the indoor/outdoor fan speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We set the contextual  variables to be ambient temperature and the ambient humidity, which are the two major ambient factors influencing  the performance of vapor compression systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The effects of these set points and contextual variables on power  and discharge temperature are not easy to model, and no simple closed-form representation exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In practice, we  assign a setpoint θ under contexts z, wait for an adequate amount of time until the power signal resides within a  95% settling zone, and use that power value as ℓ((θ, z)) and the corresponding discharge temperature as Td((θ, z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Implementation Details  We use a high-fidelity model of the dynamics of a prototype VCS3 written in the Modelica language [24] to collect  data and optimize the set-points on-the-fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' A complete model description is available in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The model was first  developed in the Dymola [10] environment, and then exported as a functional mockup unit (FMU) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Its current  version comprises 12,114 differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  We sample the system state each second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' To leave enough time  for the system state to converge to the steady state, we update the set-points every 180s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Bayesian optimization is  implemented in GPy [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  We define our set-point search space Θ := [200, 350]×[300, 450]×[500, 850], in expansion valve counts, indoor fan  rpm (revolutions per minute), and outdoor fan rpm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We aim to keep the discharge temperature below  ˆTd = 333 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' these constraints are set according to domain knowledge [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We initialize the simulator at an expansion  valve position of 340 counts, an indoor fan speed of 440 rpm, and an outdoor fan speed of 840 rpm, which is known  to be a feasible set-point based on experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Constraint violations are penalized with the function ci(s) = s2, s ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The quadratic nature of the violation  cost implies that minor violations are not as heavily penalized as larger ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The reason for this is that small  violations over a small period of time are unlikely to prove deleterious to the long-term health of the VCS, whereas  large violations could have more significant effects, even over short periods of time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' for instance, damage to motor  winding insulation or exceeding mechanical limits on the pressure vessel of the compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' These constraints have  been incorporated into the selection of the thresholds ˆTd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Of course, the threshold values and a parameterized  violation cost could be considered to be hyperparameters, and could be optimized via further experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We  3Note that while the behavior of this model has been validated against a real VCS, the numerical values and/or performance presented  in this work are not representative of any product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  9  VAPOR COMPRESSION SYSTEM (VCS)  comp freq, CF  discharge temp  LNTERNAL VARIABLES TO BE  ②  zone temp, T  compressor  ASSIGNED:  evaporator  EXPANSION VALVE POSITION  condenser  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  INDOOR FAN SPEED  indoor fan  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  OUTDOOR FAN SPEED  speed, IFS  outdoor fan I  speed, OFS  heat load  EEV position  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' ambient air  3  temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  VACBO  zone  electronic  expansion valv  POWER (OBJECTIVE)  DISCHARGE TEMPERATURE (CONSTRAINED VARIABLE)  AMBIENT CONDITIONS (TEMPERATURE, HUMIDITY)choose the RBF kernel for our problem, which is commonly used in Bayesian optimization [12], and compare our  method to two other state-of-the-art BO methods, namely, safe BO [5,11] and generic constrained BO (cBO) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' To  ensure a fair comparison, we also extend the other two state-of-the-art BO methods to the contextual setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' More  specifically, we augment the input space of Gaussian processes with contextual variables and inherit the algorithms  of both safe BO and generic cBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Based on our domain experience and prior knowledge, we set the kernel variance to  be 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='0, respectively), the kernel lengthscales in control parameters to be [50, 60, 70] ([20, 24, 28], respectively) for  the objective (constraint) and the kernel lengthscales in contextual variables to be [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='06] ([1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='06], respectively)  for the objective (constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  We showcase the effect of varying Bi, i = 1 by selecting B1 = 0, 10, and 20, and set S1,t = ai + bi t  T , where  ai + bi = 1 and ai ≥ 0 and bi ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='This choice allows the algorithm to use an increasing fraction of the total budget  minus the violation cost incurred in the previous steps when approaching the sampling limit T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Intuitively, such a  choice of budget sequence gradually increases the violation risk level without sampling too aggressively in the initial  several steps and allows us to reduce the budget based on the violation cost in the previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Proposition 1  gives a conservative way of choosing ϵt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In this case study, setting ϵt to a small constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='01 works well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We use  “VACBO B” to indicate violation aware contextual BO with budget B1 = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Generation of Context Sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We consider both artificial recurring contexts and real-world contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='1, we use artificially generated recurring contexts to showcase the effectiveness of our algorithm in optimizing  the power consumption while managing the discharge temperature violation well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='2, we apply our algorithm  to more realistic setting with real-world historic context sequences measured in Zurich, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='1  Artificial Recurring Contexts  0  100  200  300  Time/min  302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='6  302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='8  303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='0  303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='2  303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='4  303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='6  Ambient Temperature/K  The First Contextual Variable  0  100  200  300  Time/min  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='625  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='650  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='675  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='700  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='725  Ambient Humidity  The Second Contextual Variable  Figure 3: Recurring contexts used in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  To showcase the performance improvement brought by our method, we first apply an artificial sequence of  recurring contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We periodically choose context zt from a pre-defined context list (zp  i )i∈[n] with additive Gaussian  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 3 shows the change of the two contextual variables with respect to time in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  10  0  100  200  300  Time/min  400  450  500  550  600  Power/W  The Objective Value  0  100  200  300  Time/min  310  315  320  325  330  335  340  Discharge Temperature/K  The Constraint Variable  Threshold  Fixed  CEI  VACBO  Figure 4: The evolution of power and discharge temperature with respect to time under recurring contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  11  0  200  400  600  Average Power/W  0  50  100  150  200  Maximum Violation/K  Fixed Solution  VACBO  CEI  Safe BO  Figure 5: Comparison of average power and maximum constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Results and Discussion  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 4 illustrates the experimental results of violation-aware contextual Bayesian optimization compared to a fixed  solution, which fixes the tuning parameters to default feasible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' The fixed solution can maintain the feasibility  of discharge temperature constraint all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  However, without any set-point exploration and optimization,  the fixed solution maintains a relatively high operating power all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  In contrast, our VACBO algorithm  strategically explores the parameter space and optimizes the power function, with only small short-term tolerable  violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' With more and more samples collected, the VACBO algorithm achieves lower and lower power, significantly  reducing the energy consumption as compared to the fixed-parameter solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  40  60  80  Time/min  301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='5  302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='0  302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='5  303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='0  Ambient Temperature/K  The First Contextual Variable  40  60  80  Time/min  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='525  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='550  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='575  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='600  Ambient Humidity  The Second Contextual Variable  Figure 6: The ambient temperature and the ambient humidity in Zurich, Switzerland, starting at 1:30 PM on 2nd,  July, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  We then compare our VACBO algorithm to two state-of-the-art Bayesian optimization methods with constraints,  safe Bayesian optimization [32] and generic constrained Bayesian optimization [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 5 gives the comparison of  average power and the maximum discharge temperature violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Our VACBO method reduces the average power  12  by about 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='2% as compared to fixing the setpoints, while managing the discharge temperature well as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In comparison, generic constrained BO incurs discharge temperature that violates the constraint by as large  as more than 150K, which is very dangerous for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' As compared to the fixed solution, safe BO shows a  very minor improvement in terms of the average power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='2  Real-world Contexts  To further demonstrate the effectiveness of our method under real-world contexts, we use the ambient temperature  and the ambient humidity in Zurich, Switzerland as the contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 6 shows the context data used for the real-world contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 7 shows the evolution of power and discharge  temperature with respect to time under real-world contexts for a set of fixed set-points, the generic constrained  Bayesian optimization method (CEI [13,14]), and VACBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Similar to the results under recurring contexts, without  exploration and optimization of the set-point parameter space, the fixed solution keeps operating with high power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  We further show the average power and maximum discharge temperature violation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Again, VACBO achieves  significant power reduction compared to fixed set-point solution and other state-of-the-art method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Furthermore, our  method incurs small and tolerable constraint violation (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='5K), in sharp contrast to the significant violation (≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='0K)  of generic contextual constrained BO (CEI method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  2000  3000  4000  5000  Time/s  400  450  500  550  Power/W  The Objective Value  2000  3000  4000  5000  Time/s  326  328  330  332  334  Discharge Temperature/K  The Constraint Variable  Fixed  CEI  VACBO  Threshold  Figure 7: The evolution of power and discharge temperature with respect to time under real-world contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  5  Conclusions  In this paper, we design a sample-efficient and violation-aware contextual Bayesian optimization (VACBO) algorithm  13  0  100  200  300  400  500  Average Power/W  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='5  Maximum Violation/K  Fixed Solution  VACBO  CEI  Safe BO  Figure 8: The average power and maximum discharge temperature violation for different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  to solve the closed-loop control performance optimization problem with unmodeled constraints and time-varying  contextual factors, by leveraging the fact that small violations over a short period only incur limited costs during the  optimization process in many applications such as for vapor-compression cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' We strategically trade the budgeted  violations for faster convergence by solving a tractable auxiliary problem with probabilistic budget constraints at  each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Our experiments on a VCS show that, as compared to existing safe BO and generic constrained BO, our  method simultaneously exhibits improved optimization performance and manages the violation cost well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  Acknowledgements  This research was supported by the Swiss National Science Foundation under NCCR Automation, grant agreement  51NF40 180545, and in part by the Swiss Data Science Center, grant agreement C20-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  References  [1] Ali Baheri and Chris Vermillion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Waypoint optimization using Bayesian optimization: A case study in airborne  wind energy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' In 2020 Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Control Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' (ACC), pages 5102–5017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content='  [2] Ali Baheri and Christopher Vermillion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
+page_content=' Altitude optimization of airborne wind energy systems: A Bayesian  optimization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFLT4oBgHgl3EQfgi-Y/content/2301.12099v1.pdf'}
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+Generalized Toric Polygons, T-branes, and 5d SCFTs
+Antoine Bourget,1 Andr´es Collinucci,2 Sakura Sch¨afer-Nameki3
+1Universit´e Paris-Saclay, CNRS, CEA, Institut de physique th´eorique,
+91191, Gif-sur-Yvette, France
+1Laboratoire de Physique de l’´Ecole Normale Sup´erieure, PSL University,
+24 rue Lhomond, 75005 Paris, France
+2Service de Physique Th´eorique et Math´ematique, Universit´e Libre de Bruxelles and
+International Solvay Institutes, Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium
+3Mathematical Institute, University of Oxford,
+Woodstock Road, Oxford, OX2 6GG, United Kingdom
+Abstract: 5d Superconformal Field Theories (SCFTs) are intrinsically strongly-coupled
+UV fixed points, whose realization hinges on string theoretic methods: they can be con-
+structed by compactifying M-theory on local Calabi-Yau threefold singularities or alterna-
+tively from the world-volume of 5-brane-webs in type IIB string theory. There is a cor-
+respondence between 5-brane-webs and toric Calabi-Yau threefolds, however this breaks
+down when multiple 5-branes are allowed to end on a single 7-brane. In this paper, we
+extend this connection and provide a geometric realization of brane configurations includ-
+ing 7-branes. A web with 7-branes defines a so-called generalized toric polygon (GTP),
+which corresponds to combinatorial data that is obtained by removing vertices along exter-
+nal edges of a toric polygon. We identify the geometries associated to GTPs as non-toric
+deformations of toric Calabi-Yau threefolds and provide a precise, algebraic description
+of the geometry, when 7-branes are introduced along a single edge. The key ingredients
+in our analysis are T-branes in a type IIA frame, which includes D6-branes. We show
+that performing Hanany-Witten moves for the 7-branes on the type IIB side corresponds
+to switching on semisimple vacuum expectation values on the worldvolume of D6-branes,
+which in turn uplifts to complex structure deformations of the Calabi-Yau geometries. We
+test the proposal by computing the crepant resolutions of the deformed geometries, thereby
+checking consistency with the expected properties of the SCFTs.
+arXiv:2301.05239v1  [hep-th]  12 Jan 2023
+
+Contents
+1
+Introduction and Summary
+1
+2
+T-branes and Kraft-Procesi Transitions
+7
+2.1
+T-brane Basics
+7
+2.2
+Kraft-Procesi Transitions
+10
+3
+Example: GTPs for Tn and Related Models
+13
+3.1
+The Setup
+13
+3.2
+Tachyon Condensation Picture
+15
+3.3
+Example: T2
+17
+3.4
+General Case: Tn
+20
+3.5
+Interpretation in terms of Generalized Toric Polygons
+22
+3.6
+Hanany-Witten Moves
+24
+4
+General Discussion
+28
+4.1
+White Dots along a Single Edge
+28
+4.2
+Rectangular Box
+30
+4.3
+Generic Triangle
+31
+5
+Testing the Proposal: Resolution of Singularities
+33
+5.1
+Crepant Resolution of Tn
+33
+5.2
+Resolution of Deformations of Tn
+34
+A Basic algebraic notions for sln
+38
+1
+Introduction and Summary
+The existence and characterization of interacting superconformal field theories in five space-
+time dimensions (5d SCFTs) is a remarkable prediction of string theory [1]. Two approaches
+have emerged that allow the construction of 5d SCFTs within the framework of string the-
+ory: the low energy limit of M-theory on R1,4 times a local Calabi-Yau threefold [2], and
+the world-volume of a brane-web in type IIB string theory [3–12]. The properties of the
+theory are encoded in the geometry of the threefold in the first case, and in the charges of
+the external (p, q)-5-branes in the second case. A middle ground is the type IIA realization,
+which involves both geometry and branes [13].
+When the CY is toric, there is a precise dictionary between the brane-web and M-theory
+realization: the charges of the external (p, q)-5-branes can be encoded into an integral
+polygon, which in turn can be seen as the intersection of a toric three-dimensional fan in
+– 1 –
+
+←→
+Figure 1. Example of toric polygon (left) and dual brane-web (right) in which lines denote 5-branes
+and circles denote 7-branes. The 7-branes on which stacks of 5-branes are spaced to emphasize how
+the 5-brane end, here exactly one 5-brane ends on each 7-brane. This geometry encodes a 5d SCFT
+of rank 3.
+R3
+x,y,z with the plane {z = 1}. The toric threefold X constructed from this fan is such that
+the 5d SCFT obtained from M-theory on X coincides with that on the world-volume of
+the brane-web [14]. An example is shown in figure 1.
+A systematic geometric exploration and classification of 5d SCFTs was started in
+[15–38]. These studies reveal many detailed properties of the 5d SCFTs, such as their
+UV enhanced flavor symmetry, their Coulomb branch (modeled in terms of the crepant
+resolutions of the Calabi-Yau singularities), but also refined information such as their
+generalized symmetries [39–44].
+What remains somewhat obscure in this framework is
+the derivation of the full quantum corrected Higgs branch – though some progress in the
+context of isolated hypersurface Calabi-Yau singularities can be made [32–34, 45, 46].
+Not surprisingly, these explorations reveal that only a small class of 5d SCFTs have a
+realization in terms of toric Calabi-Yau threefolds. If such a toric realization exists, then
+the geometry of the moduli space of supersymmetric vacua, in particular the Higgs branch
+(but also Coulomb branch and mixed branches) can be computed exactly – irrespective of
+whether the singularity is isolated or not. The key tool is the connection between the toric
+geometries and brane-webs, where in the latter these moduli space questions have been
+determined in [47–53]. In this paper, we report progress in generalizing these methods to
+a larger class of 5d SCFTs, which have not necessarily a toric description.
+In [16], a generalization of toric polygons was introduced: a toric Calabi-Yau threefold
+can be described in terms of a convex polygon in a square integral lattice embedded into
+a 2-plane. The polygon associated to a toric geometry has the property that all lattice
+points along the edges are part of the toric data (corresponding to vertices). We will refer
+to these as black dots. Generalizing this, [16] proposed to also allow some vertices along
+the edges of the polygon to be unoccupied, which we will refer to as white dots. In the
+dual description, allowing such white dots corresponds in the web to several 5-branes that
+end on the same 7-brane. For a stack of n 5-branes, the boundary condition is encoded in
+an integer partition λ of n. Figure 2 gives an example of a configuration that translates
+to the [3, 2, 2, 1] partition of 8. This combinatorial data will be referred to as Generalized
+Toric Polygons (GTP), and generalizes the standard toric description. In this framework,
+the standard toric case considered in the previous paragraph corresponds to a boundary
+condition where for each charge (p, q), there is an equal number of (p, q)-5-branes and
+– 2 –
+
+Figure 2. Correspondence between white dots on GTPs (left) and boundary conditions of (p, q)
+5-branes on (p, q) 7-branes (middle). Here we have (p, q) = (1, 0) and the partition λ = [3, 2, 2, 1]
+of n = 8. When we draw brane-webs we usually ignore the detached 7-branes and separate the
+7-branes on a stack of 5-branes to show the boundary conditions (right).
+(p, q)-7-branes, with one 5-brane ending on one 7-brane, i.e. the partition is λ = [1n]. This
+generalization and its implications for characterizing the moduli space of supersymmetric
+vacua using magnetic quivers were explored in great detail in [32, 45, 53–63]. Note that
+O5 orientifold planes can also be included in the webs [8, 64–66], but we do not consider
+this possibility here.
+Another way of interpreting GTPs is as non-convex would-be toric polygons. These
+make sense when certain parameters, which map out the extended Coulomb branch (i.e.
+gauge couplings and masses of hypers), are turned on, but the non-convexity prevents one
+from considering the SCFT limit (i.e. from passing to the origin of the extended Coulomb
+branch).
+From the dual brane-web point of view, this is resolved using a combination
+of Hanany-Witten moves and 7-brane monodromies, and this plays a prominent role in
+the brane-web manipulations of [6, 8, 9, 11, 64, 67–71]. Importantly, not all non-convex
+polygons can be transformed into GTPs in this way, and it is in general a hard question
+to decide whether a given polygon can be transformed in this way or not. For this reason,
+it is simpler to take the GTPs as our starting point.
+GTPs can be thought of as generalizations of toric geometries. However, unlike the
+precise dictionary between the combinatorial data of a toric polygon and the algebraic
+geometry of the corresponding Calabi-Yau, no such dictionary exists thus far for GTPs.
+The main purpose of this paper is to develop initial steps in order to close this gap.
+See figure 3. In particular, in the following, we will determine the algebraic geometric
+description of GTPs, which have white dots along a single edge.
+Summary.
+We now give a schematic summary of the main ideas involved in our pro-
+posal. Consider a length n edge of a toric polygon, which we can assume to have vertical
+orientation. In the brane-web, this corresponds to n parallel semi-infinite D5-branes. In the
+associated toric threefold, there is an asymptotic region that approaches C2/Zn×C. Indeed
+after a transverse T-duality, the D5-branes become D6-branes, which uplift to n-centered
+– 3 –
+
+Convex Integral Polygon
+Toric CY3
+Brane-web
+5d SCFT
+Toric Geometry
+Dual
+M-theory
+Worldvolume
+GTP
+Non-Toric CY3
+Brane-web with
+7-brane b.c.
+5d SCFT
+Dual
+M-theory
+Worldvolume
+Figure 3. Summary of the main question addressed in this paper. On the left hand side, one
+starts from a convex polygon with integral vertices. It defines a 5d SCFT in two distinct ways:
+from M-theory on the associated toric CY, and from the dual brane-web. If on the contrary, as
+shown on the right hand side, the polygon is not convex, i.e. is a generalized toric polygon (GTP),
+the toric description is lost. However there still exists a dual brane-web, which has non-trivial
+boundary conditions on 7-branes, and thus a 5d SCFT. The central goal of this paper is to develop
+a map (the dashed line in the diagram) from GTPs to (non-toric) geometry.
+Taub-NUT spaces. At strong string coupling gs → ∞, this becomes C2/Zn. Denoting
+two longitudinal directions as the w-complex plane, we arrive at a local C2/Zn × C patch.
+M-theory on this geometry gives us N = 1 7d SYM with SU(n) gauge group, and we can
+represent it as the singular hypersurface in C4 given by
+uv = zn
+(1.1)
+with the w-coordinate tagging along. The three adjoint scalars φi=1,2,3 on the worldvolume
+of the D6-branes can be grouped into a complex scalar Φ = φ1+iφ2, and the remaining real
+one ϕ = φ3. In the M-theory uplift, Φ encodes algebraic deformations to the hypersurface,
+and ϕ encodes K¨ahler volumes of resolutions. This grouping is of course arbitrary, and
+correlates with the arbitrariness of choosing a complex structure on the noncompact K3 in
+M-theory.
+Having seen this, we can recast the hypersurface as a spectral equation for the com-
+plexified adjoint Higgs field
+uv = det(1n z − Φ(w)) .
+(1.2)
+Switching on constant vevs along the Cartan subalgebra of su(n) will deform the equation
+and unfold the singularity into a deformed K3 times the w-plane. However, switching on w-
+dependent vevs will turn this into a bona fide noncompact CY threefold. The geometry will
+be more or less desingularized, depending on the Casimir invariants of Φ that are switched
+on. The claim of this paper, is that white dots correspond to nilpotent elements in Φ. Note,
+that switching on a nilpotent Φ means that the spectral equation remains unchanged. In
+other words, the D6-branes do not actually move, and the uplifted geometry underlying
+the M-theory construction remains undeformed. This phenomenon is known as a T-brane
+[72–74]. It is a non-Abelian bound state of branes, whereby the worldvolume gauge group
+is (partially) Higgsed, but the geometry of the branes is unaltered. However, the physics
+is of course impacted by this T-brane, as we shall see momentarily.
+– 4 –
+
+Figure 4. White dot and brane transition for a length 2 edge of a toric polygon.
+To give a simple example, take a vertical edge with n + 1 dots, and replace the second
+black dot from the top with a white dot as shown in figure 4. In terms of the 5-branes, this
+corresponds to forming a bound state between the two uppermost branes, and sending a
+suspended 5-brane segment to infinity. From the D6-brane viewpoint, the bound state is
+understood as switching on a vev for Φ along the minimal nilpotent orbit of su(n)
+⟨Φ⟩ =
+�
+�
+�
+0 1
+0 0
+...
+�
+�
+� .
+(1.3)
+This binds the first two D6-branes, and partially Higgses
+su(n) → s (u(1) ⊕ u(n − 2)) .
+(1.4)
+More generally, we said above that a distribution of white dots on the edge is encoded in
+a partition λ of n. Our claim is that each such partition λ of n translates into a vev for
+Φ along an element in the nilpotent orbit Oλ of su(n) that is uniquely characterized by λ.
+The unbroken 7d gauge group on the D6-branes, which will correspond to a subgroup of
+the total 5d flavor group, is then broken to the commutant of this nilpotent element.
+So far, our discussion parallels the picture developed several years ago in [75, 76], in
+the 6d SCFT context, whereby geometric data (about elliptic fibrations) was supplemented
+by nilpotent orbits, which would partially Higgs an original theory and trigger various
+RG flows. At this point, the reader might object that simply claiming that a white dot
+translates to a nilpotent vev is not very interesting or verifiable, since that data will be
+invisible to the geometry. While this is true, from the 5-brane-web perspective we know
+that a white dot opens up the possibility to perform Hanany-Witten type transitions that
+were not possible in the presence of black dots only. For instance, the following GTPs are
+related by such a transition where one of the three leftmost 7-branes is moved to the right:
+↔
+(1.5)
+– 5 –
+
+Such HW type transitions provide non-trivial tests of our proposal: HW moves correspond
+to changing the positions of branes, which in turn will impact the dual geometry. We
+identify the subset of complex structure deformations that are associated to these nilpotent
+vevs of Φ. They are realized in terms of vevs of Φ along a slice transverse to the nilpotent
+vev (inside the full Lie algebra, not the nilpotent cone), known as the Slodowy slice. By
+switching on such a vev, a subset of possible Casimir invariants will become non-zero,
+leading to a deformation of the geometry, which is given by the spectral equation of the
+the Higgs field.
+For instance, in the simple case of su(2), we take the initial nilpotent vev along the
+minimal orbit
+Φ0 =
+�
+0 1
+0 0
+�
+.
+(1.6)
+The M-theory uplifted geometry corresponds to C2/Z2.
+The Slodowy slice is given by
+matrices of the form
+Φ =
+�
+0 1
+a 0
+�
+,
+with
+a ∈ C .
+(1.7)
+The characteristic polynomial of this Higgs field is now non-trivial, and the M-theory
+geometry deforms as follows
+uv = z2
+−→
+uv = z2 + a .
+(1.8)
+The present paper elucidates this for all GTPs, which allow for a IIA-description, i.e.
+whose white dots are along a single edge of the GTP. Dually, all 7-branes are parallel, i.e.
+mutually local. The toy-model where Slodowy slices appear is generalized in the following
+way. Higgs branches are symplectic singularities [77], to which one can associate a Hasse
+diagram of symplectic leaves [78–80]. In terms of this diagram, the T-brane data select a
+new, lowest leaf of the foliation, and the transverse slice to that leaf is the total space of a
+fibration over the complex structure moduli space of the deformed Calabi-Yau threefold. See
+for instance figure 5, where the deformations of the T4 5d SCFT, realized on the threefold
+W1W2W3 = Z4, are displayed, along with the effect on the Higgs branches.
+In future work we will aim to generalize this to arbitrary GTPs, with mutually non-
+local 7-branes.
+We conjecture that the above picture of transverse slices in the Higgs
+branch extends to this situation. Eventually we hope to develop a succinct description of
+the algebraic geometry of GTPs, as they exist for toric polygons: A precise map between
+the combinatorial data and the basic algebraic geometry, such as the set of divisors, curves,
+intersection numbers.
+Plan.
+In the rest of the paper, we spell out the details of the construction. An essential
+tool is the T-brane, which is reviewed in section 2. The bulk of the construction is then
+carried out explicitly in a representative example – that of the Tn SCFTs – in section 3,
+before generalizing to any GTP with white dots on a single edge in section 4. As a first
+check, we reproduce there the transition (1.5). Finally, in section 5 we provide consistency
+checks, by computing the resolutions of the deformed threefold geometries. This shows
+– 6 –
+
+a3
+a1
+a7
+e6
+e7
+α ̸= 0
+a1
+a7
+e6
+e7
+α ̸= 0
+β ̸= 0
+a7
+e6
+e7
+W1W2W3 = Z4
+W1W2W3 = Z2(Z2 + αW1)
+W1W2W3 = (Z2 + αW1)(Z2 + βW1)
+Figure 5.
+Three GTPs are shown on the first line, and below the algebraic equations characterizing
+the associated Calabi-Yau threefold geometry. The model on the left is a toric threefold. The other
+two, non-toric GTPs, are characterized in terms of deformations. Each of these geometries defines
+a 5d SCFT. The Hasse diagrams of symplectic singularities for the Higgs branch of these 5d SCFTs
+are shown below. The vertices represent symplectic leaves. For transverse slices we use a standard
+notation where the closure of the minimal nilpotent orbit of a simple Lie algebra is denoted using
+the lowercase form of the name of the algebra, e.g. e7 for algebra E7. In red are drawn the effects
+of the deformations.
+agreement of the UV flavor symmetry of the SCFT with the one expected from the brane-
+web (and resulting Higgs branch).
+2
+T-branes and Kraft-Procesi Transitions
+2.1
+T-brane Basics
+Consider n parallel D7-branes in type IIB string theory on flat space.
+The transverse
+space is the complex plane with coordinate z, which has coordinate ring R = C[z]. We
+call z1, . . . , zn the positions of the n branes. The stack of branes can be described as a
+D9/D9-brane tachyon condensate, which is defined mathematically as the cokernel of the
+tachyon map
+R⊕n
+R⊕n ,
+T
+(2.1)
+where T = Diag(z − z1, . . . , z − zn). This means that the D7-branes correspond to the
+sheaf1 S in the short exact sequence
+0
+R⊕n
+R⊕n
+S
+0 .
+T
+(2.2)
+The matter on the system of D7-branes is described by fluctuations of the tachyon, δT,
+which are defined up to linearized gauge transformations. This corresponds to a self-Ext1
+1We will use the formulation of branes modulo tachyon condensation in terms of the derived category
+of coherent sheaves throughout this paper. Some introduction to this topic can be found in [81–83].
+– 7 –
+
+computation for the complex (2.1), i.e. morphisms between the complex and the shifted
+version of that same complex,
+R⊕n
+R⊕n
+R⊕n
+R⊕n
+αL
+δT
+T
+αR
+T
+(2.3)
+up to homotopies,
+δT ∼ δT − T · αL + αR · T .
+(2.4)
+Concretely, this means δT is valued in the quotient ring of n×n matrices Matn(R) modulo
+the two matrix ideals in R generated by left and right multiplication by T,
+δT ∈ Matn(R)/(T·, ·T) .
+(2.5)
+Note also that the tachyon map T can be expressed as a matrix given a choice of basis
+for the D9 gauge bundle and the D9 gauge bundle. These choices are independent, which
+means algebraically that only the equivalence class of T under the equivalence relation
+T ∼ GL · T · G−1
+R
+GL, GR ∈ GL(n, C[z])
+(2.6)
+matters. A canonical representative of each such equivalence class is given by the Smith
+Normal Form (SNF) computed in the ring R = C[z], i.e. any matrix T with entries in
+polynomials in z is equivalent under (2.6) to a unique diagonal matrix
+SNF(T) := diag(p1, . . . , pr, 0, . . . , 0) ,
+(2.7)
+where the pi are monic polynomials2 in z such that p1|p2| . . . |pr.
+Example.
+To illustrate the discussion of the previous paragraph, consider the case
+n = 2. The tachyon matrix is
+T =
+�
+z − z1
+0
+0
+z − z2
+�
+(2.8)
+and the fluctuations belong to
+δT ∈
+�
+R
+(z−z1)
+R
+(z−z1,z−z2)
+R
+(z−z1,z−z2)
+R
+(z−z2)
+�
+≃
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+C 0
+0 C
+�
+z1 ̸= z2
+�
+C C
+C C
+�
+z1 = z2
+(2.9)
+For any value of z1, z2 there is U(1) adjoint matter on each D7-brane, and for z1 = z2 the
+U(1)2 gauge symmetry enhances to U(2), and one can have fluctuations in the adjoint of
+U(2). From the U(1)2 perspective this is simply bifundamental matter.
+2Unicity is guaranteed up to multiplication by units in the ring; demanding that the polynomials be
+monic, i.e. have coefficient 1 for the term of highest degree, fixes this redundancy.
+– 8 –
+
+Consider the case z1 = z2 = 0. We can then activate an off-diagonal term:
+T[1,1] =
+�
+z 0
+0 z
+�
+�→
+T[2] =
+�
+z 1
+0 z
+�
+.
+(2.10)
+After this activation, the fluctuations are reduced to
+δT[2] ∈
+�
+0 0
+1 0
+�
+C ⊕
+�
+1 0
+0 −1
+�
+C .
+(2.11)
+The SNF reveals the same structure in a slightly different guise. Indeed
+SNF
+�
+z − z1
+0
+0
+z − z2
+�
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+0
+0 (z − z1)(z − z2)
+�
+z1 ̸= z2
+�
+z − z1
+0
+0
+z − z1
+�
+z1 = z2
+(2.12)
+so the fluctuations are valued in
+δT ∈
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0 C ⊕ zC
+�
+z1 ̸= z2
+�
+C C
+C C
+�
+z1 = z2
+.
+(2.13)
+Activating the off-diagonal term translates using the SNF into
+SNF
+�
+z − z1
+1
+0
+z − z2
+�
+=
+�
+1
+0
+0 (z − z1)(z − z2)
+�
+,
+(2.14)
+valid for z1 = z2 and z1 ̸= z2 alike. In particular, putting both branes at the origin,
+SNF
+�
+z 1
+0 z
+�
+=
+�
+1 0
+0 z2
+�
+,
+(2.15)
+where we see the appearance of an infrared trivial complex
+R
+R
+∼= 0
+1
+(2.16)
+and a so-called ’thick brane’
+R
+R .
+z2
+(2.17)
+Multivariable polynomials.
+The ring of polynomials in one variable C[z] has the prop-
+erty that every ideal is principal. In particular, it is a B´ezout ring, which means by definition
+that any ideal generated by finitely many generators is principal. The ring C[x1, . . . , xn]
+for n > 1 on the other hand is not a B´ezout ring : it has non-principal finitely generated
+ideals (for example, the ideal generated by x1 and x2). It turns out the SNF is best defined
+in B´ezout rings. By [84, Theorem 2.1] an SNF does not exist for matrices with coefficients
+– 9 –
+
+in C[x1, . . . , xn].
+Thus, at face value it seems not possible to use the SNF to describe
+intersecting branes.
+However, this is not a weakness but a feature, as we now demonstrate. Consider the
+case of two variables, x and z.
+Physically, this means we are considering branes that
+share an R1,5 and wrap complex curves in the (x, z)-plane. Consider a stack of n branes
+at x = 0 and a stack of m branes at z = 0. This is described by the diagonal matrix
+diag(x, . . . , x, z, . . . , z). We can activate non-diagonal terms, that we call Q and ˜Q as they
+correspond to strings that yield hypermultiplets at low energy
+T =
+�
+x1n
+�Q
+Q
+z1m
+�
+.
+(2.18)
+In order to pick a canonical diagonal form for this matrix, we need to make a choice of
+main variable. Let us pick z. This means we extend the non-B´ezout ring C[x, z] to the ring
+C(x)[z], which is B´ezout as it is a polynomial ring in one variable over the field C(x). This
+is simply telling us that poles in x have to be included. We now describe the SNF over this
+ring. Assume first that the eigenvalues of Q �Q are all distinct, call them λ1, . . . , λm. Then
+the SNF is
+diag
+�
+x, . . . , x, 1, . . . , 1,
+m
+�
+i=1
+�
+z − λi
+x
+��
+.
+(2.19)
+If some of the eigenvalues coincide, the form of the SNF changes. We can collect this
+information, which is insensitive to the detailed properties of the eigenvalues, by simply
+stating that the SNF is
+�
+x1n
+0
+0
+z1n − Q �
+Q
+x
+�
+λi ̸= λj .
+(2.20)
+Thus, from the point of view of the stack of m branes at z = 0, the presence of the other
+stack is felt as a pole for the complex adjoint-valued Higgs field living on the brane at
+z = 0, [85–87]. Note that the situation is symmetric and one could have chosen the other
+stack as the base one. This is exactly the same arbitrariness we made when writing the
+ring as a B´ezout ring.
+2.2
+Kraft-Procesi Transitions
+Nilpotent orbits for sln are in one-to-one correspondence with partitions of n, and are
+partially ordered by inclusion of their closure [88]. The nilpotent orbit associated to a
+partition λ of n is denoted Oλ. The partial order corresponds to the well-known dominance
+ordering for partitions,3 and it can be represented by a Hasse diagram, which indicates
+the covering relation associated to this partial order.
+The diagram thus obtained also
+corresponds to the stratification of the nilpotent cone (the set of all nilpotent matrices)
+into symplectic leaves [77, 78, 89]. Elementary degenerations between adjacent nilpotent
+orbits are called Kraft-Procesi transitions. In the case of sln nilpotent orbits, these can be
+3The dominance ordering is defined as follows. If λ and µ are two partitions of n, we say that λ ≥ µ if
+and only if for all j,
+j�
+i=1
+λj ≥
+j�
+i=1
+µj.
+– 10 –
+
+either closures of minimal slm nilpotent orbits or Kleinian singularities C2/Zm for m ≤ n.
+This can be implemented in brane setups [90, 91], where nilpotent orbit closures are realized
+as Higgs or Coulomb branches of 3d N = 4 quiver theories.
+Slodowy Slices.
+Consider the case where T = z · 1n + M and M is a nilpotent matrix.
+Equation (2.4) shows that
+δT ∼ δT + (αR − αL)z + (αRM − MαL) ,
+δT ∈ Matn(R) .
+(2.21)
+Define α+ := 1
+2(αR + αL) and α− := 1
+2(αR − αL) this gives
+δT ∼ δT + 2α−z + [α+, M] + {α−, M} ,
+δT ∈ Matn(R) .
+(2.22)
+The image of the map
+Matn(R) → Matn(R)
+α− �→ 2α−z + {α−, M}
+(2.23)
+contains all of Matn(zR),4 so we can use α− to eliminate all z-dependence in δT. We still
+have the freedom to use α+, which defines an equivalence relation
+δT ∼ δT + [α+, M] ,
+δT ∈ Matn(C) .
+(2.24)
+The cokernel of the adjoint action by M has dimension
+dλ :=
+�
+i
+(2i − 1)λi ,
+(2.25)
+where λ is the partition that specifies the nilpotent orbit of M.
+If one considers only
+traceless matrices, δT then depends only on dλ − 1 parameters.
+Note that being in the cokernel of ad(M) corresponds to commuting with the other
+nilpotent element in the sl2-triple generated by M. Thus M+δT parameterizes the Slodowy
+slice SM transverse to M (see Appendix A for definitions and a proof of this statement).
+Note that indeed that
+∀M ∈ Oλ(sln) ,
+dim SM = dλ − 1 .
+(2.26)
+Kraft-Procesi Transitions.
+Consider two partitions λ and µ, which are immediately
+adjacent in the partial order – one says that λ covers µ if they are adjacent and λ > µ.
+Then λ and µ differ only in two entries, say with indices i < j, with λi − 1 = µi and
+λi+1 + 1 = µi+1, and one of the two following transitions occurs:
+Condition
+Transition name
+Transverse slice
+j = i + 1
+Aλi−λj−1
+C2/Zλi−λj
+µi = µj
+ai−j−1
+Omin(sl(i − j, C))
+(2.27)
+4This is easily proved by a recursive argument on the degree.
+– 11 –
+
+The first corresponds to a Kleinian singularity, whereas the second is the closure of a
+minimal nilpotent orbit. The equivalence class of tachyon matrices for a partition µ =
+[µ1, . . . , µr] (with µ1 ≥ · · · ≥ µr) is characterized by a common SNF, i.e.
+SNF(Tµ) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+...
+1
+zµr
+...
+zµ1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(2.28)
+Starting from the SNF of partition µ, the Kraft-Procesi transition is realized using
+SNF
+�
+zµj αzµj−1
+0
+zµi
+�
+=
+�
+zµj−1
+0
+0
+zµi+1
+�
+=
+�
+zλj
+0
+0 zλi
+�
+(2.29)
+for α ̸= 0.
+Note that this is precisely the tachyon matrix formalism analog of the way Kraft-
+Procesi transitions are realized in Hanany-Witten brane systems for 3d N = 4 quiver
+theories in [90].
+Examples.
+The equality (2.15) corresponds to the covering of partition [1, 1] by [2],
+whereby two branes are combined into a thick brane. A less trivial case is the covering of
+[2, 2] by [3, 1], where we do not simply have two branes being combined. Rather, one of
+the two thick branes needs to be broken. This is realized in our framework using (2.29) as
+follows:
+SNF
+�
+z2 αz
+0 z2
+�
+=
+�
+z1 0
+0 z3
+�
+(2.30)
+for α ̸= 0.
+When the nilpotent orbit Hasse diagram is linear, one can build matrices that encode
+all partitions at once. This is the case for n ≤ 5, where the matrices are given by
+�
+z a1
+0 z
+�
+,
+�
+�
+�
+z a1 0
+0 z a2
+0 0
+z
+�
+�
+� ,
+�
+�
+�
+�
+�
+z a1
+0
+0
+0 z za3 + a4 0
+0 0
+z
+a2
+0 0
+0
+z
+�
+�
+�
+�
+� ,
+(2.31)
+�
+�
+�
+�
+�
+�
+�
+z
+a1
+0
+0
+0
+0
+z
+za3
+0 a2a3a4
+0
+0
+z
+a2
+0
+−z3a6
+0
+z2a1a3a6
+z
+−za4
+−za2a4a5 (a1a2a3a6 + 1) 0 a2
+1a2
+2a2
+3a4a5a6 0
+z
+�
+�
+�
+�
+�
+�
+�
+.
+(2.32)
+This means that the SNF of a matrix above with a1, · · · , ar non-zero gives precisely the
+r-th partition of n, the partitions being totally ordered.
+– 12 –
+
+3
+Example: GTPs for Tn and Related Models
+In this section, we use an intermediate step in correspondence between M-theory on a
+CY threefold and IIB 5-brane-webs: IIA on a resolved C2/Zn singularity with D6-branes.
+This discussion follows the philosophy of [13]. We start in this section with the instructive
+example of Tn, and consider its description as well as GTPs obtained by adding white dots
+along a single edge.
+3.1
+The Setup
+Consider the toric local Calabi-Yau defined by the toric diagram with vertices at coordinates
+(0, 0), (n, 0) and (n, n), drawn here for n = 5:
+W1
+W3
+W2
+(3.1)
+The generators of the dual cone are
+(−1, 0, 0) ↔ W1
+(3.2)
+(1, −1, n) ↔ W2
+(3.3)
+(0, 1, 0) ↔ W3
+(3.4)
+(0, 0, 1) ↔ Z .
+(3.5)
+The first three generators are vectors normal to the 2-dimensional facets on the fan, drawn
+in blue arrows when projected on the CY plane in (3.1). As an algebraic variety, the toric
+threefold is simply a hypersurface in C4:
+W1W2W3 = Zn
+⊂
+C4⟨W1, W2, W3, Z⟩ .
+(3.6)
+This space has non-isolated singularities, specified by the following intersecting ideals:
+Ising = (W1, W2, Z) ∩ (W1, W3, Z) ∩ (W2, W3, Z) .
+(3.7)
+Along each such ideal, there is a family of An−1-singularities. These are in one-to-one
+correspondence with the three edges of the toric graph.
+The singular threefold admits three different (albeit linearly dependent) C∗-actions
+which act on the coordinates with the following weights:
+W1 W2 W3 Z
+C∗
+1
+0
+1
+−1 0
+C∗
+2
+1
+0
+−1 0
+C∗
+3
+1
+−1
+0
+0
+(3.8)
+– 13 –
+
+As explained in toric language in [13], we can define projections πi, for i = 1, 2, 3, with
+respect to these actions, and this will bring us down to IIA. Let (i, j, k) be a permutation of
+(1, 2, 3). The way to reduce along a particular C∗-action C∗
+i is to pick the pair of ‘charged’
+coordinates Wj and Wk, and setup a C∗-fibration over an new complex coordinate Vjk as
+follows:
+C∗
+i : C[W1, W2, W3, Z] ∼= C[W1, W2, W3, Z, Vjk]
+(WjWk − Vjk)
+.
+(3.9)
+Now we can rewrite the threefold in the following presentation:
+C[W1, W2, W3, Z]
+(W1W2W3 − Zn)
+∼=
+C[W1, W2, W3, Z, Vjk]
+(WjWk − Vjk; WiVjk − Zn) .
+(3.10)
+The IIA reduction is achieved by reducing over the S1 ⊂ C∗ action in each case. The non-
+compact part R ⊂ C∗ becomes a transverse direction to the D6-branes. The projection is
+simply defined as dropping the pair of coordinates (Wj, Wk), leaving us with a local K3
+with an An−1 Klein singularity
+C[Wi, Vjk, Z]
+(WiVjk − Zn) .
+(3.11)
+To simplify the notations, we switch to the more standard
+X := Wi
+Y := Vjk
+(3.12)
+so that the An−1 singularity (a local K3) is described by
+XY = Zn .
+(3.13)
+There are D6-branes on the locus defined by the ideal
+ID6 = (Y, Zn) ,
+(3.14)
+which we call the D6 ideal. It is, first of all a stack of n non-compact D6-branes. However,
+since this passes through the singularity, more is at play here, and we need to resolve the
+local K3 to refine our understanding.
+In order to describe the resolution of the An−1 orbifold (3.13), we introduce homoge-
+neous coordinates (z1, e1, . . . , en−1, z2) with n − 1 C∗-actions
+z1 e1 e2 e3 . . . en−3 en−2 en−1 z2
+C∗
+1
+1 −2 1
+0 . . .
+0
+0
+0
+0
+C∗
+2
+0
+1 −2 1 . . .
+0
+0
+0
+0
+...
+C∗
+n−2 0
+0
+0
+0 . . .
+1
+−2
+1
+0
+C∗
+n−1 0
+0
+0
+0 . . .
+0
+1
+−2
+1
+(3.15)
+The coordinates are homogeneous with respect to the n − 1 projective actions, by which
+the space is quotiented. Each row gives the list of weights of the coordinate with respect to
+each such action. Just as one must excise particular loci when creating standard projective
+space, so must one excise a number of loci here.
+– 14 –
+
+· · ·
+e1 = 0
+e2 = 0
+en−2 = 0
+en−1 = 0
+z1 = 0
+z2 = 0
+Figure 6. Geometry of the resolved An−1 singularity. Each curved line is a P1 while the straight
+lines are the non compact divisors z1,2 = 0.
+Specifically, if a coordinate is set to zero, then only one its two ‘neighbors’ are allowed
+to vanish. See figure 6. For example, we can have e2 = 0, and e3 = 0, or e2 = 0 and
+e1 = 0, but the pair (e1, e3) does not form a valid ideal for a vanishing locus. In terms of
+the coordinates (X, Y, Z), we have
+X =
+n
+�
+i=0
+en−i
+i
+,
+Y =
+n
+�
+i=0
+ei
+i ,
+Z =
+n
+�
+i=0
+ei
+(3.16)
+where we have introduced e0 := z1 and en := z2. The locus ei = 0 corresponds to the
+i-th exceptional P1. The loci z1 = 0 and z2 = 0 correspond to noncompact holomorphic
+curves intersecting the first and last P1, respectively.
+Line bundles over this space are
+characterized by their first Chern class, which is encoded as O(k1, . . . , kn−1). A section of
+this bundle is a polynomial of homogeneous multi-degree (k1, . . . , kn−1).
+The brane locus (3.14) is now given by
+ID6 =
+� n
+�
+i=0
+ei
+i,
+n
+�
+i=0
+en
+i
+�
+=
+� n
+�
+i=0
+ei
+i
+�
+.
+(3.17)
+The interpretation is as follows: there are i D6-brane wrapping the i-th P1, and n non-
+compact D6-branes on the curve z2 = 0, which intersects the (n−1)-th sphere at one point.
+At the SCFT point, all the P1’s shrink to zero size. On the Coulomb branch, where the
+K¨ahler volumes are non zero, the effective theory can be read from the ideal (3.17). It is
+described by the quiver:
+U(1)
+U(2)
+· · ·
+U(n − 2)
+U(n − 1)
+n
+(3.18)
+3.2
+Tachyon Condensation Picture
+The theory (3.18) is encoded by via the tachyon condensation/coherent sheaf language as
+follows. First recall that in terms of complexes, one can describe the branes as follows:
+• A brane Bi wrapped on the i-th P1 given by ei = 0 can be described as the cokernel
+of the complex of line bundles:
+Bi :
+O(−ei)
+O ,
+ei
+(3.19)
+– 15 –
+
+where O is the structure sheaf over the local K3, and O(−ei) is the dual of the line
+bundle O(ei), of which ei is a section. For instance, O(−e1) = O(2, −1, 0, . . . , 0).
+• A noncompact ‘flavor brane’ BF at the locus z2 = 0, intersecting the rightmost P1
+(given by en−1 = 0), is given by the following complex:
+Bn :
+O(−z2)
+O .
+z2
+(3.20)
+• More generally, a noncompact D6-brane intersecting the i-th exceptional P1 will be
+given by the zero-locus of a section of O(0, . . . , 0, 1, 0, . . . , 0), where the ‘1’ is the i-th
+entry.
+Define N = 1
+2n(n+1) D8-branes with gauge bundle FD8 := O⊕N and N anti-D8-branes
+with gauge bundle
+FD8 := O(2, −1, 0, · · · , 0) ⊕ O(−1, 2, −1, · · · , 0)⊕2 ⊕ · · · ⊕ O(0, · · · , 0, −1)⊕n .
+(3.21)
+We then define the tachyon map
+T : FD8 → FD8
+(3.22)
+as a diagonal matrix
+T = Diag(e1 · 11, e2 · 12, . . . , en−1 · 1n−1, z2 · 1n) .
+(3.23)
+Note that det T = Y , so that the equations of the threefold are
+WjWk = det T = Y ,
+XY = Zn .
+(3.24)
+from which one recover the original equation WiWjWk = Zn.
+The resulting D6-brane
+system is defined as the cokernel S := cok(T) of this map, which is the locus where T fails
+to be invertible. So S is reducible:
+S =
+n
+�
+i=1
+O⊕i
+ei ,
+(3.25)
+where Op means the structure sheaf with support over p = 0. There is an exact sequence
+FD8
+FD8
+S
+0 .
+T
+(3.26)
+Fluctuations.
+The fluctuations around this background are computed as self-extensions,
+in the same way as in section 2. This means that the fluctuations δT of the tachyon T
+belong to the self-extension group,
+δT ∈ Ext1(S, S) = HomD(K3)(S, S[1]) ,
+(3.27)
+where we consider the Homs in the derived category of coherent sheaves on K3 D(K3),
+and the [1] means that we shift the complex one step to the left. The matrix δT can be
+decomposed in blocks, and there will be non-zero fluctuations just above and below the
+diagonal.
+In practice, the fluctuations are subjected to three conditions, that we will be using
+repeatedly in the following sections:
+– 16 –
+
+(i) The C∗ weights of the columns of T and δT need to be compatible with (3.23) ;
+(ii) The fluctuations are regular sections of the relevant sheaves (no pole on the support
+locus) ;
+(iii) The components of δT are subject to the identifications (2.5).
+3.3
+Example: T2
+For concreteness we work out in detail the case n = 2 for Tn before treating the general
+case. The tachyon matrix is
+O(2) ⊕ O(−1)⊕2
+O⊕3
+T
+,
+T =
+�
+e1
+0
+0
+z2 · 12
+�
+(3.28)
+and we give names to the blocks in the fluctuation δT in correspondence with the quiver
+δT =
+�
+Φ1 �Q1
+Q1 Φ2
+�
+1
+2
+Q1
+�Q1
+Φ1
+Φ2
+(3.29)
+The three conditions listed above give
+δT ∈
+�
+Γ(O(−2)(e1))
+Γ(O(1)(e1,z2)) · C1×2
+Γ(O(−2)(e1,z2)) · C2×1
+Γ(O(1)(z2)) · C2×2
+�
+=
+�
+0
+C1×2 · z1
+C2×1 · 1
+z2
+1 C2×2 · z1
+�
+.
+(3.30)
+Here, Γ indicates that we take sections of the bundles, and Cn×m is the set of n×m matrix
+of complex numbers. The last equality is easily checked, e.g. for the lower-left entry, the
+regular sections are generated by rational functions in z1 alone, having C∗-weight −2, and
+poles are allowed as the support is the intersection between two curves where z1 ̸= 0. In
+terms of Ext groups between B1 (see (3.19)) and B2 (see (3.20)), we can write (using e.g.
+a spectral sequence argument)
+Q1 ∈ Ext1(B1, B2) = H0(O(e1)e1,z2) =
+� 1
+z2
+1
+�
+.
+(3.31)
+�Q1 ∈ Ext1(B2, B1) = H0(O(z2)e1,z2) = ⟨z1⟩ .
+(3.32)
+To summarize, the background tachyon plus fluctuation is given by
+T + δT =
+�
+e1
+�q1z1
+q1
+z2
+1
+z2 · 12 + z1ϕ2
+�
+,
+(3.33)
+where we have pulled out all dependencies in z1, e1 and z2, so that �q1 ∈ C1×2, q1 ∈ C2×1
+and ϕ2 ∈ C2×2 are pure constants:
+Q1 = q1
+z2
+1
+,
+�Q1 = �q1z1 ,
+Φ2 = ϕ2z1 .
+(3.34)
+– 17 –
+
+e1 = 0
+z1 = 0
+z2 = 0
+Φ
+ϕ1
+Q1, ˜Q1
+e1 = 0
+z1 = 0
+z2 = 0
+ϕ2 = − M
+X
+ϕ1
+Figure 7. Intersecting branes before and after the transformation that maps the off-diagonal Higgs-
+field entries Q1, ˜Q1 to diagonal ones with a pole. The orange dot signals the pole in the Higgs field
+at X = 0.
+Note that the term q1
+z2
+1 ensures that (3.33) is defined on the locus z1 ̸= 0
+We can perform basis changes from the left and right, using (2.6), as follows:
+�
+1
+0
+− q1
+z2
+1e1
+12
+�
+·
+�
+e1
+�q1z1
+q1
+z2
+1
+z2 · 12
+�
+·
+�
+1
+− �q1z1
+e1
+0
+12
+�
+=
+�
+e1
+0
+0
+z2 · 12 −
+M
+z1e1
+�
+(3.35)
+In the last step we have introduced the meson matrix
+M := q1�q1 .
+(3.36)
+In the 5d effective field theory, F-term conditions impose that M be nilpotent. This can
+also be demonstrated mathematically via the so-called cone construction in the derived
+category of coherent sheaves. See [83] for examples of this mechanism.
+This diagonalization shows that giving a vev to the meson field can be subsumed into
+a shift of vev of the adjoint field φ on the flavor branes, with a pole. Using the coordinate
+X = z2
+1e1 (see (3.16)) on the z2 = 0 plane, we can write this as
+T + δT ∼
+�
+e1
+0
+0
+z2 · 12 + z1 · ϕ2
+�
+with
+ϕ2 = −M
+X .
+(3.37)
+The transformation from (3.33) to (3.37) means that we can regard in an appropriate
+regime the intersecting branes in figure 7 as a stack of two branes on z2 = 0 with a
+complex codimension-one defect on its world-volume at e1 = 0. This is in agreement with
+the findings of [85, 87], where the authors find that a vev of bifundamental fields at the
+intersection of two branes can be subsumed into a pole for the adjoint of one of the two
+branes. In the picture, we trade the description in figure 7 on the left with the right.
+Up to a change of basis we can take M in canonical Jordan form.
+There are two
+possibilities, corresponding to the two partitions of n = 2:
+M[12] =
+�
+0 0
+0 0
+�
+and
+M[2] =
+�
+0 1
+0 0
+�
+.
+(3.38)
+– 18 –
+
+Consider the latter case,
+T[2] :=
+�
+e1
+t[2]
+�
+:=
+�
+�
+�
+e1 0
+0
+0 z2 −
+1
+z1e1
+0 0
+z2
+�
+�
+� .
+(3.39)
+We still have det T[2] = ez2
+2 = Y . So the brane configuration has not changed geometrically.
+Accordingly, the M-theory uplift is still given by (3.24). However, the tachyon matrix shows
+us that the flavor brane no longer carries an SU(2) group. This is the hallmark of a T-
+brane: A non-abelian bound state of branes that does not realize the gauge group that it
+would naively have given its geometry. This T-brane effect is the IIA counterpart of the
+change in boundary conditions in the dual type IIB brane-webs
+(3.40)
+In this particular case, once the D5-segment has been sent away, a Hanany-Witten move
+where the 7-brane detaches completely becomes possible:
+(3.41)
+How does this translate into the IIA language, i.e. in terms of the tachyon field? Let
+us see what deformations are available, starting from the new vacuum defined by (3.39).
+Using the results of section 2, the fluctuations of t[2] are
+δt[2] = z1 ·
+�
+0 0
+α 0
+�
+,
+α ∈ C .
+(3.42)
+Now in order to see how this affects the geometry, we add this perturbation to T[2]
+T[2] + δt[2] =
+�
+�
+�
+e
+0
+0
+0 z2 − 1
+z1e
+0 αz1
+z2
+�
+�
+� .
+(3.43)
+Now the geometry (3.24) is deformed to
+WjWk = det T = Y + α ,
+XY = Z2 .
+(3.44)
+– 19 –
+
+which reduces to the hypersurface
+W1W2W3 = Z2 + αWi .
+(3.45)
+So the CY threefold is fully desingularized. From the IIA perspective, we see that two flavor
+branes have recombined with one gauge brane, to give rise to a noncompact brane that can
+escape the singular locus. This is in full agreement with the 5-brane-web picture, whereby
+the 7-brane moves off to the left, becomes fully detached from the NS5-branes, and can
+escape to infinity, see (3.41). This corresponds precisely to removing the A1 singularity.
+To summarize, the white dot is represented in the IIA picture as a nilpotent vev with
+poles on the flavor D6-stack. The further Hanany-Witten move that actually deforms the
+M-theory geometry is implemented by switching on a further vev on the flavor stack along
+the Slodowy slice with respect to the initial singular nilpotent vev.
+3.4
+General Case: Tn
+The general Tn case is very similar to the T2 example treated above. The fluctuations
+around the tachyon background are denoted by fields as follows:
+1
+2
+. . .
+n − 2
+n − 1
+n
+Q1
+Qn−2
+Qn−1
+�Q1
+�Qn−2
+�Qn−1
+Φ1
+Φ2
+Φn−2
+Φn−1
+(3.46)
+Generalizing (3.31) and (3.32), we find that the 5d hypermultiplets that reside at the
+intersection of the curves ei = 0 and ei+1 = 0 are described by
+Qi ∈ Ext1(Bi, Bi+1) = H0(O(ei)(ei,ei+1)) =
+� Y
+Zi+1 ei
+�
+=
+��
+j̸=i
+ej−i−1
+j
+�
+.
+(3.47)
+and
+˜
+Qi ∈ Ext1(Bi+1, Bi) = H0(O(ei+1)(ei,ei+1)) =
+�Zi
+Y ei+1
+�
+=
+� �
+j̸=i+1
+ei−j
+j
+�
+.
+(3.48)
+On the other hand, one can check that Ext1(Bi, Bj) = 0 for |i − j| > 1. Therefore the
+tachyon matrix T, plus the fluctuations δT, fit schematically (we will provide the explicit
+form below) into the matrix
+T + δT =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+e1 · 11
+�Q1
+Q1
+e2 · 12 ...
+... ... ...
+en−1 · 1n−1
+�Qn−1
+Qn−1
+z2 · 1n
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(3.49)
+As in (3.34), we introduce the notation
+Qi = qi ·
+Y
+Zi+1 ei
+˜Qi = ˜qi · Zi
+Y ei+1 ,
+(3.50)
+– 20 –
+
+such that qi and ˜qi are constants. We can also have fluctuations on the diagonal, which we
+write as
+Φi = ϕi ·
+Y
+Zi+1 ei .
+(3.51)
+With this notation the F-terms at the ith node can be written
+�q1q1 = 0,
+and
+qi�qi = �qi+1qi+1
+i = 1, . . . , n − 2 .
+(3.52)
+We now come back to the reason why (3.49) is only a schematic form.
+The i-the
+hyper (Qi, �Qi) is only well defined at the (ei, ei+1) intersection, but has poles in the nearby
+patches. Hence, (3.49) is not well-defined over the whole target space. This is due to the
+projective nature of the resolved K3. Therefore, we must study it patch by patch. A good
+local affine coordinate on the locus ei = 0 for the hemisphere where ei+1 (respectively ei−1)
+does not vanish is Y
+Zi (respectively Zi
+Y ):
+ei = 0
+ei+1 = 0
+Coordinate Y
+Zi
+Coordinate Zi+1
+Y
+(3.53)
+Note in particular that for i = N (respectively i = 0), this is compatible with the affine
+coordinate on the locus z2 = 0 (resp. z1 = 0) being simply X (resp. Y ), as chosen in
+(3.37).
+In the patch that contains the intersection {ei = 0} ∩ {ei+1 = 0}, the tachyon fluctua-
+tion is expressed as
+δT =
+�
+Φi
+�Qi
+Qi Φi+1
+�
+∈
+�
+Ci×i ·
+Y
+Zi+1 ei
+Ci×(i+1) · Zi
+Y ei+1
+C(i+1)×i ·
+Y
+Zi+1 ei C(i+1)×(i+1) · Zi
+Y ei+1
+�
+(3.54)
+Using line and row transformations, one finds
+�
+1i
+�qi Zi
+Y
+−qi
+Y
+Zi+1 1i+1 − qi�qi
+Z
+� �
+ei
+�
+1i − �qiqi
+Z
+�
+0
+0
+ei+1 · 1i+1
+� �
+1i
+−�qi
+Ziei+1
+Y ei
+qi
+Y ei
+Zi+1ei+1 1i+1 − qi�qi
+Z
+�
+=
+�
+ei · 1i
+0
+0
+ei+1
+�
+1i+1 − qi�qi
+Z
+�
+�
+.
+(3.55)
+Effectively, this transforms a pole of the form
+�
+ei · 1i +
+Y
+Zi+1 eiϕi
+0
+0
+ei+1 · 1i+1
+�
+,
+ϕi = −�qiqi
+(Y/Zi)
+(3.56)
+into a pole of the form
+�
+ei · 1i
+0
+0
+ei+11i+1 + Zi
+Y ei+1ϕi+1
+�
+,
+ϕi+1 =
+−qi�qi
+(Zi+1/Y ) .
+(3.57)
+– 21 –
+
+Let us introduce the n × n meson matrix
+M = qn−1�qn−1 .
+(3.58)
+As argued for the T2 previously, this meson matrix must be nilpotent. We can characterize
+the tachyon fluctuation entirely by the last pole (3.57) for i = n − 1, which gives
+ϕn = −M
+X ,
+(3.59)
+consistently with what we found for the n = 2 case in (3.37). In summary, the bifun-
+damental matter between the various branes can be subsumed under a shift of the 7d
+SU(n)-adjoint Higgs ϕn, as a simple pole with residue equal to a nilpotent element M.
+Since Mn = 0, we still have
+det T =
+n
+�
+i=1
+ei
+i = Y .
+(3.60)
+Hence, the brane locus remains unscathed despite the activation of the matter fields.
+3.5
+Interpretation in terms of Generalized Toric Polygons
+We saw in the last subsection that the geometry of the threefold can be affected by the
+presence of a pole of the form (3.59).
+This pole can be encoded in two ways: in the
+nilpotent meson matrix M, or in the list of hypermultiplet deformations qi and �qi satisfying
+the relations (3.52). Let us call Q the set
+Q = {q = (q1, �q1, . . . , qn−1, �qn−1) ∈ C2×1 × . . . C(n−1)×n|(3.52)} .
+(3.61)
+We also call N the set of nilpotent n × n complex matrices. The map
+m : Q → N
+(3.62)
+q �→ M = qn−1�qn−1
+is certainly not injective, as given a nilpotent matrix M, there are infinitely many families
+{qi, �qi}i=1,...,n−1 mapping to it. Let us define
+r : Q → Nn−1
+(3.63)
+q �→ (ri = rank(qi�qi))i=1,...,n−1 .
+For a given M ∈ N, the ranks of the bilinears in qi�qi that map to M are not fixed.
+In other words, r(m−1(M)) contains more than one element. However, there is a unique
+element of rmin ∈ r(m−1(M)) that minimizes the sum of the entries. If M ∈ Oλ, then r0
+is given by the partial sums of the transpose of λ,
+rmin =
+�
+�
+n
+�
+j=n+1−i
+λT
+j
+�
+�
+i=1,...,n−1
+.
+(3.64)
+– 22 –
+
+Figure 8. On top, we depict a part of the resolved GTP for the T9 theory, with white dots on
+the right edge, along with an internal triangulation consistent with the white dots. The presence of
+white dots propagates into the interior of the GTP, thereby limiting the possible resolutions. Each
+column in the GTP corresponds to a boundary condition for D5-branes ending on D7-branes. This
+information is encoded in the ranks ri of the matrices appearing in δT. On the bottom, we show
+the associated D5-brane boundary conditions (where each circle denotes a D7-brane).
+Note that this coincides with the ranks of the linear quiver
+rmin
+1
+rmin
+2
+· · ·
+rmin
+n−1
+n
+(3.65)
+whose 3d N = 4 Coulomb branch is the corresponding nilpotent orbit closure. The ranks
+in (3.64) represent the minimal deformations needed in δT to produce the pole (3.59). In
+simple cases, the quivers (3.65) appear as embedded in the magnetic quiver describing the
+Higgs branch of the 5d theory. For a more precise statement about the embedding, the
+reader is referred to [55].
+In the GTP, these ranks can be encoded with white dots inside the polygon, using
+again the notation introduced in [16]. For instance, for n = 9 and partition λ = [4, 3, 2], we
+can draw the configuration shown in figure 8. The minimal ranks correspond to the number
+of white dots in each column: here we get rmin = (0, 0, 0, 0, 0, 1, 3, 6). These numbers give
+the brane configuration which saturates the s-rule, as illustrated in the lower part of figure
+8.
+– 23 –
+
+...
+...
+H
+H′
+Pole
+Figure 9.
+Schematic form of a Hasse diagram of the Higgs branch H viewed as a symplectic
+singularity.
+The effect of the pole prescription corresponds geometrically to imposing a higher
+dimensional base leaf (the transverse slice is shown in red – here it is minimal, but it does not have
+to be in general). The remaining Higgs branch H′ is the transverse slice.
+Impact on the Hasse diagram.
+The vevs of Higgs branch operators in the 5d theory
+can be projected on the space of complex structure deformations of the geometry. The
+fact that the pole (3.59) freezes some of these deformations means geometrically that the
+resulting pole-deformed Higgs branch is the slice in the initial Higgs branch transverse to
+these imposed deformations. This can be represented as follows.
+The Higgs branch H with no pole is a symplectic singularity, which can be depicted
+using its Hasse diagram of singularities. The effect of the pole is to freeze certain defor-
+mations, represented here as a forced choice of a higher dimensional bottom symplectic
+leaf. The resulting Higgs branch H′ is the transverse slice to that leaf. See figure 9 for a
+schematic depiction.
+The analysis above applies to the theory in any phase. In the case of the IIA reduction
+on the resolved An−1 singularity shown in figure 6, the Higgs branch of the Tn theory
+becomes the nilpotent cone of sln. The transverse slices are then identified with the Slodowy
+slices. The example n = 4 is illustrated in figure 10. If M ∈ Oλ for λ some partition of n,
+then the corresponding Higgs branch is the Slodowy slice Sλ ∩ N. Moving on to the SCFT
+phase, the Higgs branch is no longer a nilpotent orbit closure, but the general picture stays
+the same: the Higgs branch is restricted to a transverse slice. This is what we already
+mentioned in the introduction, see figure 5. The Hasse diagrams drawn in figure 5 can be
+reproduced independently using the quiver subtraction algorithm [92, 93] on the magnetic
+quivers extracted from the GTPs.
+3.6
+Hanany-Witten Moves
+In the previous section, we defined the IIA counterpart of a ‘white dot’ as a nilpotent
+residue for the adjoint complex scalar on the flavor D6-branes. The nilpotency implies
+– 24 –
+
+0
+3
+4
+5
+6
+a3
+a1
+A1
+A3
+M ∈ O[14]
+M ∈ O[2,12]
+M ∈ O[22]
+M ∈ O[3,1]
+Figure 10. Diagram for the nilpotent cone of sl4. The dots are nilpotent orbits. When M belongs
+to a given orbit, the Higgs branch of the resulting theory is the transverse Slodowy slice, represented
+by a bracket on the right.
+that there will be no repercussions on the geometry of the branes, and hence, the M-theory
+CY will not be deformed. This is the hallmark of a ‘T-brane’ [74].
+We would now like to determine how the T-brane configuration impacts the physics
+of the 5d theory. This is done using the brane-web language, with Hanany-Witten moves,
+as we have explained in detail in section 3.3. The key point is that the nilpotent orbit of
+M determines which Hanany-Witten move can be performed in order to detach any of the
+7-branes.
+In the IIA setup, flavor D6-branes are now allowed to move across exceptional P1s,
+thereby changing the quiver structure. The way this is seen at the level of the Higgs field,
+is that by activating vevs along the Slodowy (transverse) slice to the nilpotent vev with
+poles, the characteristic polynomial of the tachyon matrix actually becomes deformed.
+This can be seen by computing the self-Ext group Ext1 �
+B(nil)
+F
+, B(nil)
+F
+�
+of the nilpotent
+configuration on the flavor brane. Using the computation from section 2, we are looking
+for δTF such that
+δTF ∼ δTF + z2
+Z [M, g] .
+(3.66)
+This is equivalent to requiring that δTF be on a transverse slice to M, gauge equivalent
+to the Slodowy slice. We will choose a gauge such that δTF be the companion matrix to
+M as in [94], which is referred to as a ‘reconstructible Higgs’ in [74]. The details on how
+this is done are given in Appendix A. Say Tnil is in the maximal nilpotent orbit, then the
+‘Hanany-Witten’ tachyon will take the form
+THW = Tnil + δTF = z2
+Z
+�
+�
+�
+�
+�
+�
+�
+�
+Z
+1
+Z
+1
+... ... ...
+Z
+1
+(−)n−1anX (−)n−2an−1X
+−a2X Z
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(3.67)
+– 25 –
+
+where the ai are constants (the fluctuation δTF is proportional to X as a consequence of
+(3.54) with i = N − 1). This matrix has determinant
+det(THW) =
+�z2
+Z
+�n �
+Zn + a2XZn−2 + . . . + anX
+�
+.
+(3.68)
+More generally, for M in the [λ1, λ2, . . . , λr] partition, we take a block diagonal matrix
+where each block will take the form:
+(THW)i = z2
+Z
+�
+�
+�
+�
+�
+�
+�
+�
+Z
+1
+Z
+1
+... ... ...
+Z
+1
+(−)λi−1a(i)
+λi X (−)λi−2a(i)
+λi−1X
+−a(i)
+2 X Z
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(3.69)
+Note that by doing so, we are not using the full Slodowy slice Sλ (which contains non-
+block-diagonal matrices) but instead restrict to the intersection S0
+λ = Sλ ∩ lλ with the Levi
+subalgebra lλ, see Appendix A. This is physically justified, as it guarantees that the flavor
+symmetry will not be further broken by the Hanany-Witten moves than it already has been
+by the white dots. The missing parameters, in Sλ − S0
+λ, are associated to the non splitting
+of the flavor branes, illustrated on an example below in (3.77).
+Putting together all these blocks, and the non-flavor part of the tachyon matrix, the
+end result of the full deformation is
+Y �→ 1
+X ·
+r
+�
+i=1
+�
+�Zλi + X
+�
+�
+λi−2
+�
+j=0
+a(i)
+λi−jZj
+�
+�
+�
+� ,
+(3.70)
+where �
+i λi = n.
+Actually it can be argued that only the coefficients a(i)
+λi affect the physics near the
+singularity: coefficient a(i)
+λi−j for j ̸= 0 correspond to a shift of
+Zλi−j
+X
+, which is using
+(3.53) the coordinate along a P1 with which the brane has zero intersection. Therefore the
+equation simplifies to
+Y �→ 1
+X ·
+r
+�
+i=1
+�
+Zλi + Xa(i)�
+,
+(3.71)
+where we have renamed a(i) := a(i)
+λi . Reverting to the original notations (W1, W2, W3, Z)
+for the coordinates in C4, see (3.12), one finally gets
+W1W2W3 =
+r
+�
+i=1
+�
+Zλi + a(i)W1
+�
+.
+(3.72)
+Examples.
+Let us work out a few examples. Equation (3.70) is worked out explicitly
+for T3 and T4 in table 1. The factorization allows to read off the corresponding quivers,
+which are the magnetic quivers for nilpotent orbits of sl(3) and sl(4) [95], as expected.
+– 26 –
+
+Partition
+det T
+Factorization
+[13]
+Y
+e1e2
+2(z3
+2)
+[2, 1]
+Y + a2Z
+e1e2(z2)(e2z2
+2 + a2z1)
+[3]
+Y + a2Z + a3
+smooth
+Partition
+det T
+Factorization
+[14]
+Y
+e1e2
+2e3
+3(z4
+2)
+[2, 12]
+Y + a2Z2
+e1e2
+2e2
+3(e3z2
+2 + a2z2
+1e1)(z2
+2)
+[22]
+Y + (a(1)
+2
++ a(2)
+2 )Z2 + a(1)
+2 a(2)
+2 X
+e1e2
+2e3(e3z2
+2 + a(1)
+2 z2
+1e1)(e3z2
+2 + a(2)
+2 z2
+1e1)
+[3, 1]
+Y + a2Z2 + a3Z
+e1e2e3(e2e2
+3z3
+2 + a2z2
+1e1e2e3z2 + a3z1)(z2)
+[4]
+Y + a2Z2 + a3Z + a4
+smooth
+Table 1. Equations defining the brane configurations in T3 and T4 with white dots on one edge
+after HW moves. In the last column, the equation is rewritten in terms of the toric variables for the
+resolution, and maximally factorized. The factor in orange give the ranks of the low energy quiver
+while the terms within brackets give the flavor ranks.
+We can illustrate the problem discussed in the previous paragraph with partition [2, 1].
+The generic matrix in the Slodowy slice would give rise to the final block for the tachyon
+matrix given by
+�
+�
+�
+Z
+1
+0
+−a2X Z
+αX
+βX
+0 Z + γX
+�
+�
+� ,
+(3.73)
+The resulting, deformed, equation then reads
+det T = e1e2
+�
+a2γe1z3
+1 + a2z1z2 + γe1e2z2
+1z2
+2 + αβe1z3
+1 + e2z3
+2
+�
+,
+(3.74)
+from which one reads off a quiver with two abelian nodes and a single flavor brane system:
+e1 = 0
+e2 = 0
+z1 = 0
+z2 = 0
+(3.75)
+This system, drawn in teal, intersects both P1 divisors at e1 = 0 and e2 = 0. This breaks
+the global isometry u(1) of the nilpotent Slodowy slice S[2,1] ∩ N. By restricting to the
+Levi subalgebra, i.e. sending α, β and γ to zero, one gets one more factorization:
+det T = e1e2(z2)
+�
+a2z1 + e2z2
+2
+�
+.
+(3.76)
+– 27 –
+
+e1 = 0
+e2 = 0
+z1 = 0
+z2 = 0
+(3.77)
+The flavor branes are now disjoint and can be moved independently:
+e1 = 0
+e2 = 0
+z1 = 0
+z2 = 0
+(3.78)
+Each flavor brane intersects a single P1, and the global symmetry of the resulting Higgs
+branch (still in the resolved phase) is read to be s(u(1) ⊕ u(1)) = u(1).
+4
+General Discussion
+4.1
+White Dots along a Single Edge
+Having discussed the case of Tn in detail, we move to a generic toric polygon and its GTP
+deformations. Let P be a convex polygon in R2 with vertices in Z2. Pick an edge on P,
+and denote by n its length (i.e. the edge contains exactly n + 1 points in Z2). Using an
+SL(2, Z) transformation, one can assume without loss of generality that this edge extends
+between the vertices (0, 0) and (0, n), and that all vertices of P other than (0, 0) and (0, n)
+have coordinates (i, j) with i < 0:
+· · ·
+· · ·
+(0, 0)
+(0, n)
+(4.1)
+In the toric threefold, the length n edge is translated into the presence of a dimension 1
+singular stratum with transverse slice of type An−1. In the brane-web picture, this means
+that n D5-branes extend to infinity, and the boundary condition, encoded in how they end
+on D7-branes, can again be encoded in a T-brane datum. This can be done explicitly using
+the IIA reduction as in section 3 for any given polygon (and we do so for the example of a
+– 28 –
+
+rectangular P in the next section), even though it would be extremely cumbersome to try
+to give general formulas that would apply to any polygon. The general idea, however, is
+clear: in the Tn example discussed in section 3, the undeformed equation W1W2W3 = Zn
+yields the An−1 singularity transverse to the line W2 = W3 = Z = 0 by setting W1 equal
+to a constant, say W1 = c, giving
+cW2W3 = Zn .
+(4.2)
+It is then this equation that is modified by (i) adding a nilpotent pole of the form (3.59)
+with M in the prescribed nilpotent orbit, and (ii) performing HW moves to deform the
+geometry, in effect replacing
+Zn →
+�
+i
+(Zλi + a(i)W1) .
+(4.3)
+The general case is obtained by mimicking this procedure: among the equations for the
+toric threefold corresponding to the polygon P, the line with transverse singularity An−1
+can be parametrized by a coordinate W1, and the transverse slice is again given by (4.2).
+This is one of the equations defining the toric threefold, called a ”boundary equation” in
+[96]. This equation should then be replaced by
+cW2W3 =
+�
+i
+(Zλi + a(i)W1) .
+(4.4)
+This is in agreement with [96, Theorem 4.8]. A useful mnemonic is that in the toric diagram
+(3.1), the deformation of the edge labelled by W1 involves an An−1 equation involving the
+coordinates labelling the adjacent edges, here W2 and W3, with the degree n polynomial in
+Z deformed by powers of W1. In the following subsections, we give examples to illustrate
+the scope of our results, and also show certain limitations.
+Successive deformations.
+Before discussing the examples, we want to address a
+natural question: given a GTP with white dots on an edge of length n characterized by
+a partition µ, is it possible to deform it to the same GTP with partition λ instead? One
+necessary condition is that λ > µ, and it is also sufficient if the polygon is large enough.5
+In this case, one can deform the nilpotent pole according to the general rules spelled out
+in section 2.
+For instance, if n = 4, we can use the explicit form (2.31). Going from partition [2, 12]
+to [22] is straightforward as it just involves merging two Jordan blocks, while going from
+5If the GTP is small, then certain deformations could lead to violations of the S-rule.
+– 29 –
+
+[22] to [3, 1] requires using higher powers of z (recall (2.29)):
+�
+�
+�
+�
+�
+z 1 0 0
+0 z 0 0
+0 0 z 0
+0 0 0 z
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+z 1 0 0
+0 z 0 0
+0 0 z α
+0 0 0 z
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+z 1 0 0
+0 z zβ 0
+0 0 z α
+0 0 0 z
+�
+�
+�
+�
+�
+α ̸= 0
+β ̸= 0
+(4.5)
+However it is worth mentioning that this can be done on the T-brane before HW defor-
+mations are added. Crucially, the two operations do not commute in general. Here for
+instance the deformed equations for T4 with partition [2, 2] and [3, 1], given in table 1,
+cannot be deformed from one to the other.
+4.2
+Rectangular Box
+Consider a rectangular box of size n × m,
+m
+n
+W1
+W3
+W2
+W4
+(4.6)
+The 5d SCFT this describes admits several well-known low energy gauge theory deforma-
+tions. The toric threefold singularity is not a hypersurface, but it is described by the pair
+of equations
+W1W3 = Zm ,
+W2W4 = Zn .
+(4.7)
+We can add white dots on the right segment, labeled by W1, exactly as in section 3. The
+deformed equations are
+W1W3 = Zm ,
+W2W4 =
+�
+i
+(Zλi + a(i)W1) .
+(4.8)
+Note that again, the coordinate W1 is used to deform the An−1 equation involving the
+coordinates labeling the adjacent edges W2 and W4.
+– 30 –
+
+Example.
+The following example is a useful check of this proposal, as the resulting
+model is related to T3. Consider the case m = 3, n = 2. Placing a white dot on the length
+2 edge yields the equations
+W1W3 = Z3 ,
+W2W4 = Z3 + aW1 .
+(4.9)
+We can now use the second equation to solve for W1, and we get the hypersurface
+W2W3W4 = Z2(aZ + W3) .
+(4.10)
+In the limit where a → ∞, the D7-brane, in the web interpretation, has crossed the whole
+brane system from right to left. We are left with
+W2W3W4 = Z3 ,
+(4.11)
+and we have reproduced the prediction (1.5)
+↔
+(4.12)
+This is a consistency check on the validity of our proposal.
+4.3
+Generic Triangle
+In this subsection we consider more examples which involve threefolds what are not hyper-
+surfaces in C4. Consider the case where the toric polygon is a triangle, of arbitrary shape.
+Pick one edge of length n. Then an SL(2, Z) transformation can bring the triangle to a
+frame where its three vertices are:
+{(0, 0); (0, n); (−a, b)} ,
+a ∈ Z>0, b ∈ Z .
+(4.13)
+This means the toric fan is generated by the three vectors (0, 0, 1), (0, n, 1) and (−a, b, 1).
+Among the generators of the dual cone we have
+(−1, 0, 0) ↔ W1
+(4.14)
+�n − b
+d2
+, − a
+d2
+, an
+d2
+�
+↔ W2
+(4.15)
+� b
+d1
+, a
+d1
+, 0
+�
+↔ W3
+(4.16)
+(0, 0, 1) ↔ Z .
+(4.17)
+In order to write the equation of the threefold, we have introduced d1 = gcd(a, b), d2 =
+gcd(a, n − b) and d3 = gcd(a, b, n − b). One of the equations for the toric threefold is
+W n/d3
+1
+W d2/d3
+2
+W d1/d3
+3
+= Zan/d3 ,
+(4.18)
+but in general there are other equations, as there are other generators in the dual cone. In
+the case of Tn (3.1), equation (4.18) is sufficient and reduces to (3.6), but this is a special
+case. To illustrate this phenomenon, we consider an example.
+– 31 –
+
+Example.
+We take the case a = 2 and b = 0.
+2
+2n
+W1
+W2
+W3
+(4.19)
+The dual cone is generated by 5 vectors:6
+(−1, 0, 0) ↔ W1
+(4.20)
+(n, −1, 2n) ↔ W2
+(4.21)
+(0, 1, 0) ↔ W3
+(4.22)
+(0, 0, 1) ↔ Z
+(4.23)
+(1, 0, 2) ↔ T
+(4.24)
+and the threefold is described by two equations in C5:
+W n
+1 W2W3 = Z2n ,
+W1T = Z2 .
+(4.25)
+The first equation corresponds to (4.18). The singular locus associated to the length 2n
+edge is Z = W2 = W3 = T = 0, parametrized by W1. Setting as before W1 = c, we
+can eliminate T and we find as expected an equation cnW2W3 = Z2n that we can deform.
+Therefore, the system of equations for the triangle (4.19) with one white dot on the long
+edge according to our conjecture is
+↔
+�
+W n
+1 W2W3 = Z2n−2(Z2 + aW1)
+W1T = Z2
+(4.26)
+6There is an algorithmic way to see that the fifth vector (1, 0, 2) is needed, an no other. This is the
+computation of the Hilbert basis of the semi-group σ∨ ∩M, using standard notation in toric geometry. This
+Hilbert basis is unique, and in the present case it has 5 elements, given here.
+– 32 –
+
+5
+Testing the Proposal: Resolution of Singularities
+To test the proposal for the deformation of singularities, we compute the resolutions for
+the deformed singularities. For the starting point, which we take to be a toric variety, the
+resolution is easily obtained in a combinatorial fashion, by a complete triangulation of the
+toric polygon. However, no such computational simplification exists yet for the resolution
+of GTPs. In view of this, we therefore need to revert to resolving the actual algebraic
+varieties. We will carry this out for the Tn theories.
+5.1
+Crepant Resolution of Tn
+The simplest type of singularities are the hypersurfaces that realize the Tn theories. The
+flavor symmetry algebra is su(n)3 except for n = 3, where we expect e6. This can be
+easily detected from the toric geometry by extracting the set of curves that are complete
+intersections between compact and non-compact (i.e. flavor) divisors. The resulting so-
+called combined fiber diagram (CFD) [26–28] is easily read off from the toric polygon [80].
+Here we will take the more laborious path of resolving the hypersurface singularity, in
+preparation for resolving the deformed singularities. The hypersurface in C4 is
+W1W2W3 = Zn .
+(5.1)
+The first resolution for all of these singularities is simply to remove the locus W1 =
+W2 = W3 = Z = 0, which is achieved by inserting a P3 with projective coordinates
+[W1, W2, W3, Z] (for a detailed exposition of resolutions from a physicist’s perspective, see
+[97]). Denoting the exceptional section of the blowup by δ1 the equation, after proper
+transform, which ensures that the resolution is crepant, becomes
+W1W2W3 = Znδn−3
+1
+.
+(5.2)
+This can be further resolved by consecutively blowing up the loci W1 = W2 = W3 = δi = 0
+i.e. by inserting another projective space with coordinates
+[W1, W2, W3, δi] ,
+i = 1, · · · , ⌊n − 3i⌋ ,
+(5.3)
+which implies that the coordinates cannot vanish at the same time (and thus the above is
+a relation in the Stanley-Reissner ideal of the hypersurface). This is iterated until we reach
+W1W2W3 = Zn
+⌊n−3i⌋
+�
+i=1
+δn−3i
+i
+.
+(5.4)
+The remaining blowups will resolve the local singularities along W1 = 0, W2 = 0 and
+W3 = 0 respectively, by small resolutions, i.e.
+[Wi, Z] ,
+or
+[Wi, δi] .
+(5.5)
+Let us denote the compact divisors by Si and the non-compact divisors by Da. We then
+find from these resolutions the intersection matrix
+Ga =
+�
+i
+Si · Da · Da
+(5.6)
+– 33 –
+
+T3
+T3 with deformation [1, 2]
+Figure 11.
+CFDs: The intersection graph for T3 (left hand side) and the deformation of T3
+described by the partition [1, 2] (right).
+The nodes are curves that are complete intersections
+between � Si (i.e. the sum of all compact divisors) and the non-compact divisors. The green nodes
+are −2 self-intersection curves, which correspond to roots of the flavor symmetry algebra, white are
+−1 curves, which can be thought of as bifundamental matter. On the left we see the su(3)3 flavor
+symmetry and the matter in the (3, 3, 1) etc, which enhances to e6. On the right the theory is rank
+0.
+to be precisely the adjacency matrix of the CFD for Tn: i.e. three su(n) Cartan matrices,
+pairwise connected by −1 curves. We have shown examples in figures 11 and 12.
+5.2
+Resolution of Deformations of Tn
+Deformations of T3.
+By including the deformations, some of the above resolutions get
+obstructed. Lets consider the simplest case of T3, which itself is the rank 1 Seiberg theory
+with flavor symmetry e6. After a single blowup [W1, W2, W3, Z] we get one compact divisor
+δ1 = 0, and three A2 singularities. Resolving yields the CFD shown in figure 11 on the left
+hand side. The representations are (3, 3, 1), and cyclic permutations, and thus we get the
+known enhancement to e6 7,
+(3, 3, 1) ⊕ (1, 3, 3) ⊕ (3, 1, 3) −→
+27 .
+(5.7)
+Adding the deformation results in W1W2W3 = Z(Z2 + W1), which does not allow for a big
+resolution. There are small resolutions, along e.g. W1 = Z = 0, indicating that this is a
+rank 0 theory.
+Deformations of T4.
+More interestingly we can start with T4. All deformations involving
+a single edge and yielding a positive rank theory are shown in table 2. The deformation
+that is associated to the partition [2, 12] along one edge is the hypersurface
+W1W2W3 = Z2(Z2 + W1) .
+(5.8)
+7From the geometry we are also able to extract the flavor symmetry group [42, 98], which however will
+not play a role in the present paper.
+– 34 –
+
+Partitions
+Rank
+Free
+Symmetry
+Equation
+Fig.
+[14]
+3
+0
+su(4)3
+W1W2W3 = Z4
+12
+[2, 12]
+2
+0
+su(8) ⊕ su(2)
+W1W2W3 = Z2(Z2 + W1)
+12
+[22]
+1
+0
+e7
+W1W2W3 = (Z2 + aW1)(Z2 + bW1)
+12
+Table 2. The partition that defines the distribution of white dots in the GTP, the rank of the 5d
+SCFT, the flavor symmetry algebra, as well as hypersurface equation for the deformations of T4.
+The first resolution (5.3) results in
+W1W2W3 = δ2Z4 + Z2W1 .
+(5.9)
+Continuing the blowup results in the intersection matrix between the sum of compact
+divisors �
+i Si and each of the non-compact ones
+��
+i
+Si
+�
+· Da · Db =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−1 1
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+1 −1 0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0 −2 1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+1 −1 1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1 −2 0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0 −2 0
+1
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0 −2 1
+0
+0
+0
+1
+0
+0
+0
+0
+0
+1
+1 −1 0
+0
+0
+0
+1
+0
+0
+0
+1
+1
+0
+0 −2 0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0 −1 1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1 −2 1
+0
+1
+0
+0
+0
+0
+1
+0
+0
+0
+1 −2
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ab
+.
+(5.10)
+This is shown in figure 12, from which the manifest flavor symmetry algebra is read off
+from the collection of −2 curves
+su(4)2 ⊕ su(2) ⊕ u(1) .
+(5.11)
+The additional u(1) follows from the general rule on flavor symmetry algebras from CFDs
+as laid out in [31]. The matter is in the representations, which combine naturally into the
+representations of the enhanced symmetry algebra as follows
+(4, 4, 1) ⊕ (6, 1, 1) ⊕ (1, 6, 1) ⊕ (4, 1, 2) ⊕ (1, 4, 2)
+−→
+(28, 1) ⊕ (8, 2) ,
+(5.12)
+where we identify the enhanced flavor symmetry algebra as
+gUV = su(8) ⊕ su(2) .
+(5.13)
+Similarly for the partition [22], the manifest symmetry is su(4)2, while the actual
+symmetry is e7. In the resolution, see figure 12 on the right hand side, where we see the
+representation
+(4, 4) ⊕ (4, 4) ⊕ 2 × (6, 1) ⊕ 2 × (1, 6)
+−→
+56 .
+(5.14)
+These indeed arise from the branching of the 56 of e7 to su(4)2.
+– 35 –
+
+T4
+T4 with deformation [12, 2]
+T4 with deformation [22]
+Figure 12.
+CFDs: The intersection graph for T4 (left hand side) and the deformation of T4
+described by the partition [122] (middle) and deformation of T4 with partition [22]. The nodes are
+curves that are complete intersections between � Si (i.e. the sum of all compact divisors) and the
+non-compact divisors. The green nodes are −2 self-intersection curves, which correspond to roots of
+the flavor symmetry algebra, white are −1 curves, which can be thought of as bifundamental matter.
+On the left we see the su(4)3 flavor symmetry and the matter in the (4, 4, 1) etc representations.
+In the middle, we see the manifest flavor symmetry su(4)2 ⊕ su(2) ⊕ u(1), but the matter shown
+results in the enhancement to su(8) ⊕ u(1). On the right the flavor symmetry enhances to e7.
+Partitions
+Rank
+Symmetry
+Equation
+Fig.
+[15]
+6
+su(5)3
+W1W2W3 = Z5
+13
+[2, 13]
+5
+su(5)2 ⊕ su(3) ⊕ u(1)
+W1W2W3 = Z3(Z2 + W1)
+13
+[22, 1]
+4
+su(5)2 ⊕ su(2) ⊕ u(1)
+W1W2W3 = Z(Z2 + W1)2
+15
+[3, 12]
+3
+su(10) ⊕ su(2)
+W1W2W3 = Z2(Z3 + W1)
+13
+[3, 2]
+2
+su(10)
+W1W2W3 = (Z2 + W1)(Z3 + W1)
+14
+Table 3.
+Deformations of the T5 model: the partition that defines the GTP, the rank of the
+expected 5d SCFT, the flavor symmetry algebra, and the deformed hypersurface equation.
+Deformations of T5.
+Finally consider the T5 model with deformations added along a
+single edge. These are summarized in table 3.
+First we recompute the resolution of the undeformed T5 model, which is given in figure
+13. The flavor symmetry algebra and the representation of the hypers is
+gUV = su(5)3 :
+(5, 5, 1) ⊕ (1, 5, 5) ⊕ (5, 1, 5) .
+(5.15)
+The CFD is shown in figure 13. Adding a single white dot results in a theory with flavor
+algebra su(5)2 ⊕ su(3) ⊕ u(1), with the CFD shown in the middle in figure 13, which has
+bifundamental matter consistent with these flavor symmetry algebras.
+Adding two white dots along a single edge in the partition [3, 12] we obtain the right
+hand figure 13. The representations under su(5)2 ⊕ su(2) are
+(5, 5, 1) ⊕ (1, 5, 2) ⊕ (5, 1, 2) ⊕ (10, 1, 1) ⊕ (1, 10, 1)
+−→
+(45, 1) ⊕ (10, 2) (5.16)
+– 36 –
+
+T5
+T5 with deformation [13, 2]
+T5 with deformation [12, 3]
+Figure 13. CFDs: The intersection graph of −2 (green) and (−1) (white) curves, i.e. the CFD,
+for T5 (LHS), the partition [2, 13] in the middle, and [3, 12] on the right hand side.
+which is consistent with the flavor symmetry enhancement to
+gUV = su(10) ⊕ su(2) .
+(5.17)
+Similarly for [3, 2] we computed the resolution and find the CFD shown in figure 14, which
+shows the representations under su(5)2 to be
+(5, 5) ⊕ (10, 1) ⊕ (1, 10)
+−→
+45 ,
+(5.18)
+which is consistent with the enhancement to su(10). Finally consider figure 15, where we
+manifestly see the su(5) but the su(2) is decomposed into u(1).
+For partition [22, 1], although the su(2) factor of the global symmetry is not directly
+visible in the resolution, an educated guess allows to read the following representations of
+su(5) ⊕ su(5) ⊕ su(2) on figure 15:
+(5, 1, 1) ⊕ (1, 5, 1) ⊕ (10, 1, 2) ⊕ (5, 10, 2) ⊕ (5, 5, 1) .
+(5.19)
+Again, the flavor symmetry algebra is consistent with the one predicted from GTPs.
+Acknowledgments
+We would like to thank Cyril Closset for collaboration in early stages of this work. AB and
+SSN thank Hendrik S¨uß for discussions of related questions. This work, in particular AC
+and SSN, was supported and enabled by the Fondation Wiener-Anspach. This research
+is further supported by IISN-Belgium (convention 4.4503.15).
+AB is supported by the
+ERC Consolidator Grant 772408-Stringlandscape, and by the LabEx ENS-ICFP: ANR-
+10-LABX-0010/ANR-10-IDEX-0001-02 PSL*. SSN is supported in part by the “Simons
+Collaboration on Special Holonomy in Geometry, Analysis and Physics” and the EPSRC
+Open Fellowship EP/X01276X/1.
+– 37 –
+
+T5 with deformation [2,3]
+Figure 14. CFDs: The intersection graph for T5 with the deformation labeled by the partition
+[2, 3]. We see the su(5)2 ⊕ u(1) and the matter in the (5, 5) ⊕ (10, 1) ⊕ (1, 10) which enhance into
+45 of su(10).
+T5 with deformation [1, 22]
+Figure 15. CFDs: The intersection graph for T5 with the deformation labeled by the partition
+[1, 22].
+A
+Basic algebraic notions for sln
+This appendix gathers a few elementary algebraic concepts needed in the bulk of the paper.
+In the following, we take g = sln(C), which we represent as n × n complex matrices with
+vanishing trace, and we pick the diagonal matrices for the Cartan subalgebra h.
+Nilpotent Orbits and Triples.
+Let e be a nilpotent element of g. By the Jacobson-
+Morozov theorem, one can construct an sl2 triple (e, f, h), i.e. a triple of elements of g with
+h ∈ h satisfying the commutation relations
+[e, f] = h ,
+[h, e] = 2e ,
+[h, f] = −2f .
+(A.1)
+More precisely, there exists an embedding
+ρ : sl2 −→ g ,
+(A.2)
+– 38 –
+
+such that ρ(e) defines a nilpotent element of g. In our case, nilpotent orbits are charac-
+terized by partitions of n. For the partition λ = (λ1, . . . , λr), there is a canonical triple,
+where e is in the Jordan form specified by λ, h is diagonal and f is lower diagonal, defined
+as follows. Consider first the maximal orbit, λ = (n). In this case we define the canonical
+triple e = Jn, h = Hn and f = ˜Jn where
+(Jn)i,j = δi+1,j ,
+(Hn)i,j = δi,j(n + 1 − 2i) ,
+( ˜Jn)i,j = δi−1,jj(n − j) .
+(A.3)
+Then for any partition λ the canonical triple is the block diagonal
+e = Diag(Jλi) ,
+f = Diag( ˜Jλi) ,
+h = Diag(Hλi) .
+(A.4)
+Slodowy Slices.
+Given a triple (e, f, h), one constructs the space
+Se = e + gf ,
+(A.5)
+where gf is the centralizer of f in g. This is called the Slodowy slice transverse to e. Note
+that this should not be confused with the nilpotent Slodowy slice, which is the intersection
+of the Slodowy slice with the nilpotent cone.
+When (e, f, h) is the canonical nilpotent
+element (A.4) associated to partition λ, we call Sλ the Slodowy slice.
+In order to construct explicitly Sλ, we first need the set of matrices which commute
+with ˜Jn. These are the matrices S(a1, . . . , an) with a1, . . . , an ∈ C, defined by8
+(S(a1, . . . , an))ij :=
+n−1
+�
+k=0
+ak+1δi−k,j
+i−1
+�
+l=j
+l(n − l) .
+(A.7)
+Then the centralizer gf is a set of block matrices of sizes λi × λj, where the block (i, j)
+depends on min(λi, λj) parameters. We will not need its explicit form here. We only need
+the form of the diagonal blocks (i = j), which are of the form
+S(a(i)
+1 , . . . , a(i)
+λi ) .
+(A.8)
+Using this, we can compute the dimension of the Slodowy slices, which is
+dim Sλ = −1 +
+�
+1≤i,j,≤r
+min(λi, λj) = −(n + 1) + 2
+r
+�
+i=1
+iλi
+(A.9)
+and
+dim Sλ ∩ N = dim Sλ − (n − 1) = −2n + 2
+r
+�
+i=1
+iλi .
+(A.10)
+8The matrices S(a1, . . . , an) for low values of n are:
+S(a1, a2) =
+�
+a1 0
+a2 a1
+�
+,
+S(a1, a2, a3) =
+�
+�
+�
+a1
+0
+0
+2a2 a1
+0
+4a3 2a2 a1
+�
+�
+� ,
+S(a1, a2, a3, a4) =
+�
+�
+�
+�
+�
+a1
+0
+0
+0
+3a2
+a1
+0
+0
+12a3 4a2
+a1
+0
+36a4 12a3 3a2 a1
+�
+�
+�
+�
+� .
+(A.6)
+– 39 –
+
+The dimension of the nilpotent cone N is n(n − 1), and therefore we can deduce that the
+dimension of the nilpotent orbit of e is
+dim Oλ = n(n + 1) − 2
+r
+�
+i=1
+iλi ,
+(A.11)
+which matches known results.
+Levi subalgebras and Slodowy slices.
+A Borel subalgebra b of g is a maximal
+solvable subalgebra.
+The standard choice, which we make, is to pick for b the upper
+triangular matrices. The simple roots αi can be indexed by i ∈ I := {1, . . . , n − 1}, and
+to every subset Θ of I one can associate a parabolic subalgebra pΘ of g containing b. This
+parabolic subalgebra decomposes as a direct sum
+pΘ = lΘ ⊕ nΘ
+(A.12)
+where lΘ is the Levi subalgebra and nΘ is the nilradical of pΘ.
+Let λ = (λ1, . . . , λr) be a partition of n. We associate to this partition the set
+Θλ = I − {λ1, λ1 + λ2, . . . , λ1 + · · · + λr−1} .
+(A.13)
+This way, one can construct the corresponding Levi subalgebra lλ := lΘλ. We then intro-
+duce the intersection
+S0
+λ := Sλ ∩ lλ .
+(A.14)
+With out choices, lλ is simply the algebra of block diagonal matrices, with blocks of
+respective sizes λ1, . . . , λr, and S0
+λ is the set of traceless matrices which are block diagonal,
+with diagonal blocks Jλi +S(a1, a2, . . . , aλi). The dimension is dim S0
+λ = n−1, independent
+of λ.
+Companion matrix.
+It is easy to check that the matrix Jn + S(0, a2, . . . , an) can be
+conjugated to a matrix of the form
+C(b2, . . . , bn) :=
+�
+�
+�
+�
+�
+�
+�
+�
+0
+1
+0
+1
+... ... ...
+0
+1
+(−)n−1bn (−)n−2bn−1
+· · ·
+−b2 0
+�
+�
+�
+�
+�
+�
+�
+�
+(A.15)
+where the bi are known functions of the ai. Note that the change of basis depends explicitly
+on the ai. A matrix in the form (A.15) is called a companion matrix. It has the convenient
+property that its characteristic polynomial (evaluated at −z) is
+det(z1n + C) = zn +
+n
+�
+i=2
+bizn−i .
+(A.16)
+Any matrix in S0
+λ can then be conjugated to a block diagonal matrix with blocks of the
+form C(b2, . . . , bλi).
+– 40 –
+
+Cokernel of ad(e).
+Consider a finite dimensional representation ρ : sl(2) → V of sl(2).
+One can decompose V into irreducible representations Vi of dimensions ni. In each irrep,
+the image of ρ(e) has codimension 1, and the cokernel is spanned by the weight space of
+lowest weight, which by definition is precisely the kernel of ρ(f). Therefore in the Lie
+algebra g seen as an sl(2) representation defined by a triple (e, f, h), the cokernel of ad(e)
+is gf.
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diff --git a/pNE4T4oBgHgl3EQfvQ0x/content/tmp_files/load_file.txt b/pNE4T4oBgHgl3EQfvQ0x/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf,len=1819
+page_content='Generalized Toric Polygons, T-branes, and 5d SCFTs  Antoine Bourget,1 Andr´es Collinucci,2 Sakura Sch¨afer-Nameki3  1Universit´e Paris-Saclay, CNRS, CEA, Institut de physique th´eorique,  91191, Gif-sur-Yvette, France  1Laboratoire de Physique de l’´Ecole Normale Sup´erieure, PSL University,  24 rue Lhomond, 75005 Paris, France  2Service de Physique Th´eorique et Math´ematique, Universit´e Libre de Bruxelles and  International Solvay Institutes, Campus Plaine C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' 231, B-1050 Bruxelles, Belgium  3Mathematical Institute, University of Oxford,  Woodstock Road, Oxford, OX2 6GG, United Kingdom  Abstract: 5d Superconformal Field Theories (SCFTs) are intrinsically strongly-coupled  UV fixed points, whose realization hinges on string theoretic methods: they can be con-  structed by compactifying M-theory on local Calabi-Yau threefold singularities or alterna-  tively from the world-volume of 5-brane-webs in type IIB string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' There is a cor-  respondence between 5-brane-webs and toric Calabi-Yau threefolds, however this breaks  down when multiple 5-branes are allowed to end on a single 7-brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In this paper, we  extend this connection and provide a geometric realization of brane configurations includ-  ing 7-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' A web with 7-branes defines a so-called generalized toric polygon (GTP),  which corresponds to combinatorial data that is obtained by removing vertices along exter-  nal edges of a toric polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We identify the geometries associated to GTPs as non-toric  deformations of toric Calabi-Yau threefolds and provide a precise, algebraic description  of the geometry, when 7-branes are introduced along a single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The key ingredients  in our analysis are T-branes in a type IIA frame, which includes D6-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We show  that performing Hanany-Witten moves for the 7-branes on the type IIB side corresponds  to switching on semisimple vacuum expectation values on the worldvolume of D6-branes,  which in turn uplifts to complex structure deformations of the Calabi-Yau geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We  test the proposal by computing the crepant resolutions of the deformed geometries, thereby  checking consistency with the expected properties of the SCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='05239v1  [hep-th]  12 Jan 2023  Contents  1  Introduction and Summary  1  2  T-branes and Kraft-Procesi Transitions  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1  T-brane Basics  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2  Kraft-Procesi Transitions  10  3  Example: GTPs for Tn and Related Models  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1  The Setup  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2  Tachyon Condensation Picture  15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='3  Example: T2  17  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='4  General Case: Tn  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='5  Interpretation in terms of Generalized Toric Polygons  22  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='6  Hanany-Witten Moves  24  4  General Discussion  28  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1  White Dots along a Single Edge  28  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2  Rectangular Box  30  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='3  Generic Triangle  31  5  Testing the Proposal: Resolution of Singularities  33  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1  Crepant Resolution of Tn  33  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2  Resolution of Deformations of Tn  34  A Basic algebraic notions for sln  38  1  Introduction and Summary  The existence and characterization of interacting superconformal field theories in five space-  time dimensions (5d SCFTs) is a remarkable prediction of string theory [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Two approaches  have emerged that allow the construction of 5d SCFTs within the framework of string the-  ory: the low energy limit of M-theory on R1,4 times a local Calabi-Yau threefold [2], and  the world-volume of a brane-web in type IIB string theory [3–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The properties of the  theory are encoded in the geometry of the threefold in the first case, and in the charges of  the external (p, q)-5-branes in the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' A middle ground is the type IIA realization,  which involves both geometry and branes [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  When the CY is toric, there is a precise dictionary between the brane-web and M-theory  realization: the charges of the external (p, q)-5-branes can be encoded into an integral  polygon, which in turn can be seen as the intersection of a toric three-dimensional fan in  – 1 –  ←→  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Example of toric polygon (left) and dual brane-web (right) in which lines denote 5-branes  and circles denote 7-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The 7-branes on which stacks of 5-branes are spaced to emphasize how  the 5-brane end, here exactly one 5-brane ends on each 7-brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This geometry encodes a 5d SCFT  of rank 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  R3  x,y,z with the plane {z = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The toric threefold X constructed from this fan is such that  the 5d SCFT obtained from M-theory on X coincides with that on the world-volume of  the brane-web [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' An example is shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  A systematic geometric exploration and classification of 5d SCFTs was started in  [15–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' These studies reveal many detailed properties of the 5d SCFTs, such as their  UV enhanced flavor symmetry, their Coulomb branch (modeled in terms of the crepant  resolutions of the Calabi-Yau singularities), but also refined information such as their  generalized symmetries [39–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  What remains somewhat obscure in this framework is  the derivation of the full quantum corrected Higgs branch – though some progress in the  context of isolated hypersurface Calabi-Yau singularities can be made [32–34, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Not surprisingly, these explorations reveal that only a small class of 5d SCFTs have a  realization in terms of toric Calabi-Yau threefolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' If such a toric realization exists, then  the geometry of the moduli space of supersymmetric vacua, in particular the Higgs branch  (but also Coulomb branch and mixed branches) can be computed exactly – irrespective of  whether the singularity is isolated or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The key tool is the connection between the toric  geometries and brane-webs, where in the latter these moduli space questions have been  determined in [47–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In this paper, we report progress in generalizing these methods to  a larger class of 5d SCFTs, which have not necessarily a toric description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  In [16], a generalization of toric polygons was introduced: a toric Calabi-Yau threefold  can be described in terms of a convex polygon in a square integral lattice embedded into  a 2-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The polygon associated to a toric geometry has the property that all lattice  points along the edges are part of the toric data (corresponding to vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We will refer  to these as black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Generalizing this, [16] proposed to also allow some vertices along  the edges of the polygon to be unoccupied, which we will refer to as white dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In the  dual description, allowing such white dots corresponds in the web to several 5-branes that  end on the same 7-brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' For a stack of n 5-branes, the boundary condition is encoded in  an integer partition λ of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Figure 2 gives an example of a configuration that translates  to the [3, 2, 2, 1] partition of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This combinatorial data will be referred to as Generalized  Toric Polygons (GTP), and generalizes the standard toric description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In this framework,  the standard toric case considered in the previous paragraph corresponds to a boundary  condition where for each charge (p, q), there is an equal number of (p, q)-5-branes and  – 2 –  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Correspondence between white dots on GTPs (left) and boundary conditions of (p, q)  5-branes on (p, q) 7-branes (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Here we have (p, q) = (1, 0) and the partition λ = [3, 2, 2, 1]  of n = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' When we draw brane-webs we usually ignore the detached 7-branes and separate the  7-branes on a stack of 5-branes to show the boundary conditions (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (p, q)-7-branes, with one 5-brane ending on one 7-brane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' the partition is λ = [1n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This  generalization and its implications for characterizing the moduli space of supersymmetric  vacua using magnetic quivers were explored in great detail in [32, 45, 53–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Note that  O5 orientifold planes can also be included in the webs [8, 64–66], but we do not consider  this possibility here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Another way of interpreting GTPs is as non-convex would-be toric polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' These  make sense when certain parameters, which map out the extended Coulomb branch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  gauge couplings and masses of hypers), are turned on, but the non-convexity prevents one  from considering the SCFT limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' from passing to the origin of the extended Coulomb  branch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  From the dual brane-web point of view, this is resolved using a combination  of Hanany-Witten moves and 7-brane monodromies, and this plays a prominent role in  the brane-web manipulations of [6, 8, 9, 11, 64, 67–71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Importantly, not all non-convex  polygons can be transformed into GTPs in this way, and it is in general a hard question  to decide whether a given polygon can be transformed in this way or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' For this reason,  it is simpler to take the GTPs as our starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  GTPs can be thought of as generalizations of toric geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' However, unlike the  precise dictionary between the combinatorial data of a toric polygon and the algebraic  geometry of the corresponding Calabi-Yau, no such dictionary exists thus far for GTPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The main purpose of this paper is to develop initial steps in order to close this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  See figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In particular, in the following, we will determine the algebraic geometric  description of GTPs, which have white dots along a single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  We now give a schematic summary of the main ideas involved in our pro-  posal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Consider a length n edge of a toric polygon, which we can assume to have vertical  orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In the brane-web, this corresponds to n parallel semi-infinite D5-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In the  associated toric threefold, there is an asymptotic region that approaches C2/Zn×C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Indeed  after a transverse T-duality, the D5-branes become D6-branes, which uplift to n-centered  – 3 –  Convex Integral Polygon  Toric CY3  Brane-web  5d SCFT  Toric Geometry  Dual  M-theory  Worldvolume  GTP  Non-Toric CY3  Brane-web with  7-brane b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  5d SCFT  Dual  M-theory  Worldvolume  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Summary of the main question addressed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' On the left hand side, one  starts from a convex polygon with integral vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' It defines a 5d SCFT in two distinct ways:  from M-theory on the associated toric CY, and from the dual brane-web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' If on the contrary, as  shown on the right hand side, the polygon is not convex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' is a generalized toric polygon (GTP),  the toric description is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' However there still exists a dual brane-web, which has non-trivial  boundary conditions on 7-branes, and thus a 5d SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The central goal of this paper is to develop  a map (the dashed line in the diagram) from GTPs to (non-toric) geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Taub-NUT spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' At strong string coupling gs → ∞, this becomes C2/Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Denoting  two longitudinal directions as the w-complex plane, we arrive at a local C2/Zn × C patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  M-theory on this geometry gives us N = 1 7d SYM with SU(n) gauge group, and we can  represent it as the singular hypersurface in C4 given by  uv = zn  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1)  with the w-coordinate tagging along.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The three adjoint scalars φi=1,2,3 on the worldvolume  of the D6-branes can be grouped into a complex scalar Φ = φ1+iφ2, and the remaining real  one ϕ = φ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In the M-theory uplift, Φ encodes algebraic deformations to the hypersurface,  and ϕ encodes K¨ahler volumes of resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This grouping is of course arbitrary, and  correlates with the arbitrariness of choosing a complex structure on the noncompact K3 in  M-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Having seen this, we can recast the hypersurface as a spectral equation for the com-  plexified adjoint Higgs field  uv = det(1n z − Φ(w)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2)  Switching on constant vevs along the Cartan subalgebra of su(n) will deform the equation  and unfold the singularity into a deformed K3 times the w-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' However, switching on w-  dependent vevs will turn this into a bona fide noncompact CY threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The geometry will  be more or less desingularized, depending on the Casimir invariants of Φ that are switched  on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The claim of this paper, is that white dots correspond to nilpotent elements in Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Note,  that switching on a nilpotent Φ means that the spectral equation remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In  other words, the D6-branes do not actually move, and the uplifted geometry underlying  the M-theory construction remains undeformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This phenomenon is known as a T-brane  [72–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' It is a non-Abelian bound state of branes, whereby the worldvolume gauge group  is (partially) Higgsed, but the geometry of the branes is unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' However, the physics  is of course impacted by this T-brane, as we shall see momentarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 4 –  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' White dot and brane transition for a length 2 edge of a toric polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  To give a simple example, take a vertical edge with n + 1 dots, and replace the second  black dot from the top with a white dot as shown in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In terms of the 5-branes, this  corresponds to forming a bound state between the two uppermost branes, and sending a  suspended 5-brane segment to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' From the D6-brane viewpoint, the bound state is  understood as switching on a vev for Φ along the minimal nilpotent orbit of su(n)  ⟨Φ⟩ =  �  �  �  0 1  0 0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='3)  This binds the first two D6-branes, and partially Higgses  su(n) → s (u(1) ⊕ u(n − 2)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='4)  More generally, we said above that a distribution of white dots on the edge is encoded in  a partition λ of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Our claim is that each such partition λ of n translates into a vev for  Φ along an element in the nilpotent orbit Oλ of su(n) that is uniquely characterized by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The unbroken 7d gauge group on the D6-branes, which will correspond to a subgroup of  the total 5d flavor group, is then broken to the commutant of this nilpotent element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  So far, our discussion parallels the picture developed several years ago in [75, 76], in  the 6d SCFT context, whereby geometric data (about elliptic fibrations) was supplemented  by nilpotent orbits, which would partially Higgs an original theory and trigger various  RG flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' At this point, the reader might object that simply claiming that a white dot  translates to a nilpotent vev is not very interesting or verifiable, since that data will be  invisible to the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' While this is true, from the 5-brane-web perspective we know  that a white dot opens up the possibility to perform Hanany-Witten type transitions that  were not possible in the presence of black dots only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' For instance, the following GTPs are  related by such a transition where one of the three leftmost 7-branes is moved to the right:  ↔  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='5)  – 5 –  Such HW type transitions provide non-trivial tests of our proposal: HW moves correspond  to changing the positions of branes, which in turn will impact the dual geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We  identify the subset of complex structure deformations that are associated to these nilpotent  vevs of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' They are realized in terms of vevs of Φ along a slice transverse to the nilpotent  vev (inside the full Lie algebra, not the nilpotent cone), known as the Slodowy slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' By  switching on such a vev, a subset of possible Casimir invariants will become non-zero,  leading to a deformation of the geometry, which is given by the spectral equation of the  the Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  For instance, in the simple case of su(2), we take the initial nilpotent vev along the  minimal orbit  Φ0 =  �  0 1  0 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='6)  The M-theory uplifted geometry corresponds to C2/Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The Slodowy slice is given by  matrices of the form  Φ =  �  0 1  a 0  �  ,  with  a ∈ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='7)  The characteristic polynomial of this Higgs field is now non-trivial, and the M-theory  geometry deforms as follows  uv = z2  −→  uv = z2 + a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='8)  The present paper elucidates this for all GTPs, which allow for a IIA-description, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  whose white dots are along a single edge of the GTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Dually, all 7-branes are parallel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  mutually local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The toy-model where Slodowy slices appear is generalized in the following  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Higgs branches are symplectic singularities [77], to which one can associate a Hasse  diagram of symplectic leaves [78–80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In terms of this diagram, the T-brane data select a  new, lowest leaf of the foliation, and the transverse slice to that leaf is the total space of a  fibration over the complex structure moduli space of the deformed Calabi-Yau threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' See  for instance figure 5, where the deformations of the T4 5d SCFT, realized on the threefold  W1W2W3 = Z4, are displayed, along with the effect on the Higgs branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  In future work we will aim to generalize this to arbitrary GTPs, with mutually non-  local 7-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  We conjecture that the above picture of transverse slices in the Higgs  branch extends to this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Eventually we hope to develop a succinct description of  the algebraic geometry of GTPs, as they exist for toric polygons: A precise map between  the combinatorial data and the basic algebraic geometry, such as the set of divisors, curves,  intersection numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  In the rest of the paper, we spell out the details of the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' An essential  tool is the T-brane, which is reviewed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The bulk of the construction is then  carried out explicitly in a representative example – that of the Tn SCFTs – in section 3,  before generalizing to any GTP with white dots on a single edge in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' As a first  check, we reproduce there the transition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Finally, in section 5 we provide consistency  checks, by computing the resolutions of the deformed threefold geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This shows  – 6 –  a3  a1  a7  e6  e7  α ̸= 0  a1  a7  e6  e7  α ̸= 0  β ̸= 0  a7  e6  e7  W1W2W3 = Z4  W1W2W3 = Z2(Z2 + αW1)  W1W2W3 = (Z2 + αW1)(Z2 + βW1)  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Three GTPs are shown on the first line, and below the algebraic equations characterizing  the associated Calabi-Yau threefold geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The model on the left is a toric threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The other  two, non-toric GTPs, are characterized in terms of deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Each of these geometries defines  a 5d SCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The Hasse diagrams of symplectic singularities for the Higgs branch of these 5d SCFTs  are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The vertices represent symplectic leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' For transverse slices we use a standard  notation where the closure of the minimal nilpotent orbit of a simple Lie algebra is denoted using  the lowercase form of the name of the algebra, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' e7 for algebra E7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In red are drawn the effects  of the deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  agreement of the UV flavor symmetry of the SCFT with the one expected from the brane-  web (and resulting Higgs branch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  2  T-branes and Kraft-Procesi Transitions  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1  T-brane Basics  Consider n parallel D7-branes in type IIB string theory on flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The transverse  space is the complex plane with coordinate z, which has coordinate ring R = C[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We  call z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , zn the positions of the n branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The stack of branes can be described as a  D9/D9-brane tachyon condensate, which is defined mathematically as the cokernel of the  tachyon map  R⊕n  R⊕n ,  T  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1)  where T = Diag(z − z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , z − zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This means that the D7-branes correspond to the  sheaf1 S in the short exact sequence  0  R⊕n  R⊕n  S  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  T  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2)  The matter on the system of D7-branes is described by fluctuations of the tachyon, δT,  which are defined up to linearized gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This corresponds to a self-Ext1  1We will use the formulation of branes modulo tachyon condensation in terms of the derived category  of coherent sheaves throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Some introduction to this topic can be found in [81–83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 7 –  computation for the complex (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' morphisms between the complex and the shifted  version of that same complex,  R⊕n  R⊕n  R⊕n  R⊕n  αL  δT  T  αR  T  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='3)  up to homotopies,  δT ∼ δT − T · αL + αR · T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='4)  Concretely, this means δT is valued in the quotient ring of n×n matrices Matn(R) modulo  the two matrix ideals in R generated by left and right multiplication by T,  δT ∈ Matn(R)/(T·, ·T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='5)  Note also that the tachyon map T can be expressed as a matrix given a choice of basis  for the D9 gauge bundle and the D9 gauge bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' These choices are independent, which  means algebraically that only the equivalence class of T under the equivalence relation  T ∼ GL · T · G−1  R  GL, GR ∈ GL(n, C[z])  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='6)  matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' A canonical representative of each such equivalence class is given by the Smith  Normal Form (SNF) computed in the ring R = C[z], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' any matrix T with entries in  polynomials in z is equivalent under (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='6) to a unique diagonal matrix  SNF(T) := diag(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , pr, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , 0) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='7)  where the pi are monic polynomials2 in z such that p1|p2| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' |pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  To illustrate the discussion of the previous paragraph, consider the case  n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The tachyon matrix is  T =  �  z − z1  0  0  z − z2  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='8)  and the fluctuations belong to  δT ∈  �  R  (z−z1)  R  (z−z1,z−z2)  R  (z−z1,z−z2)  R  (z−z2)  �  ≃  �  �  �  �  �  �  �  �  �  �  �  �  �  �  C 0  0 C  �  z1 ̸= z2  �  C C  C C  �  z1 = z2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='9)  For any value of z1, z2 there is U(1) adjoint matter on each D7-brane, and for z1 = z2 the  U(1)2 gauge symmetry enhances to U(2), and one can have fluctuations in the adjoint of  U(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' From the U(1)2 perspective this is simply bifundamental matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  2Unicity is guaranteed up to multiplication by units in the ring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' demanding that the polynomials be  monic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' have coefficient 1 for the term of highest degree, fixes this redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 8 –  Consider the case z1 = z2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We can then activate an off-diagonal term:  T[1,1] =  �  z 0  0 z  �  �→  T[2] =  �  z 1  0 z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='10)  After this activation, the fluctuations are reduced to  δT[2] ∈  �  0 0  1 0  �  C ⊕  �  1 0  0 −1  �  C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='11)  The SNF reveals the same structure in a slightly different guise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Indeed  SNF  �  z − z1  0  0  z − z2  �  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  1  0  0 (z − z1)(z − z2)  �  z1 ̸= z2  �  z − z1  0  0  z − z1  �  z1 = z2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='12)  so the fluctuations are valued in  δT ∈  �  �  �  �  �  �  �  �  �  �  �  �  �  �  0  0  0 C ⊕ zC  �  z1 ̸= z2  �  C C  C C  �  z1 = z2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='13)  Activating the off-diagonal term translates using the SNF into  SNF  �  z − z1  1  0  z − z2  �  =  �  1  0  0 (z − z1)(z − z2)  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='14)  valid for z1 = z2 and z1 ̸= z2 alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In particular, putting both branes at the origin,  SNF  �  z 1  0 z  �  =  �  1 0  0 z2  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='15)  where we see the appearance of an infrared trivial complex  R  R  ∼= 0  1  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='16)  and a so-called ’thick brane’  R  R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  z2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='17)  Multivariable polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The ring of polynomials in one variable C[z] has the prop-  erty that every ideal is principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In particular, it is a B´ezout ring, which means by definition  that any ideal generated by finitely many generators is principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The ring C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , xn]  for n > 1 on the other hand is not a B´ezout ring : it has non-principal finitely generated  ideals (for example, the ideal generated by x1 and x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' It turns out the SNF is best defined  in B´ezout rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' By [84, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1] an SNF does not exist for matrices with coefficients  – 9 –  in C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , xn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Thus, at face value it seems not possible to use the SNF to describe  intersecting branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  However, this is not a weakness but a feature, as we now demonstrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Consider the  case of two variables, x and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Physically, this means we are considering branes that  share an R1,5 and wrap complex curves in the (x, z)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Consider a stack of n branes  at x = 0 and a stack of m branes at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is described by the diagonal matrix  diag(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , x, z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We can activate non-diagonal terms, that we call Q and ˜Q as they  correspond to strings that yield hypermultiplets at low energy  T =  �  x1n  �Q  Q  z1m  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='18)  In order to pick a canonical diagonal form for this matrix, we need to make a choice of  main variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Let us pick z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This means we extend the non-B´ezout ring C[x, z] to the ring  C(x)[z], which is B´ezout as it is a polynomial ring in one variable over the field C(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This  is simply telling us that poles in x have to be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We now describe the SNF over this  ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Assume first that the eigenvalues of Q �Q are all distinct, call them λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Then  the SNF is  diag  �  x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , x, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , 1,  m  �  i=1  �  z − λi  x  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='19)  If some of the eigenvalues coincide, the form of the SNF changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We can collect this  information, which is insensitive to the detailed properties of the eigenvalues, by simply  stating that the SNF is  �  x1n  0  0  z1n − Q �  Q  x  �  λi ̸= λj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='20)  Thus, from the point of view of the stack of m branes at z = 0, the presence of the other  stack is felt as a pole for the complex adjoint-valued Higgs field living on the brane at  z = 0, [85–87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Note that the situation is symmetric and one could have chosen the other  stack as the base one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is exactly the same arbitrariness we made when writing the  ring as a B´ezout ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2  Kraft-Procesi Transitions  Nilpotent orbits for sln are in one-to-one correspondence with partitions of n, and are  partially ordered by inclusion of their closure [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The nilpotent orbit associated to a  partition λ of n is denoted Oλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The partial order corresponds to the well-known dominance  ordering for partitions,3 and it can be represented by a Hasse diagram, which indicates  the covering relation associated to this partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The diagram thus obtained also  corresponds to the stratification of the nilpotent cone (the set of all nilpotent matrices)  into symplectic leaves [77, 78, 89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Elementary degenerations between adjacent nilpotent  orbits are called Kraft-Procesi transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In the case of sln nilpotent orbits, these can be  3The dominance ordering is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' If λ and µ are two partitions of n, we say that λ ≥ µ if  and only if for all j,  j�  i=1  λj ≥  j�  i=1  µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 10 –  either closures of minimal slm nilpotent orbits or Kleinian singularities C2/Zm for m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  This can be implemented in brane setups [90, 91], where nilpotent orbit closures are realized  as Higgs or Coulomb branches of 3d N = 4 quiver theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Slodowy Slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Consider the case where T = z · 1n + M and M is a nilpotent matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='4) shows that  δT ∼ δT + (αR − αL)z + (αRM − MαL) ,  δT ∈ Matn(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='21)  Define α+ := 1  2(αR + αL) and α− := 1  2(αR − αL) this gives  δT ∼ δT + 2α−z + [α+, M] + {α−, M} ,  δT ∈ Matn(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='22)  The image of the map  Matn(R) → Matn(R)  α− �→ 2α−z + {α−, M}  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='23)  contains all of Matn(zR),4 so we can use α− to eliminate all z-dependence in δT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We still  have the freedom to use α+, which defines an equivalence relation  δT ∼ δT + [α+, M] ,  δT ∈ Matn(C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='24)  The cokernel of the adjoint action by M has dimension  dλ :=  �  i  (2i − 1)λi ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='25)  where λ is the partition that specifies the nilpotent orbit of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  If one considers only  traceless matrices, δT then depends only on dλ − 1 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Note that being in the cokernel of ad(M) corresponds to commuting with the other  nilpotent element in the sl2-triple generated by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Thus M+δT parameterizes the Slodowy  slice SM transverse to M (see Appendix A for definitions and a proof of this statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Note that indeed that  ∀M ∈ Oλ(sln) ,  dim SM = dλ − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='26)  Kraft-Procesi Transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Consider two partitions λ and µ, which are immediately  adjacent in the partial order – one says that λ covers µ if they are adjacent and λ > µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Then λ and µ differ only in two entries, say with indices i < j, with λi − 1 = µi and  λi+1 + 1 = µi+1, and one of the two following transitions occurs:  Condition  Transition name  Transverse slice  j = i + 1  Aλi−λj−1  C2/Zλi−λj  µi = µj  ai−j−1  Omin(sl(i − j, C))  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='27)  4This is easily proved by a recursive argument on the degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 11 –  The first corresponds to a Kleinian singularity, whereas the second is the closure of a  minimal nilpotent orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The equivalence class of tachyon matrices for a partition µ =  [µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , µr] (with µ1 ≥ · · · ≥ µr) is characterized by a common SNF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  SNF(Tµ) =  �  �  �  �  �  �  �  �  �  �  �  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  1  zµr  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  zµ1  �  �  �  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='28)  Starting from the SNF of partition µ, the Kraft-Procesi transition is realized using  SNF  �  zµj αzµj−1  0  zµi  �  =  �  zµj−1  0  0  zµi+1  �  =  �  zλj  0  0 zλi  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='29)  for α ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Note that this is precisely the tachyon matrix formalism analog of the way Kraft-  Procesi transitions are realized in Hanany-Witten brane systems for 3d N = 4 quiver  theories in [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='15) corresponds to the covering of partition [1, 1] by [2],  whereby two branes are combined into a thick brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' A less trivial case is the covering of  [2, 2] by [3, 1], where we do not simply have two branes being combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Rather, one of  the two thick branes needs to be broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is realized in our framework using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='29) as  follows:  SNF  �  z2 αz  0 z2  �  =  �  z1 0  0 z3  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='30)  for α ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  When the nilpotent orbit Hasse diagram is linear, one can build matrices that encode  all partitions at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is the case for n ≤ 5, where the matrices are given by  �  z a1  0 z  �  ,  �  �  �  z a1 0  0 z a2  0 0  z  �  �  � ,  �  �  �  �  �  z a1  0  0  0 z za3 + a4 0  0 0  z  a2  0 0  0  z  �  �  �  �  � ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='31)  �  �  �  �  �  �  �  z  a1  0  0  0  0  z  za3  0 a2a3a4  0  0  z  a2  0  −z3a6  0  z2a1a3a6  z  −za4  −za2a4a5 (a1a2a3a6 + 1) 0 a2  1a2  2a2  3a4a5a6 0  z  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='32)  This means that the SNF of a matrix above with a1, · · · , ar non-zero gives precisely the  r-th partition of n, the partitions being totally ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 12 –  3  Example: GTPs for Tn and Related Models  In this section, we use an intermediate step in correspondence between M-theory on a  CY threefold and IIB 5-brane-webs: IIA on a resolved C2/Zn singularity with D6-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  This discussion follows the philosophy of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We start in this section with the instructive  example of Tn, and consider its description as well as GTPs obtained by adding white dots  along a single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1  The Setup  Consider the toric local Calabi-Yau defined by the toric diagram with vertices at coordinates  (0, 0), (n, 0) and (n, n), drawn here for n = 5:  W1  W3  W2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1)  The generators of the dual cone are  (−1, 0, 0) ↔ W1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2)  (1, −1, n) ↔ W2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='3)  (0, 1, 0) ↔ W3  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='4)  (0, 0, 1) ↔ Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='5)  The first three generators are vectors normal to the 2-dimensional facets on the fan, drawn  in blue arrows when projected on the CY plane in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' As an algebraic variety, the toric  threefold is simply a hypersurface in C4:  W1W2W3 = Zn  ⊂  C4⟨W1, W2, W3, Z⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='6)  This space has non-isolated singularities, specified by the following intersecting ideals:  Ising = (W1, W2, Z) ∩ (W1, W3, Z) ∩ (W2, W3, Z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='7)  Along each such ideal, there is a family of An−1-singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' These are in one-to-one  correspondence with the three edges of the toric graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The singular threefold admits three different (albeit linearly dependent) C∗-actions  which act on the coordinates with the following weights:  W1 W2 W3 Z  C∗  1  0  1  −1 0  C∗  2  1  0  −1 0  C∗  3  1  −1  0  0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='8)  – 13 –  As explained in toric language in [13], we can define projections πi, for i = 1, 2, 3, with  respect to these actions, and this will bring us down to IIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Let (i, j, k) be a permutation of  (1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The way to reduce along a particular C∗-action C∗  i is to pick the pair of ‘charged’  coordinates Wj and Wk, and setup a C∗-fibration over an new complex coordinate Vjk as  follows:  C∗  i : C[W1, W2, W3, Z] ∼= C[W1, W2, W3, Z, Vjk]  (WjWk − Vjk)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='9)  Now we can rewrite the threefold in the following presentation:  C[W1, W2, W3, Z]  (W1W2W3 − Zn)  ∼=  C[W1, W2, W3, Z, Vjk]  (WjWk − Vjk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' WiVjk − Zn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='10)  The IIA reduction is achieved by reducing over the S1 ⊂ C∗ action in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The non-  compact part R ⊂ C∗ becomes a transverse direction to the D6-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The projection is  simply defined as dropping the pair of coordinates (Wj, Wk), leaving us with a local K3  with an An−1 Klein singularity  C[Wi, Vjk, Z]  (WiVjk − Zn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='11)  To simplify the notations, we switch to the more standard  X := Wi  Y := Vjk  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='12)  so that the An−1 singularity (a local K3) is described by  XY = Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='13)  There are D6-branes on the locus defined by the ideal  ID6 = (Y, Zn) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='14)  which we call the D6 ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' It is, first of all a stack of n non-compact D6-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' However,  since this passes through the singularity, more is at play here, and we need to resolve the  local K3 to refine our understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  In order to describe the resolution of the An−1 orbifold (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='13), we introduce homoge-  neous coordinates (z1, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , en−1, z2) with n − 1 C∗-actions  z1 e1 e2 e3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' en−3 en−2 en−1 z2  C∗  1  1 −2 1  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  0  0  0  0  C∗  2  0  1 −2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  0  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  C∗  n−2 0  0  0  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  1  −2  1  0  C∗  n−1 0  0  0  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  0  1  −2  1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='15)  The coordinates are homogeneous with respect to the n − 1 projective actions, by which  the space is quotiented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Each row gives the list of weights of the coordinate with respect to  each such action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Just as one must excise particular loci when creating standard projective  space, so must one excise a number of loci here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 14 –  · ·  e1 = 0  e2 = 0  en−2 = 0  en−1 = 0  z1 = 0  z2 = 0  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Geometry of the resolved An−1 singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Each curved line is a P1 while the straight  lines are the non compact divisors z1,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Specifically, if a coordinate is set to zero, then only one its two ‘neighbors’ are allowed  to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' See figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' For example, we can have e2 = 0, and e3 = 0, or e2 = 0 and  e1 = 0, but the pair (e1, e3) does not form a valid ideal for a vanishing locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In terms of  the coordinates (X, Y, Z), we have  X =  n  �  i=0  en−i  i  ,  Y =  n  �  i=0  ei  i ,  Z =  n  �  i=0  ei  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='16)  where we have introduced e0 := z1 and en := z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The locus ei = 0 corresponds to the  i-th exceptional P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The loci z1 = 0 and z2 = 0 correspond to noncompact holomorphic  curves intersecting the first and last P1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Line bundles over this space are  characterized by their first Chern class, which is encoded as O(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , kn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' A section of  this bundle is a polynomial of homogeneous multi-degree (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , kn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The brane locus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='14) is now given by  ID6 =  � n  �  i=0  ei  i,  n  �  i=0  en  i  �  =  � n  �  i=0  ei  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='17)  The interpretation is as follows: there are i D6-brane wrapping the i-th P1, and n non-  compact D6-branes on the curve z2 = 0, which intersects the (n−1)-th sphere at one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  At the SCFT point, all the P1’s shrink to zero size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' On the Coulomb branch, where the  K¨ahler volumes are non zero, the effective theory can be read from the ideal (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' It is  described by the quiver:  U(1)  U(2)  · ·  U(n − 2)  U(n − 1)  n  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='18)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2  Tachyon Condensation Picture  The theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='18) is encoded by via the tachyon condensation/coherent sheaf language as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' First recall that in terms of complexes, one can describe the branes as follows:  A brane Bi wrapped on the i-th P1 given by ei = 0 can be described as the cokernel  of the complex of line bundles:  Bi :  O(−ei)  O ,  ei  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='19)  – 15 –  where O is the structure sheaf over the local K3, and O(−ei) is the dual of the line  bundle O(ei), of which ei is a section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' For instance, O(−e1) = O(2, −1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  A noncompact ‘flavor brane’ BF at the locus z2 = 0, intersecting the rightmost P1  (given by en−1 = 0), is given by the following complex:  Bn :  O(−z2)  O .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  z2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='20)  More generally, a noncompact D6-brane intersecting the i-th exceptional P1 will be  given by the zero-locus of a section of O(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , 0), where the ‘1’ is the i-th  entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Define N = 1  2n(n+1) D8-branes with gauge bundle FD8 := O⊕N and N anti-D8-branes  with gauge bundle  FD8 := O(2, −1, 0, · · · , 0) ⊕ O(−1, 2, −1, · · · , 0)⊕2 ⊕ · · · ⊕ O(0, · · · , 0, −1)⊕n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='21)  We then define the tachyon map  T : FD8 → FD8  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='22)  as a diagonal matrix  T = Diag(e1 · 11, e2 · 12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , en−1 · 1n−1, z2 · 1n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='23)  Note that det T = Y , so that the equations of the threefold are  WjWk = det T = Y ,  XY = Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='24)  from which one recover the original equation WiWjWk = Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The resulting D6-brane  system is defined as the cokernel S := cok(T) of this map, which is the locus where T fails  to be invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' So S is reducible:  S =  n  �  i=1  O⊕i  ei ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='25)  where Op means the structure sheaf with support over p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' There is an exact sequence  FD8  FD8  S  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  T  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='26)  Fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The fluctuations around this background are computed as self-extensions,  in the same way as in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This means that the fluctuations δT of the tachyon T  belong to the self-extension group,  δT ∈ Ext1(S, S) = HomD(K3)(S, S[1]) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='27)  where we consider the Homs in the derived category of coherent sheaves on K3 D(K3),  and the [1] means that we shift the complex one step to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The matrix δT can be  decomposed in blocks, and there will be non-zero fluctuations just above and below the  diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  In practice, the fluctuations are subjected to three conditions, that we will be using  repeatedly in the following sections:  – 16 –  (i) The C∗ weights of the columns of T and δT need to be compatible with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='23) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (ii) The fluctuations are regular sections of the relevant sheaves (no pole on the support  locus) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (iii) The components of δT are subject to the identifications (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='3  Example: T2  For concreteness we work out in detail the case n = 2 for Tn before treating the general  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The tachyon matrix is  O(2) ⊕ O(−1)⊕2  O⊕3  T  ,  T =  �  e1  0  0  z2 · 12  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='28)  and we give names to the blocks in the fluctuation δT in correspondence with the quiver  δT =  �  Φ1 �Q1  Q1 Φ2  �  1  2  Q1  �Q1  Φ1  Φ2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='29)  The three conditions listed above give  δT ∈  �  Γ(O(−2)(e1))  Γ(O(1)(e1,z2)) · C1×2  Γ(O(−2)(e1,z2)) · C2×1  Γ(O(1)(z2)) · C2×2  �  =  �  0  C1×2 · z1  C2×1 · 1  z2  1 C2×2 · z1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='30)  Here, Γ indicates that we take sections of the bundles, and Cn×m is the set of n×m matrix  of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The last equality is easily checked, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' for the lower-left entry, the  regular sections are generated by rational functions in z1 alone, having C∗-weight −2, and  poles are allowed as the support is the intersection between two curves where z1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In  terms of Ext groups between B1 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='19)) and B2 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='20)), we can write (using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  a spectral sequence argument)  Q1 ∈ Ext1(B1, B2) = H0(O(e1)e1,z2) =  � 1  z2  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='31)  �Q1 ∈ Ext1(B2, B1) = H0(O(z2)e1,z2) = ⟨z1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='32)  To summarize, the background tachyon plus fluctuation is given by  T + δT =  �  e1  �q1z1  q1  z2  1  z2 · 12 + z1ϕ2  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='33)  where we have pulled out all dependencies in z1, e1 and z2, so that �q1 ∈ C1×2, q1 ∈ C2×1  and ϕ2 ∈ C2×2 are pure constants:  Q1 = q1  z2  1  ,  �Q1 = �q1z1 ,  Φ2 = ϕ2z1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='34)  – 17 –  e1 = 0  z1 = 0  z2 = 0  Φ  ϕ1  Q1, ˜Q1  e1 = 0  z1 = 0  z2 = 0  ϕ2 = − M  X  ϕ1  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Intersecting branes before and after the transformation that maps the off-diagonal Higgs-  field entries Q1, ˜Q1 to diagonal ones with a pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The orange dot signals the pole in the Higgs field  at X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Note that the term q1  z2  1 ensures that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='33) is defined on the locus z1 ̸= 0  We can perform basis changes from the left and right, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='6), as follows:  �  1  0  − q1  z2  1e1  12  � �  e1  �q1z1  q1  z2  1  z2 · 12  � �  1  − �q1z1  e1  0  12  �  =  �  e1  0  0  z2 · 12 −  M  z1e1  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='35)  In the last step we have introduced the meson matrix  M := q1�q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='36)  In the 5d effective field theory, F-term conditions impose that M be nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This can  also be demonstrated mathematically via the so-called cone construction in the derived  category of coherent sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' See [83] for examples of this mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  This diagonalization shows that giving a vev to the meson field can be subsumed into  a shift of vev of the adjoint field φ on the flavor branes, with a pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Using the coordinate  X = z2  1e1 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='16)) on the z2 = 0 plane, we can write this as  T + δT ∼  �  e1  0  0  z2 · 12 + z1 · ϕ2  �  with  ϕ2 = −M  X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='37)  The transformation from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='33) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='37) means that we can regard in an appropriate  regime the intersecting branes in figure 7 as a stack of two branes on z2 = 0 with a  complex codimension-one defect on its world-volume at e1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is in agreement with  the findings of [85, 87], where the authors find that a vev of bifundamental fields at the  intersection of two branes can be subsumed into a pole for the adjoint of one of the two  branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In the picture, we trade the description in figure 7 on the left with the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Up to a change of basis we can take M in canonical Jordan form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  There are two  possibilities, corresponding to the two partitions of n = 2:  M[12] =  �  0 0  0 0  �  and  M[2] =  �  0 1  0 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='38)  – 18 –  Consider the latter case,  T[2] :=  �  e1  t[2]  �  :=  �  �  �  e1 0  0  0 z2 −  1  z1e1  0 0  z2  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='39)  We still have det T[2] = ez2  2 = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' So the brane configuration has not changed geometrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Accordingly, the M-theory uplift is still given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' However, the tachyon matrix shows  us that the flavor brane no longer carries an SU(2) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is the hallmark of a T-  brane: A non-abelian bound state of branes that does not realize the gauge group that it  would naively have given its geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This T-brane effect is the IIA counterpart of the  change in boundary conditions in the dual type IIB brane-webs  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='40)  In this particular case, once the D5-segment has been sent away, a Hanany-Witten move  where the 7-brane detaches completely becomes possible:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='41)  How does this translate into the IIA language, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' in terms of the tachyon field?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Let  us see what deformations are available, starting from the new vacuum defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Using the results of section 2, the fluctuations of t[2] are  δt[2] = z1 ·  �  0 0  α 0  �  ,  α ∈ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='42)  Now in order to see how this affects the geometry, we add this perturbation to T[2]  T[2] + δt[2] =  �  �  �  e  0  0  0 z2 − 1  z1e  0 αz1  z2  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='43)  Now the geometry (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='24) is deformed to  WjWk = det T = Y + α ,  XY = Z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='44)  – 19 –  which reduces to the hypersurface  W1W2W3 = Z2 + αWi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='45)  So the CY threefold is fully desingularized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' From the IIA perspective, we see that two flavor  branes have recombined with one gauge brane, to give rise to a noncompact brane that can  escape the singular locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is in full agreement with the 5-brane-web picture, whereby  the 7-brane moves off to the left, becomes fully detached from the NS5-branes, and can  escape to infinity, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This corresponds precisely to removing the A1 singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  To summarize, the white dot is represented in the IIA picture as a nilpotent vev with  poles on the flavor D6-stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The further Hanany-Witten move that actually deforms the  M-theory geometry is implemented by switching on a further vev on the flavor stack along  the Slodowy slice with respect to the initial singular nilpotent vev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='4  General Case: Tn  The general Tn case is very similar to the T2 example treated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The fluctuations  around the tachyon background are denoted by fields as follows:  1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  n − 2  n − 1  n  Q1  Qn−2  Qn−1  �Q1  �Qn−2  �Qn−1  Φ1  Φ2  Φn−2  Φn−1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='46)  Generalizing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='31) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='32), we find that the 5d hypermultiplets that reside at the  intersection of the curves ei = 0 and ei+1 = 0 are described by  Qi ∈ Ext1(Bi, Bi+1) = H0(O(ei)(ei,ei+1)) =  � Y  Zi+1 ei  �  =  ��  j̸=i  ej−i−1  j  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='47)  and  ˜  Qi ∈ Ext1(Bi+1, Bi) = H0(O(ei+1)(ei,ei+1)) =  �Zi  Y ei+1  �  =  � �  j̸=i+1  ei−j  j  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='48)  On the other hand, one can check that Ext1(Bi, Bj) = 0 for |i − j| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Therefore the  tachyon matrix T, plus the fluctuations δT, fit schematically (we will provide the explicit  form below) into the matrix  T + δT =  �  �  �  �  �  �  �  �  �  e1 · 11  �Q1  Q1  e2 · 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  en−1 · 1n−1  �Qn−1  Qn−1  z2 · 1n  �  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='49)  As in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='34), we introduce the notation  Qi = qi ·  Y  Zi+1 ei  ˜Qi = ˜qi · Zi  Y ei+1 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='50)  – 20 –  such that qi and ˜qi are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We can also have fluctuations on the diagonal, which we  write as  Φi = ϕi ·  Y  Zi+1 ei .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='51)  With this notation the F-terms at the ith node can be written  �q1q1 = 0,  and  qi�qi = �qi+1qi+1  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , n − 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='52)  We now come back to the reason why (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='49) is only a schematic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The i-the  hyper (Qi, �Qi) is only well defined at the (ei, ei+1) intersection, but has poles in the nearby  patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='49) is not well-defined over the whole target space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is due to the  projective nature of the resolved K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Therefore, we must study it patch by patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' A good  local affine coordinate on the locus ei = 0 for the hemisphere where ei+1 (respectively ei−1)  does not vanish is Y  Zi (respectively Zi  Y ):  ei = 0  ei+1 = 0  Coordinate Y  Zi  Coordinate Zi+1  Y  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='53)  Note in particular that for i = N (respectively i = 0), this is compatible with the affine  coordinate on the locus z2 = 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' z1 = 0) being simply X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Y ), as chosen in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  In the patch that contains the intersection {ei = 0} ∩ {ei+1 = 0}, the tachyon fluctua-  tion is expressed as  δT =  �  Φi  �Qi  Qi Φi+1  �  ∈  �  Ci×i ·  Y  Zi+1 ei  Ci×(i+1) · Zi  Y ei+1  C(i+1)×i ·  Y  Zi+1 ei C(i+1)×(i+1) · Zi  Y ei+1  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='54)  Using line and row transformations, one finds  �  1i  �qi Zi  Y  −qi  Y  Zi+1 1i+1 − qi�qi  Z  � �  ei  �  1i − �qiqi  Z  �  0  0  ei+1 · 1i+1  � �  1i  −�qi  Ziei+1  Y ei  qi  Y ei  Zi+1ei+1 1i+1 − qi�qi  Z  �  =  �  ei · 1i  0  0  ei+1  �  1i+1 − qi�qi  Z  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='55)  Effectively, this transforms a pole of the form  �  ei · 1i +  Y  Zi+1 eiϕi  0  0  ei+1 · 1i+1  �  ,  ϕi = −�qiqi  (Y/Zi)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='56)  into a pole of the form  �  ei · 1i  0  0  ei+11i+1 + Zi  Y ei+1ϕi+1  �  ,  ϕi+1 =  −qi�qi  (Zi+1/Y ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='57)  – 21 –  Let us introduce the n × n meson matrix  M = qn−1�qn−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='58)  As argued for the T2 previously, this meson matrix must be nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We can characterize  the tachyon fluctuation entirely by the last pole (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='57) for i = n − 1, which gives  ϕn = −M  X ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='59)  consistently with what we found for the n = 2 case in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In summary, the bifun-  damental matter between the various branes can be subsumed under a shift of the 7d  SU(n)-adjoint Higgs ϕn, as a simple pole with residue equal to a nilpotent element M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Since Mn = 0, we still have  det T =  n  �  i=1  ei  i = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='60)  Hence, the brane locus remains unscathed despite the activation of the matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='5  Interpretation in terms of Generalized Toric Polygons  We saw in the last subsection that the geometry of the threefold can be affected by the  presence of a pole of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  This pole can be encoded in two ways: in the  nilpotent meson matrix M, or in the list of hypermultiplet deformations qi and �qi satisfying  the relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Let us call Q the set  Q = {q = (q1, �q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , qn−1, �qn−1) ∈ C2×1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' C(n−1)×n|(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='52)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='61)  We also call N the set of nilpotent n × n complex matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The map  m : Q → N  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='62)  q �→ M = qn−1�qn−1  is certainly not injective, as given a nilpotent matrix M, there are infinitely many families  {qi, �qi}i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=',n−1 mapping to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Let us define  r : Q → Nn−1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='63)  q �→ (ri = rank(qi�qi))i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=',n−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  For a given M ∈ N, the ranks of the bilinears in qi�qi that map to M are not fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  In other words, r(m−1(M)) contains more than one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' However, there is a unique  element of rmin ∈ r(m−1(M)) that minimizes the sum of the entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' If M ∈ Oλ, then r0  is given by the partial sums of the transpose of λ,  rmin =  �  �  n  �  j=n+1−i  λT  j  �  �  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=',n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='64)  – 22 –  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' On top, we depict a part of the resolved GTP for the T9 theory, with white dots on  the right edge, along with an internal triangulation consistent with the white dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The presence of  white dots propagates into the interior of the GTP, thereby limiting the possible resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Each  column in the GTP corresponds to a boundary condition for D5-branes ending on D7-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This  information is encoded in the ranks ri of the matrices appearing in δT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' On the bottom, we show  the associated D5-brane boundary conditions (where each circle denotes a D7-brane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Note that this coincides with the ranks of the linear quiver  rmin  1  rmin  2  · ·  rmin  n−1  n  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='65)  whose 3d N = 4 Coulomb branch is the corresponding nilpotent orbit closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The ranks  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='64) represent the minimal deformations needed in δT to produce the pole (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In  simple cases, the quivers (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='65) appear as embedded in the magnetic quiver describing the  Higgs branch of the 5d theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' For a more precise statement about the embedding, the  reader is referred to [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  In the GTP, these ranks can be encoded with white dots inside the polygon, using  again the notation introduced in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' For instance, for n = 9 and partition λ = [4, 3, 2], we  can draw the configuration shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The minimal ranks correspond to the number  of white dots in each column: here we get rmin = (0, 0, 0, 0, 0, 1, 3, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' These numbers give  the brane configuration which saturates the s-rule, as illustrated in the lower part of figure  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 23 –  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  H  H′  Pole  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Schematic form of a Hasse diagram of the Higgs branch H viewed as a symplectic  singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The effect of the pole prescription corresponds geometrically to imposing a higher  dimensional base leaf (the transverse slice is shown in red – here it is minimal, but it does not have  to be in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The remaining Higgs branch H′ is the transverse slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Impact on the Hasse diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The vevs of Higgs branch operators in the 5d theory  can be projected on the space of complex structure deformations of the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The  fact that the pole (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='59) freezes some of these deformations means geometrically that the  resulting pole-deformed Higgs branch is the slice in the initial Higgs branch transverse to  these imposed deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This can be represented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The Higgs branch H with no pole is a symplectic singularity, which can be depicted  using its Hasse diagram of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The effect of the pole is to freeze certain defor-  mations, represented here as a forced choice of a higher dimensional bottom symplectic  leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The resulting Higgs branch H′ is the transverse slice to that leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' See figure 9 for a  schematic depiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The analysis above applies to the theory in any phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In the case of the IIA reduction  on the resolved An−1 singularity shown in figure 6, the Higgs branch of the Tn theory  becomes the nilpotent cone of sln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The transverse slices are then identified with the Slodowy  slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The example n = 4 is illustrated in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' If M ∈ Oλ for λ some partition of n,  then the corresponding Higgs branch is the Slodowy slice Sλ ∩ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Moving on to the SCFT  phase, the Higgs branch is no longer a nilpotent orbit closure, but the general picture stays  the same: the Higgs branch is restricted to a transverse slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is what we already  mentioned in the introduction, see figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The Hasse diagrams drawn in figure 5 can be  reproduced independently using the quiver subtraction algorithm [92, 93] on the magnetic  quivers extracted from the GTPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='6  Hanany-Witten Moves  In the previous section, we defined the IIA counterpart of a ‘white dot’ as a nilpotent  residue for the adjoint complex scalar on the flavor D6-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The nilpotency implies  – 24 –  0  3  4  5  6  a3  a1  A1  A3  M ∈ O[14]  M ∈ O[2,12]  M ∈ O[22]  M ∈ O[3,1]  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Diagram for the nilpotent cone of sl4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The dots are nilpotent orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' When M belongs  to a given orbit, the Higgs branch of the resulting theory is the transverse Slodowy slice, represented  by a bracket on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  that there will be no repercussions on the geometry of the branes, and hence, the M-theory  CY will not be deformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is the hallmark of a ‘T-brane’ [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  We would now like to determine how the T-brane configuration impacts the physics  of the 5d theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is done using the brane-web language, with Hanany-Witten moves,  as we have explained in detail in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The key point is that the nilpotent orbit of  M determines which Hanany-Witten move can be performed in order to detach any of the  7-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  In the IIA setup, flavor D6-branes are now allowed to move across exceptional P1s,  thereby changing the quiver structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The way this is seen at the level of the Higgs field,  is that by activating vevs along the Slodowy (transverse) slice to the nilpotent vev with  poles, the characteristic polynomial of the tachyon matrix actually becomes deformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  This can be seen by computing the self-Ext group Ext1 �  B(nil)  F  , B(nil)  F  �  of the nilpotent  configuration on the flavor brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Using the computation from section 2, we are looking  for δTF such that  δTF ∼ δTF + z2  Z [M, g] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='66)  This is equivalent to requiring that δTF be on a transverse slice to M, gauge equivalent  to the Slodowy slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We will choose a gauge such that δTF be the companion matrix to  M as in [94], which is referred to as a ‘reconstructible Higgs’ in [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The details on how  this is done are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Say Tnil is in the maximal nilpotent orbit, then the  ‘Hanany-Witten’ tachyon will take the form  THW = Tnil + δTF = z2  Z  �  �  �  �  �  �  �  �  Z  1  Z  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Z  1  (−)n−1anX (−)n−2an−1X  −a2X Z  �  �  �  �  �  �  �  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='67)  – 25 –  where the ai are constants (the fluctuation δTF is proportional to X as a consequence of  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='54) with i = N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This matrix has determinant  det(THW) =  �z2  Z  �n �  Zn + a2XZn−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' + anX  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='68)  More generally, for M in the [λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , λr] partition, we take a block diagonal matrix  where each block will take the form:  (THW)i = z2  Z  �  �  �  �  �  �  �  �  Z  1  Z  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Z  1  (−)λi−1a(i)  λi X (−)λi−2a(i)  λi−1X  −a(i)  2 X Z  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='69)  Note that by doing so, we are not using the full Slodowy slice Sλ (which contains non-  block-diagonal matrices) but instead restrict to the intersection S0  λ = Sλ ∩ lλ with the Levi  subalgebra lλ, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is physically justified, as it guarantees that the flavor  symmetry will not be further broken by the Hanany-Witten moves than it already has been  by the white dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The missing parameters, in Sλ − S0  λ, are associated to the non splitting  of the flavor branes, illustrated on an example below in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Putting together all these blocks, and the non-flavor part of the tachyon matrix, the  end result of the full deformation is  Y �→ 1  X ·  r  �  i=1  �  �Zλi + X  �  �  λi−2  �  j=0  a(i)  λi−jZj  �  �  �  � ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='70)  where �  i λi = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Actually it can be argued that only the coefficients a(i)  λi affect the physics near the  singularity: coefficient a(i)  λi−j for j ̸= 0 correspond to a shift of  Zλi−j  X  , which is using  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='53) the coordinate along a P1 with which the brane has zero intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Therefore the  equation simplifies to  Y �→ 1  X ·  r  �  i=1  �  Zλi + Xa(i)�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='71)  where we have renamed a(i) := a(i)  λi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Reverting to the original notations (W1, W2, W3, Z)  for the coordinates in C4, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='12), one finally gets  W1W2W3 =  r  �  i=1  �  Zλi + a(i)W1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='72)  Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Let us work out a few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='70) is worked out explicitly  for T3 and T4 in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The factorization allows to read off the corresponding quivers,  which are the magnetic quivers for nilpotent orbits of sl(3) and sl(4) [95], as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 26 –  Partition  det T  Factorization  [13]  Y  e1e2  2(z3  2)  [2, 1]  Y + a2Z  e1e2(z2)(e2z2  2 + a2z1)  [3]  Y + a2Z + a3  smooth  Partition  det T  Factorization  [14]  Y  e1e2  2e3  3(z4  2)  [2, 12]  Y + a2Z2  e1e2  2e2  3(e3z2  2 + a2z2  1e1)(z2  2)  [22]  Y + (a(1)  2  + a(2)  2 )Z2 + a(1)  2 a(2)  2 X  e1e2  2e3(e3z2  2 + a(1)  2 z2  1e1)(e3z2  2 + a(2)  2 z2  1e1)  [3, 1]  Y + a2Z2 + a3Z  e1e2e3(e2e2  3z3  2 + a2z2  1e1e2e3z2 + a3z1)(z2)  [4]  Y + a2Z2 + a3Z + a4  smooth  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Equations defining the brane configurations in T3 and T4 with white dots on one edge  after HW moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In the last column, the equation is rewritten in terms of the toric variables for the  resolution, and maximally factorized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The factor in orange give the ranks of the low energy quiver  while the terms within brackets give the flavor ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  We can illustrate the problem discussed in the previous paragraph with partition [2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The generic matrix in the Slodowy slice would give rise to the final block for the tachyon  matrix given by  �  �  �  Z  1  0  −a2X Z  αX  βX  0 Z + γX  �  �  � ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='73)  The resulting, deformed, equation then reads  det T = e1e2  �  a2γe1z3  1 + a2z1z2 + γe1e2z2  1z2  2 + αβe1z3  1 + e2z3  2  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='74)  from which one reads off a quiver with two abelian nodes and a single flavor brane system:  e1 = 0  e2 = 0  z1 = 0  z2 = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='75)  This system, drawn in teal, intersects both P1 divisors at e1 = 0 and e2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This breaks  the global isometry u(1) of the nilpotent Slodowy slice S[2,1] ∩ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' By restricting to the  Levi subalgebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' sending α, β and γ to zero, one gets one more factorization:  det T = e1e2(z2)  �  a2z1 + e2z2  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='76)  – 27 –  e1 = 0  e2 = 0  z1 = 0  z2 = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='77)  The flavor branes are now disjoint and can be moved independently:  e1 = 0  e2 = 0  z1 = 0  z2 = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='78)  Each flavor brane intersects a single P1, and the global symmetry of the resulting Higgs  branch (still in the resolved phase) is read to be s(u(1) ⊕ u(1)) = u(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  4  General Discussion  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1  White Dots along a Single Edge  Having discussed the case of Tn in detail, we move to a generic toric polygon and its GTP  deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Let P be a convex polygon in R2 with vertices in Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Pick an edge on P,  and denote by n its length (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' the edge contains exactly n + 1 points in Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Using an  SL(2, Z) transformation, one can assume without loss of generality that this edge extends  between the vertices (0, 0) and (0, n), and that all vertices of P other than (0, 0) and (0, n)  have coordinates (i, j) with i < 0:  · ·  · ·  (0, 0)  (0, n)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1)  In the toric threefold, the length n edge is translated into the presence of a dimension 1  singular stratum with transverse slice of type An−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In the brane-web picture, this means  that n D5-branes extend to infinity, and the boundary condition, encoded in how they end  on D7-branes, can again be encoded in a T-brane datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This can be done explicitly using  the IIA reduction as in section 3 for any given polygon (and we do so for the example of a  – 28 –  rectangular P in the next section), even though it would be extremely cumbersome to try  to give general formulas that would apply to any polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The general idea, however, is  clear: in the Tn example discussed in section 3, the undeformed equation W1W2W3 = Zn  yields the An−1 singularity transverse to the line W2 = W3 = Z = 0 by setting W1 equal  to a constant, say W1 = c, giving  cW2W3 = Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2)  It is then this equation that is modified by (i) adding a nilpotent pole of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='59)  with M in the prescribed nilpotent orbit, and (ii) performing HW moves to deform the  geometry, in effect replacing  Zn →  �  i  (Zλi + a(i)W1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='3)  The general case is obtained by mimicking this procedure: among the equations for the  toric threefold corresponding to the polygon P, the line with transverse singularity An−1  can be parametrized by a coordinate W1, and the transverse slice is again given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  This is one of the equations defining the toric threefold, called a ”boundary equation” in  [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This equation should then be replaced by  cW2W3 =  �  i  (Zλi + a(i)W1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='4)  This is in agreement with [96, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' A useful mnemonic is that in the toric diagram  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1), the deformation of the edge labelled by W1 involves an An−1 equation involving the  coordinates labelling the adjacent edges, here W2 and W3, with the degree n polynomial in  Z deformed by powers of W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In the following subsections, we give examples to illustrate  the scope of our results, and also show certain limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Successive deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Before discussing the examples, we want to address a  natural question: given a GTP with white dots on an edge of length n characterized by  a partition µ, is it possible to deform it to the same GTP with partition λ instead?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' One  necessary condition is that λ > µ, and it is also sufficient if the polygon is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='5  In this case, one can deform the nilpotent pole according to the general rules spelled out  in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  For instance, if n = 4, we can use the explicit form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Going from partition [2, 12]  to [22] is straightforward as it just involves merging two Jordan blocks, while going from  5If the GTP is small, then certain deformations could lead to violations of the S-rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 29 –  [22] to [3, 1] requires using higher powers of z (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='29)):  �  �  �  �  �  z 1 0 0  0 z 0 0  0 0 z 0  0 0 0 z  �  �  �  �  �  �  �  �  �  �  z 1 0 0  0 z 0 0  0 0 z α  0 0 0 z  �  �  �  �  �  �  �  �  �  �  z 1 0 0  0 z zβ 0  0 0 z α  0 0 0 z  �  �  �  �  �  α ̸= 0  β ̸= 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='5)  However it is worth mentioning that this can be done on the T-brane before HW defor-  mations are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Crucially, the two operations do not commute in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Here for  instance the deformed equations for T4 with partition [2, 2] and [3, 1], given in table 1,  cannot be deformed from one to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2  Rectangular Box  Consider a rectangular box of size n × m,  m  n  W1  W3  W2  W4  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='6)  The 5d SCFT this describes admits several well-known low energy gauge theory deforma-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The toric threefold singularity is not a hypersurface, but it is described by the pair  of equations  W1W3 = Zm ,  W2W4 = Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='7)  We can add white dots on the right segment, labeled by W1, exactly as in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The  deformed equations are  W1W3 = Zm ,  W2W4 =  �  i  (Zλi + a(i)W1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='8)  Note that again, the coordinate W1 is used to deform the An−1 equation involving the  coordinates labeling the adjacent edges W2 and W4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 30 –  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The following example is a useful check of this proposal, as the resulting  model is related to T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Consider the case m = 3, n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Placing a white dot on the length  2 edge yields the equations  W1W3 = Z3 ,  W2W4 = Z3 + aW1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='9)  We can now use the second equation to solve for W1, and we get the hypersurface  W2W3W4 = Z2(aZ + W3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='10)  In the limit where a → ∞, the D7-brane, in the web interpretation, has crossed the whole  brane system from right to left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We are left with  W2W3W4 = Z3 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='11)  and we have reproduced the prediction (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='5)  ↔  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='12)  This is a consistency check on the validity of our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='3  Generic Triangle  In this subsection we consider more examples which involve threefolds what are not hyper-  surfaces in C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Consider the case where the toric polygon is a triangle, of arbitrary shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Pick one edge of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Then an SL(2, Z) transformation can bring the triangle to a  frame where its three vertices are:  {(0, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' (0, n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' (−a, b)} ,  a ∈ Z>0, b ∈ Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='13)  This means the toric fan is generated by the three vectors (0, 0, 1), (0, n, 1) and (−a, b, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Among the generators of the dual cone we have  (−1, 0, 0) ↔ W1  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='14)  �n − b  d2  , − a  d2  , an  d2  �  ↔ W2  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='15)  � b  d1  , a  d1  , 0  �  ↔ W3  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='16)  (0, 0, 1) ↔ Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='17)  In order to write the equation of the threefold, we have introduced d1 = gcd(a, b), d2 =  gcd(a, n − b) and d3 = gcd(a, b, n − b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' One of the equations for the toric threefold is  W n/d3  1  W d2/d3  2  W d1/d3  3  = Zan/d3 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='18)  but in general there are other equations, as there are other generators in the dual cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In  the case of Tn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1), equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='18) is sufficient and reduces to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='6), but this is a special  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' To illustrate this phenomenon, we consider an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 31 –  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  We take the case a = 2 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  2  2n  W1  W2  W3  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='19)  The dual cone is generated by 5 vectors:6  (−1, 0, 0) ↔ W1  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='20)  (n, −1, 2n) ↔ W2  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='21)  (0, 1, 0) ↔ W3  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='22)  (0, 0, 1) ↔ Z  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='23)  (1, 0, 2) ↔ T  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='24)  and the threefold is described by two equations in C5:  W n  1 W2W3 = Z2n ,  W1T = Z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='25)  The first equation corresponds to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The singular locus associated to the length 2n  edge is Z = W2 = W3 = T = 0, parametrized by W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Setting as before W1 = c, we  can eliminate T and we find as expected an equation cnW2W3 = Z2n that we can deform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Therefore, the system of equations for the triangle (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='19) with one white dot on the long  edge according to our conjecture is  ↔  �  W n  1 W2W3 = Z2n−2(Z2 + aW1)  W1T = Z2  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='26)  6There is an algorithmic way to see that the fifth vector (1, 0, 2) is needed, an no other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is the  computation of the Hilbert basis of the semi-group σ∨ ∩M, using standard notation in toric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This  Hilbert basis is unique, and in the present case it has 5 elements, given here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 32 –  5  Testing the Proposal: Resolution of Singularities  To test the proposal for the deformation of singularities, we compute the resolutions for  the deformed singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' For the starting point, which we take to be a toric variety, the  resolution is easily obtained in a combinatorial fashion, by a complete triangulation of the  toric polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' However, no such computational simplification exists yet for the resolution  of GTPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In view of this, we therefore need to revert to resolving the actual algebraic  varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We will carry this out for the Tn theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1  Crepant Resolution of Tn  The simplest type of singularities are the hypersurfaces that realize the Tn theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The  flavor symmetry algebra is su(n)3 except for n = 3, where we expect e6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This can be  easily detected from the toric geometry by extracting the set of curves that are complete  intersections between compact and non-compact (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' flavor) divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The resulting so-  called combined fiber diagram (CFD) [26–28] is easily read off from the toric polygon [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Here we will take the more laborious path of resolving the hypersurface singularity, in  preparation for resolving the deformed singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The hypersurface in C4 is  W1W2W3 = Zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1)  The first resolution for all of these singularities is simply to remove the locus W1 =  W2 = W3 = Z = 0, which is achieved by inserting a P3 with projective coordinates  [W1, W2, W3, Z] (for a detailed exposition of resolutions from a physicist’s perspective, see  [97]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Denoting the exceptional section of the blowup by δ1 the equation, after proper  transform, which ensures that the resolution is crepant, becomes  W1W2W3 = Znδn−3  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2)  This can be further resolved by consecutively blowing up the loci W1 = W2 = W3 = δi = 0  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' by inserting another projective space with coordinates  [W1, W2, W3, δi] ,  i = 1, · · · , ⌊n − 3i⌋ ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='3)  which implies that the coordinates cannot vanish at the same time (and thus the above is  a relation in the Stanley-Reissner ideal of the hypersurface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is iterated until we reach  W1W2W3 = Zn  ⌊n−3i⌋  �  i=1  δn−3i  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='4)  The remaining blowups will resolve the local singularities along W1 = 0, W2 = 0 and  W3 = 0 respectively, by small resolutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  [Wi, Z] ,  or  [Wi, δi] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='5)  Let us denote the compact divisors by Si and the non-compact divisors by Da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We then  find from these resolutions the intersection matrix  Ga =  �  i  Si · Da · Da  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='6)  – 33 –  T3  T3 with deformation [1, 2]  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  CFDs: The intersection graph for T3 (left hand side) and the deformation of T3  described by the partition [1, 2] (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The nodes are curves that are complete intersections  between � Si (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' the sum of all compact divisors) and the non-compact divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The green nodes  are −2 self-intersection curves, which correspond to roots of the flavor symmetry algebra, white are  −1 curves, which can be thought of as bifundamental matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' On the left we see the su(3)3 flavor  symmetry and the matter in the (3, 3, 1) etc, which enhances to e6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' On the right the theory is rank  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  to be precisely the adjacency matrix of the CFD for Tn: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' three su(n) Cartan matrices,  pairwise connected by −1 curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We have shown examples in figures 11 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2  Resolution of Deformations of Tn  Deformations of T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  By including the deformations, some of the above resolutions get  obstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Lets consider the simplest case of T3, which itself is the rank 1 Seiberg theory  with flavor symmetry e6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' After a single blowup [W1, W2, W3, Z] we get one compact divisor  δ1 = 0, and three A2 singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Resolving yields the CFD shown in figure 11 on the left  hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The representations are (3, 3, 1), and cyclic permutations, and thus we get the  known enhancement to e6 7,  (3, 3, 1) ⊕ (1, 3, 3) ⊕ (3, 1, 3) −→  27 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='7)  Adding the deformation results in W1W2W3 = Z(Z2 + W1), which does not allow for a big  resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' There are small resolutions, along e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' W1 = Z = 0, indicating that this is a  rank 0 theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Deformations of T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  More interestingly we can start with T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' All deformations involving  a single edge and yielding a positive rank theory are shown in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The deformation  that is associated to the partition [2, 12] along one edge is the hypersurface  W1W2W3 = Z2(Z2 + W1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='8)  7From the geometry we are also able to extract the flavor symmetry group [42, 98], which however will  not play a role in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 34 –  Partitions  Rank  Free  Symmetry  Equation  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  [14]  3  0  su(4)3  W1W2W3 = Z4  12  [2, 12]  2  0  su(8) ⊕ su(2)  W1W2W3 = Z2(Z2 + W1)  12  [22]  1  0  e7  W1W2W3 = (Z2 + aW1)(Z2 + bW1)  12  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The partition that defines the distribution of white dots in the GTP, the rank of the 5d  SCFT, the flavor symmetry algebra, as well as hypersurface equation for the deformations of T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The first resolution (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='3) results in  W1W2W3 = δ2Z4 + Z2W1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='9)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='Continuing the blowup results in the intersection matrix between the sum of compact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='divisors �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='i Si and each of the non-compact ones  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
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+page_content='Da · Db =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
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+page_content='ab  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='10)  This is shown in figure 12, from which the manifest flavor symmetry algebra is read off  from the collection of −2 curves  su(4)2 ⊕ su(2) ⊕ u(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='11)  The additional u(1) follows from the general rule on flavor symmetry algebras from CFDs  as laid out in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The matter is in the representations, which combine naturally into the  representations of the enhanced symmetry algebra as follows  (4, 4, 1) ⊕ (6, 1, 1) ⊕ (1, 6, 1) ⊕ (4, 1, 2) ⊕ (1, 4, 2)  −→  (28, 1) ⊕ (8, 2) ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='12)  where we identify the enhanced flavor symmetry algebra as  gUV = su(8) ⊕ su(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='13)  Similarly for the partition [22], the manifest symmetry is su(4)2, while the actual  symmetry is e7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In the resolution, see figure 12 on the right hand side, where we see the  representation  (4, 4) ⊕ (4, 4) ⊕ 2 × (6, 1) ⊕ 2 × (1, 6)  −→  56 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='14)  These indeed arise from the branching of the 56 of e7 to su(4)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 35 –  T4  T4 with deformation [12, 2]  T4 with deformation [22]  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  CFDs: The intersection graph for T4 (left hand side) and the deformation of T4  described by the partition [122] (middle) and deformation of T4 with partition [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The nodes are  curves that are complete intersections between � Si (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' the sum of all compact divisors) and the  non-compact divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The green nodes are −2 self-intersection curves, which correspond to roots of  the flavor symmetry algebra, white are −1 curves, which can be thought of as bifundamental matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  On the left we see the su(4)3 flavor symmetry and the matter in the (4, 4, 1) etc representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  In the middle, we see the manifest flavor symmetry su(4)2 ⊕ su(2) ⊕ u(1), but the matter shown  results in the enhancement to su(8) ⊕ u(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' On the right the flavor symmetry enhances to e7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Partitions  Rank  Symmetry  Equation  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  [15]  6  su(5)3  W1W2W3 = Z5  13  [2, 13]  5  su(5)2 ⊕ su(3) ⊕ u(1)  W1W2W3 = Z3(Z2 + W1)  13  [22, 1]  4  su(5)2 ⊕ su(2) ⊕ u(1)  W1W2W3 = Z(Z2 + W1)2  15  [3, 12]  3  su(10) ⊕ su(2)  W1W2W3 = Z2(Z3 + W1)  13  [3, 2]  2  su(10)  W1W2W3 = (Z2 + W1)(Z3 + W1)  14  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Deformations of the T5 model: the partition that defines the GTP, the rank of the  expected 5d SCFT, the flavor symmetry algebra, and the deformed hypersurface equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Deformations of T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Finally consider the T5 model with deformations added along a  single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' These are summarized in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  First we recompute the resolution of the undeformed T5 model, which is given in figure  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The flavor symmetry algebra and the representation of the hypers is  gUV = su(5)3 :  (5, 5, 1) ⊕ (1, 5, 5) ⊕ (5, 1, 5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='15)  The CFD is shown in figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Adding a single white dot results in a theory with flavor  algebra su(5)2 ⊕ su(3) ⊕ u(1), with the CFD shown in the middle in figure 13, which has  bifundamental matter consistent with these flavor symmetry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Adding two white dots along a single edge in the partition [3, 12] we obtain the right  hand figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The representations under su(5)2 ⊕ su(2) are  (5, 5, 1) ⊕ (1, 5, 2) ⊕ (5, 1, 2) ⊕ (10, 1, 1) ⊕ (1, 10, 1)  −→  (45, 1) ⊕ (10, 2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='16)  – 36 –  T5  T5 with deformation [13, 2]  T5 with deformation [12, 3]  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' CFDs: The intersection graph of −2 (green) and (−1) (white) curves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' the CFD,  for T5 (LHS), the partition [2, 13] in the middle, and [3, 12] on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  which is consistent with the flavor symmetry enhancement to  gUV = su(10) ⊕ su(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='17)  Similarly for [3, 2] we computed the resolution and find the CFD shown in figure 14, which  shows the representations under su(5)2 to be  (5, 5) ⊕ (10, 1) ⊕ (1, 10)  −→  45 ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='18)  which is consistent with the enhancement to su(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Finally consider figure 15, where we  manifestly see the su(5) but the su(2) is decomposed into u(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  For partition [22, 1], although the su(2) factor of the global symmetry is not directly  visible in the resolution, an educated guess allows to read the following representations of  su(5) ⊕ su(5) ⊕ su(2) on figure 15:  (5, 1, 1) ⊕ (1, 5, 1) ⊕ (10, 1, 2) ⊕ (5, 10, 2) ⊕ (5, 5, 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='19)  Again, the flavor symmetry algebra is consistent with the one predicted from GTPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Acknowledgments  We would like to thank Cyril Closset for collaboration in early stages of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' AB and  SSN thank Hendrik S¨uß for discussions of related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This work, in particular AC  and SSN, was supported and enabled by the Fondation Wiener-Anspach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This research  is further supported by IISN-Belgium (convention 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='4503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  AB is supported by the  ERC Consolidator Grant 772408-Stringlandscape, and by the LabEx ENS-ICFP: ANR-  10-LABX-0010/ANR-10-IDEX-0001-02 PSL*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' SSN is supported in part by the “Simons  Collaboration on Special Holonomy in Geometry, Analysis and Physics” and the EPSRC  Open Fellowship EP/X01276X/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 37 –  T5 with deformation [2,3]  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' CFDs: The intersection graph for T5 with the deformation labeled by the partition  [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We see the su(5)2 ⊕ u(1) and the matter in the (5, 5) ⊕ (10, 1) ⊕ (1, 10) which enhance into  45 of su(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  T5 with deformation [1, 22]  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' CFDs: The intersection graph for T5 with the deformation labeled by the partition  [1, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  A  Basic algebraic notions for sln  This appendix gathers a few elementary algebraic concepts needed in the bulk of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  In the following, we take g = sln(C), which we represent as n × n complex matrices with  vanishing trace, and we pick the diagonal matrices for the Cartan subalgebra h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Nilpotent Orbits and Triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Let e be a nilpotent element of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' By the Jacobson-  Morozov theorem, one can construct an sl2 triple (e, f, h), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' a triple of elements of g with  h ∈ h satisfying the commutation relations  [e, f] = h ,  [h, e] = 2e ,  [h, f] = −2f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='1)  More precisely, there exists an embedding  ρ : sl2 −→ g ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='2)  – 38 –  such that ρ(e) defines a nilpotent element of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In our case, nilpotent orbits are charac-  terized by partitions of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' For the partition λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , λr), there is a canonical triple,  where e is in the Jordan form specified by λ, h is diagonal and f is lower diagonal, defined  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Consider first the maximal orbit, λ = (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In this case we define the canonical  triple e = Jn, h = Hn and f = ˜Jn where  (Jn)i,j = δi+1,j ,  (Hn)i,j = δi,j(n + 1 − 2i) ,  ( ˜Jn)i,j = δi−1,jj(n − j) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='3)  Then for any partition λ the canonical triple is the block diagonal  e = Diag(Jλi) ,  f = Diag( ˜Jλi) ,  h = Diag(Hλi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='4)  Slodowy Slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Given a triple (e, f, h), one constructs the space  Se = e + gf ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='5)  where gf is the centralizer of f in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This is called the Slodowy slice transverse to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Note  that this should not be confused with the nilpotent Slodowy slice, which is the intersection  of the Slodowy slice with the nilpotent cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  When (e, f, h) is the canonical nilpotent  element (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='4) associated to partition λ, we call Sλ the Slodowy slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  In order to construct explicitly Sλ, we first need the set of matrices which commute  with ˜Jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' These are the matrices S(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , an) with a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , an ∈ C, defined by8  (S(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , an))ij :=  n−1  �  k=0  ak+1δi−k,j  i−1  �  l=j  l(n − l) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='7)  Then the centralizer gf is a set of block matrices of sizes λi × λj, where the block (i, j)  depends on min(λi, λj) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We will not need its explicit form here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We only need  the form of the diagonal blocks (i = j), which are of the form  S(a(i)  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , a(i)  λi ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='8)  Using this, we can compute the dimension of the Slodowy slices, which is  dim Sλ = −1 +  �  1≤i,j,≤r  min(λi, λj) = −(n + 1) + 2  r  �  i=1  iλi  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='9)  and  dim Sλ ∩ N = dim Sλ − (n − 1) = −2n + 2  r  �  i=1  iλi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='10)  8The matrices S(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , an) for low values of n are:  S(a1, a2) =  �  a1 0  a2 a1  �  ,  S(a1, a2, a3) =  �  �  �  a1  0  0  2a2 a1  0  4a3 2a2 a1  �  �  � ,  S(a1, a2, a3, a4) =  �  �  �  �  �  a1  0  0  0  3a2  a1  0  0  12a3 4a2  a1  0  36a4 12a3 3a2 a1  �  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='6)  – 39 –  The dimension of the nilpotent cone N is n(n − 1), and therefore we can deduce that the  dimension of the nilpotent orbit of e is  dim Oλ = n(n + 1) − 2  r  �  i=1  iλi ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='11)  which matches known results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Levi subalgebras and Slodowy slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  A Borel subalgebra b of g is a maximal  solvable subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  The standard choice, which we make, is to pick for b the upper  triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The simple roots αi can be indexed by i ∈ I := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , n − 1}, and  to every subset Θ of I one can associate a parabolic subalgebra pΘ of g containing b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' This  parabolic subalgebra decomposes as a direct sum  pΘ = lΘ ⊕ nΘ  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='12)  where lΘ is the Levi subalgebra and nΘ is the nilradical of pΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Let λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , λr) be a partition of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We associate to this partition the set  Θλ = I − {λ1, λ1 + λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , λ1 + · · · + λr−1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='13)  This way, one can construct the corresponding Levi subalgebra lλ := lΘλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' We then intro-  duce the intersection  S0  λ := Sλ ∩ lλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='14)  With out choices, lλ is simply the algebra of block diagonal matrices, with blocks of  respective sizes λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , λr, and S0  λ is the set of traceless matrices which are block diagonal,  with diagonal blocks Jλi +S(a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , aλi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' The dimension is dim S0  λ = n−1, independent  of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  It is easy to check that the matrix Jn + S(0, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , an) can be  conjugated to a matrix of the form  C(b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , bn) :=  �  �  �  �  �  �  �  �  0  1  0  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  0  1  (−)n−1bn (−)n−2bn−1  · ·  −b2 0  �  �  �  �  �  �  �  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='15)  where the bi are known functions of the ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Note that the change of basis depends explicitly  on the ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' A matrix in the form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='15) is called a companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' It has the convenient  property that its characteristic polynomial (evaluated at −z) is  det(z1n + C) = zn +  n  �  i=2  bizn−i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='16)  Any matrix in S0  λ can then be conjugated to a block diagonal matrix with blocks of the  form C(b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' , bλi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  – 40 –  Cokernel of ad(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  Consider a finite dimensional representation ρ : sl(2) → V of sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  One can decompose V into irreducible representations Vi of dimensions ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' In each irrep,  the image of ρ(e) has codimension 1, and the cokernel is spanned by the weight space of  lowest weight, which by definition is precisely the kernel of ρ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Therefore in the Lie  algebra g seen as an sl(2) representation defined by a triple (e, f, h), the cokernel of ad(e)  is gf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
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+page_content=' Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Schafer-Nameki and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
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+page_content=' Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
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+page_content=' Schafer-Nameki and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Wang, Coulomb and Higgs branches from canonical  singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Part I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Hypersurfaces with smooth Calabi-Yau resolutions, JHEP 04 (2022)  061, [2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
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+page_content=' Part I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' General setup, JHEP 09 (2021) 186,  [2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
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+page_content=' Part II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
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+page_content='2902].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content='  [95] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' Hanany and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
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+page_content='04020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE4T4oBgHgl3EQfvQ0x/content/2301.05239v1.pdf'}
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+Convolutional neural networks for valid
+and efficient causal inference
+Mohammad Ghasempour∗, Niloofar Moosavi, Xavier de Luna
+Department of Statistics, USBE, Ume˚a University, Ume˚a, Sweden
+January 30, 2023
+Abstract
+Convolutional neural networks (CNN) have been successful in machine learn-
+ing applications. Their success relies on their ability to consider space invariant
+local features. We consider the use of CNN to fit nuisance models in semipara-
+metric estimation of the average causal effect of a treatment. In this setting,
+nuisance models are functions of pre-treatment covariates that need to be con-
+trolled for. In an application where we want to estimate the effect of early
+retirement on a health outcome, we propose to use CNN to control for time-
+structured covariates. Thus, CNN is used when fitting nuisance models explain-
+ing the treatment and the outcome. These fits are then combined into an aug-
+mented inverse probability weighting estimator yielding efficient and uniformly
+valid inference. Theoretically, we contribute by providing rates of convergence
+for CNN equipped with the rectified linear unit activation function and com-
+pare it to an existing result for feedforward neural networks. We also show
+when those rates guarantee uniformly valid inference. A Monte Carlo study
+is provided where the performance of the proposed estimator is evaluated and
+compared with other strategies. Finally, we give results on a study of the effect
+of early retirement on hospitalization using data covering the whole Swedish
+population.
+∗We are grateful to Xijia Liu and Jenny H¨aggstr¨om for helpful comments that have improved
+the paper.
+We acknowledge funding from the Swedish Research Council and the Marianne and
+Marcus Wallenberg Foundation. The Ume˚a SIMSAM Lab data infrastructure used in this study
+was developed with support from the Swedish Research Council, the Riksbanken Jubileumsfond and
+by strategic funds from Ume˚a University.
+Correspondence: mohammad.ghasemour@umu.se
+1
+arXiv:2301.11732v1  [stat.ML]  27 Jan 2023
+
+1
+Introduction
+Convolutional Neural Networks (CNN) have been found successful in discovering
+location-invariant patterns in speech, images, and time series data (LeCun et al.,
+1995). In particular, they have been shown to have a universal approximation prop-
+erty and be more efficient in terms of number of hidden layers than fully-connected
+multi-layer networks in high-dimensional situations (Zhou, 2020). In this paper we
+show how CNN can be useful in controlling confounding information when using rich
+observational databases in order to perform semiparametric inference on a low di-
+mensional causal parameter: we focus on average causal effects of a binary treatment
+on an outcome of interest, although our results are relevant for the semiparametric
+estimation of other low dimensional parameters of interest (see e.g. Chernozhukov
+et al., 2018).
+Augmented Inverse Probability Weighting (AIPW) estimators (also called Double
+Robust (DR) estimators, Robins et al., 1994) attain the semiparametric efficiency
+bound and yield uniformly valid inference as long as the nuisance functions of the
+confounding covariates are fitted consistenlty with fast enough convergence rates; e.g.
+all nuisance functions are estimated with order n−1/4 (Belloni et al., 2014; Farrell,
+2015; Kennedy, 2016; Moosavi et al., 2021).
+In this paper, we contribute to this
+theory by showing that CNN fits of nuisance functions achieve the n−1/4 convergence
+rate required to obtain uniformly valid inference on causal parameters.
+To show
+this we use a result obtained by Farrell et al. (2021) for general Rectified Linear
+Unit (ReLU)-based Feed-forward Neural Networks (FNN). They show that, for large
+samples, the estimation error rate of FNN are bounded by the following term, with a
+probability increasing exponentially with γ:
+�
+log n/n × complexity penalty +
+�
+(log log n + γ)/n + approximation rate.
+(1)
+We deduce the above approximation rate for CNN architectures inspired by earlier
+work by Zhou (2020). However, in contrast to the latter paper, we consider a larger
+number of free parameters by considering multi-channel convolutional neural network,
+so as to achieve a trade-off between complexity penalty and approximation rate in
+(1), and thereby obtain the convergence rate n−1/4 for the CNN fit of the nuisance
+functions. In the next section we formally define the causal parameters of interest
+using the potential outcome framework (Rubin, 1974), and introduce the assumptions
+yielding identification, and locally efficient and uniformly valid inference when using
+AIPW estimators. We also introduce the convolutional network architectures with
+which we propose to fit the nuisance functions used by AIPW, followed by our main
+theoretical results, including conditions to obtain uniformly valid inference when using
+CNN based AIPW estimation. Section 3 presents numerical experiments illustrating
+2
+
+the finite sample behaviour of this estimation strategy under different data generat-
+ing mechanisms. The proposed estimator is compared to AIPW, Targeted Maximum
+Likelihood Estimation (TMLE) (van der Laan and Rose, 2011) and Outcome Re-
+gression (OR) estimation (Tan, 2007; Moosavi et al., 2021) using fully-connected
+feed-forward ReLU based neural networks (Multilayer Perceptron (MLP) in Farrell
+et al., 2021) and Lasso to fit the nuisance functions. In Section 4, we study the effect
+of early retirement (at 62 years old), compared to retiring later in life, on morbidity
+and mortality outcomes (Barban et al., 2020). We use population wide Swedish reg-
+ister data and follow cohorts born in 1946 and 1947 for which we have a rich reservoir
+of potential pre-treatment confounders, including hospitalization and income histo-
+ries. CNN allows us to consider that such life histories may contain location-invariant
+patterns that confound the causal effects of the treatment (decision to retire early).
+2
+Theory and method
+2.1
+Causal parameters and uniformly valid inference
+The Average Causal Effect (ACE) and Average Causal Effect among the Treated
+(ACET) of a binary treatment (T) are parameters defined using potential outcomes
+(Rubin (1974)), respectively:
+τ = E(Y (1)) − E(Y (0)),
+τt = E(Y (1) − Y (0)|T = 1),
+where Y (1) and Y (0) are the outcomes that would be observed if T = 1 and T = 0,
+respectively.
+For a given individual only one of these potential outcomes can be
+observed. This intrinsic missing data problem implies that assumptions need to be
+made to identify τ and τt. For this purpose, and given a vector of observed pre-
+treatment covariates X, we assume:
+Assumption 1.
+a. No unmeasured confounders:
+(i) Y (0)
+T | X.
+(ii) Y (1)
+T | X.
+b. Overlap:
+(i) P(T = 0 | X) > 0.
+(ii) P(T = 1 | X) > 0.
+c. Consistency: The observed outcome is Y = Y (1)T + Y (0)(1 − T).
+3
+
+Note that ACET is identified if only 1a(i), 1b(i) and 1c hold. Assumption 1a
+requires that the observed vector X includes all confounders. Assumption 1b requires
+that for any value X both treatment levels have non-zero probability to occur, and
+by Assumption 1c one of the potential outcomes is observed for each individual, and
+its value is not affected by the treatment received by other individuals in the sample.
+We aim at uniformly (over the class of Data Generating Process (DGP)s, for which
+Assumption 2 hold) valid inference while using semiparametric estimation (Moosavi
+et al., 2021).
+Therefore, the following AIPW estimators of ACE are considered
+(Robins et al., 1994; Scharfstein et al., 1999):
+ˆτ = En[ ˆψ1(zi) − ˆψ0(zi)]
+where En[·] is the empirical mean operator,
+ˆψt(zi) = 1{ti = t}(yi − ˆµt(xi))
+ˆP[T = t|X = xi]
++ ˆµt(xi),
+and ˆµt(x) is an estimator of µt(x) = E(Y (t) | X = x).
+For ACET, we use the estimator
+ˆτt = En[ ˆψ1,1(zi) − ˆψ0,1(zi)],
+where
+ˆψt,t′(zi) =
+ˆP[T = t′|X = xi]
+ˆP[T = t′]
+1{ti = t}(yi − ˆµt(xi))
+ˆP[T = t|X = xi]
++ 1{ti = t′}ˆµt(xi)
+ˆP[T = t′]
+.
+We use the following assumptions for the DGPs.
+Assumption 2. We have
+a. Let {(yi, ti, xi), i = 1, . . . , n} be an i.i.d. sample from (Y, T, X).
+b. Let U = Y (t) − µt(X). There is some r > 0 for which E
+�
+|µt (xi) µt′ (xi)|1+r�
+and E
+�
+|ui|4+r�
+are bounded, for given values of t and t′.
+The estimators of the nuisance functions have yet to be introduced for these AIPW
+estimation strategies. In order to get the desired results, the proposed estimators
+should be well behaved. More precisely, the following consistency and rate conditions
+are considered for the nuisance function estimators.
+Assumption 3. Let ˆp(x) be an estimator of P[T = 1|X = xi] which only depends on
+{xi, ti}n
+i=1 (assumption of “no additional randomness”, Farrell, 2015). Moreover, for
+a given t we have
+4
+
+a. En[(ˆp(xi) − p(xi))2] = oP(1) and En[(ˆµt(xi) − µt(xi))2] = oP(1),
+b. En[(ˆµt(xi) − µt(xi))2]1/2En[(ˆp(xi) − p(xi))2]1/2 = oP(n−1/2),
+c. En[(ˆµt(xi) − µt(xi))(1 − 1{ti = t}/P[T = t|X = xi])] = oP(n−1/2).
+The following proposition describes uniform validity results obtained by Farrell
+(2018, Corollary 2 and 3) under the above regularity conditions.
+Proposition 1. For each n, let Pn be the set of distributions obeying Assumptions
+1a(i), 1b(i), 1c and 2a. Further, assume Assumption 2b holds for t = t′ = 0, and let
+ˆp(x) and ˆµ0(x) fulfill Assumption 3. Then, we have:
+sup
+P∈Pn
+����PP
+�
+τt ∈
+�
+ˆτt ± cα
+�
+ˆVt/n
+��
+− (1 − α)
+���� → 0,
+where
+ˆVt = n2
+n2
+t
+En
+�
+1(ti = 1) (yi − ˆµ0 (xi) − ˆτt)2�
++n2
+n2
+t
+En
+�
+ˆp (xi)2
+(1 − ˆp (xi))21(ti = 0) (yi − ˆµ0 (xi))2
+�
+,
+nt = Σn
+i=11(T = t) and cα = Φ−1(1 − α/2).
+Let also Assumptions 1a(ii), 1b(ii) be fulfilled and Assumption 2b hold for t, t′ ∈
+{0, 1}. Additionally, assume ˆµ1(x) fulfills Assumption 3. Then, we have:
+sup
+P∈Pn
+����PP
+�
+τ ∈
+�
+ˆτ ± cα
+�
+ˆV /n
+��
+− (1 − α)
+���� → 0,
+where
+ˆV = En
+�
+1(ti = 1) (yi − ˆµ1 (xi))2
+ˆp (xi)2
+�
++ En
+��
+ˆµ1 (xi) − En[ ˆψ1(zi)]
+�2�
++ En
+�
+1(ti = 0) (yi − ˆµ0 (xi))2
+(1 − ˆp (xi))2
+�
++ En
+��
+ˆµ0 (xi) − En[ ˆψ0(zi)]
+�2�
+.
+Remark 1. A multiplicative rate condition as Assumption 3(b) is weaker than sepa-
+rate conditions on the two nuisance model estimators. It only requires that one of the
+nuisance functions is estimated at faster rate if the other one is estimated at slower
+rate. However, using the regularity conditions in this paper and Farrell et al. (2021),
+the rate oP(n−1/4) is obtained for each of the nuisance estimators separately. This is
+to make sure Assumption 3(c) for ˆµ is also fulfilled. This assumption can be, however,
+dropped by considering sample splitting (Chernozhukov et al., 2018).
+Note, finally, that because the AIPW estimators is based on the efficient influence
+function, its asymptotic variance is equal to the semiparametric efficiency bound
+(Tsiatis, 2006).
+5
+
+2.2
+Convolutional neural networks
+We consider a specific CNN architecture with parallel hidden layers structured as
+follows. Let the input column vector be denoted by h0 := (x1, · · · , xd)′ ∈ Ω, where
+Ω ⊆ Rd. We consider L to be the number of hidden layers in which we have E number
+of parallel vectors. Let σ be the ReLU function defined on the space of real numbers
+as σ(z) = max(0, z) for z ∈ R. The vectors in each hidden layer l ∈ {1, · · · , �L} have
+size dl and are defined by hl
+e = σ(W l
+ehl−1
+e
+− be
+l ), where e ∈ {1, · · · , E}, h0
+e = h0, be
+l
+is a vector called bias vector in the neural network literature, W l
+e is a weight matrix
+defined as:
+W l
+e =
+�
+���������������������
+we
+0,l
+0
+0
+0
+· · ·
+0
+we
+1,l
+we
+0,l
+0
+0
+· · ·
+0
+...
+...
+...
+...
+· · ·
+...
+we
+S,l
+we
+S−1,l
+· · ·
+we
+0,l
+0 · · ·
+0
+0
+we
+S,l
+we
+S−1,l
+· · ·
+we
+0,l · · ·
+0
+...
+...
+...
+...
+· · ·
+...
+...
+· · ·
+0
+we
+S,l
+· · ·
+we
+0,l
+...
+· · ·
+· · ·
+0
+we
+S,l · · ·
+we
+1,l
+...
+...
+...
+...
+...
+...
+0
+· · ·
+· · ·
+· · ·
+0
+we
+S,l
+we
+S−1,l
+0
+· · ·
+· · ·
+· · ·
+· · · 0
+we
+S,l
+�
+���������������������
+,
+and the weight parameter we
+i,l is the i-the element of the filter mask (we
+0,l, · · · , we
+S,l).
+S is considered fixed through all values of l and e. We have dl = d0 + Sl, where dl is
+the size of vectors in the l-th layer and d0 = d.
+The structure of a CNN as defined above provides us with the following function
+space on Ω:
+C = {
+E
+�
+e=1
+c′
+ehL
+e : ce ∈ RdL}.
+Similar to Zhou (2020) (who, however, uses E = 1) we assume that for all values
+of e in {1, · · · E} and l in {1, · · · , L−1} the vector be
+l has the form (be
+l,1, · · · , be
+l,S, be
+l,S+1,
+· · · , be
+l,S+1, be
+l,dl−S+1, · · · , be
+l,dl)′ with the dl − 2S repeated components in the middle.
+Therefore, the total number of parameters (duplicate/shared weights are counted
+more than once), Q for the neural network fulfills the following inequality:
+Q = E(d(1 + s)(L − 1) + s(1 + s)L(L − 1)/2 + (s + 3)(d + sL)) ≍ EL2,
+(2)
+where the notation xn ≍ yn for two sequences of random variables xn and yn means
+xn = OP(yn) and yn = OP(xn).
+6
+
+2.3
+Main results
+In this section, we provide results that allow us to use the CNN architectures described
+above to fit the nuisance models needed for AIPW estimations of ACE and ACET,
+and obtain uniformly valid inference. In particular, we need to show that the rate
+conditions in Assumption 3 are fulfilled.
+Assumption 4. Assume that zi = (ui, v′
+i)′ , 1 ≤ i ≤ n are i.i.d.
+copies of Z =
+(U, V ) ∈ [−1, 1]d × V, where U is continuously distributed. For an absolute constant
+M > 0 and a target f ∗ (a nuisance function), assume V ∈ [−M, M] and ∥f ∗∥∞ ≤ M.
+Assumption 5. Let ℓ be a loss function that, for any arbitrary functions f and g
+and z a realization of Z, fulfills:
+|ℓ(f, z) − ℓ(g, z)| ≤ Cℓ|f(x) − g(x)|,
+c1E((f − ˜f)2) ≤ E(ℓ(f, Z)) − E(ℓ( ˜f, Z)) ≤ c2E((f − ˜f)2),
+where ˜f = arg min E(ℓ(f, Z)).
+Further, let F be an arbitrary set of functions. We define:
+ˆfF := arg min
+f∈F
+∥f∥∞≤M′
+n
+�
+i=1
+ℓ(f, z),
+(3)
+and
+εF,F′ := sup
+f∈F
+inf
+f′∈F′
+∥f′∥∞≤M′
+∥f − f ′∥∞ ,
+(4)
+where ∥·∥∞ is the supremum norm. The following result is a special case of Theorem
+2(b) given in Farrell et al. (2021), and gives the rate of convergence for our estimate
+ˆfC of a target f ∗ (a nuisance function). Prior to presenting the theorem, the target
+space in which we seek to find the nuisance functions need to be defined. We use the
+notation W β,∞(Ω) for the Sobolev space with smoothness β ∈ N+. We consider W to
+be the unit ball on the Sobolev space defined on [−1, 1]d; W := B1(W β,∞([−1, 1]d))
+such as:
+W := {f : max
+α,|α|≤β ∥Dαf(x)∥L∞([−1,1]d) ≤ 1},
+where ∥f∥L∞(Ω) is the essential supremum norm of the absolute value of a function f
+defined on Ω, α = (α1, · · · , αd)′, |α| = α1 + · · · + αd, and Dαf is the weak derivative.
+7
+
+Theorem 1. Let f ∗ ∈ W, and assume that Assumptions 4-5 hold. For an absolute
+constant M > 0 and M ′ = 2M, let the approximation error εW,C be defined as in (4).
+With probability at least 1 − e−γ, and for large enough n,
+En[( ˆfC − f ∗)2] ≤ C
+�
+QL log Q
+n
+log n + log log n + γ
+n
++ ε2
+W,C
+�
+,
+(5)
+where the constant C > 0 is independent of n, but may depend on d, M, and other
+fixed constants.
+Our next step is to bound the term ε2
+W,C in the above inequality on the estimation
+error. The following result is similar to Theorem 1 in Zhou (2020). However, we allow
+here the number of parallel layers to grow with n, which translates into faster growing
+function space for our specific CNN architectures, thereby a larger decay rate for the
+error.
+Theorem 2. Let 2 ≤ s ≤ d. If L ≥ 2d/(s − 1) and f ∈ W, ∥f∥∞ ≤ M and an
+integer index β > 2 + d/2, then there exist w, b and f w,b
+L,E ∈ C such that
+���f − f w,b
+L,E
+���
+C([−1,1]d) ≤ c2d/2�
+log(LE) 1
+LE
+1
+2 + 1
+d,
+where c is an absolute constant and therefore
+εW,C ≤ C
+�
+log(LE) 1
+LE
+1
+2 + 1
+d.
+Finally, our main result gives conditions for the CNN nuisance function fit to fulfill
+Assumption 3, and therefore conditions for AIPW estimation based on CNN fit to
+yield uniformly valid inference for ACE and ACET (Proposition 1).
+Theorem 3. Let the conditions in Theorem 1 hold for ui = xi, vi = yi, and f ∗ = µt.
+Moreover the same conditions are fulfilled for ui = xi, vi = ti, and f ∗ = p. Let
+also E ≍ n
+d
+2d+4, L ≍ n
+1
+4d+8 log2(n), β = βp ∧ βµ, and s and d fulfill the assumptions
+of Theorem 2. Then, the rate conditions in Assumption 3 are fulfilled for nuisance
+functions estimators (3) with F = C.
+Remark 2. The rate result for CNN based estimators in Theorem 3 corresponds to
+similar results for MLP in Farrell et al. (2021, Theorem 3). However, the conditions
+on smoothness is less strict in our case. The rate of growth for the number of pa-
+rameters considered for MLP based estimators in Farrell et al. (2021, Theorem 1) is
+n
+d
+β+d log5 n, while we have considered the growth rate n
+d+1
+2d+4 log4(n). These rates are
+similar up to a log factor if d is large and β is close to d. Notice, however, that these
+are not necessarily minimal rates required for valid inference on ACE(T).
+8
+
+3
+Simulation Study
+We perform a simulation study to evaluate the use of the CNN together with AIPW
+estimation strategy of average causal effects proposed above. We focus here on τt
+(ACET) but similar results are obtained for τ. Comparisons are made with other
+algorithms to fit nuisance functions, including, MLP, (Farrell et al., 2021) and post-
+lasso estimators (Farrell, 2015). Moreover, we also include alternative strategies to
+AIPW, including OR estimation (Tan, 2007) and targeted learning (van der Laan
+and Rose, 2011).
+The simulation results are based on 1000 replications. The neural networks initial
+weights are different for each replication, which makes the results more robust. A se-
+quence of ten covariates are independently generated having normal distributions with
+increasing means and varying variances: X1 ∼ N(100, 20), X2 ∼ N(102, 15), X3 ∼
+N(105, 13), X4 ∼ N(107, 11), X5 ∼ N(109, 8), X6 ∼ N(110, 20), X7 ∼ N(112, 15),
+X8 ∼ N(115, 13), X9 ∼ N(117, 11), and X10 ∼ N(119, 8). While these variables
+are independent, their ordering is going to be important in how the outcomes and
+treatment assignment are generated as described below in two different settings.
+Setting 1 Potential outcomes and treatment assignment are generated as follows:
+Y (0) = 1+0.001((X2−X1)2+(X4−X3)3+(X6−X5)2+(X8−X7)3+(X10−X9)2)+e0,
+Y (1) = 2−0.001((X2−X1)2+(X4−X3)3+(X6−X5)2+(X8−X7)3+(X10−X9)2)+e1,
+P(T = 1|X) =1/(1 + exp(5 × 10−6((X2 − X1)2+
+(X4 − X3)3 + (X6 − X5)2 + (X8 − X7)3 + (X10 − X9)2))).
+In this setting, polynomial functions of the difference of some neighboring vari-
+ables in the sequence are used to generate the nuisance models.
+Setting 2 Potential outcomes and treatment assignment are generated as follows:
+Y (0) = 1 + (l1(X1, X2, X3, X4) + l2(X4, X5, X6, X7) + l3(X6, X7, X8, X9)) + e0,
+Y (1) = 2 − (l1(X1, X2, X3, X4) + l2(X4, X5, X6, X7) + l3(X6, X7, X8, X9)) + e1,
+P(T = 1|X) =1/(1 + exp(0.05X5−
+0.1(l1(X1, X2, X3, X4) + l2(X4, X5, X6, X7) + l3(X6, X7, X8, X9)))),
+where
+l1(x, y, z, w) =
+�
+10,
+if y/x, z/y, and w/z > 1.15,
+0,
+otherwise,
+9
+
+E1 
+E2 
+E2 
+El-2 
+El-1 
+El-1 
+Flattened
+Fully connected
+Figure 1: Illustration of the convolutional neural network utilized in the simulations
+and the real data study. Each layer l contains different number of channels El, where
+E0 (not shown in the Figure) is the number of time series in the set of input covariates.
+Each channel (feature map) in layer l (e.g. the blue vector in the second layer) is
+formed by first applying El−1 number of kernels (convolution filters) on the vectors
+in layer l − 1 (as demonstrated by the blue arrow in the set of kernels applied on the
+first layer) and then summing the results. There exist some covariates which are not
+time series. Inclusion of those vectors in the neural network is not illustrated in the
+figure for simplification purpose. A fully connected feed-forward structure has been
+used for those covariates and the results are then combined with the results from the
+time-series data in the flattening layer demonstrated above.
+l2(x, y, z, w) =
+�
+5,
+if y/x, z/y, and w/z < 1.05,
+0,
+otherwise,
+l3(x, y, z, w) =
+�
+3,
+if (y − 1.1x)(z − 1.1y)(w − 1.1z) < 0,
+0,
+otherwise.
+Here, the output of l1 and l2 are non-zero only if an increase by a factor bigger
+than 1.15 and a decrease by a factor bigger than 1.05 is found in all consecutive
+pair of covariates in the input of size four, respectively. The value of l3 is nonzero
+if an increase by a factor bigger than 1.1 is followed by a decrease and followed
+by another increase by a factor of bigger than 1.1 or if the oscillation has the
+opposite direction.
+The following algorithms to estimate the nuisance functions are evaluated:
+CNN A convolutional neural network using the ReLU activation function, with ar-
+chitecture illustrated in Figure 1. Note that this architecture is slightly more
+flexible than the one considered in Section 2.2, hence, we expect it to perform
+at least as well in terms of approximation rate. Both for the potential out-
+come model µ0 and the propensity score p, the number of channels in the input
+10
+
+layer is one. For the outcome models 128 channels in the first hidden layer
+and 16 channels in the second hidden layer have been used, while 32 channels
+in the first hidden layer and 8 channels in the second hidden layer have been
+used for fitting the propensity score. We have implemented the CNN together
+with AIPW estimation strategy in the package DNNCausal which is available at
+https://github.com/stat4reg/DNNCausal.
+MLP A ReLU activation function based feedforward neural network with two hidden
+layers. Two different feedforward architectures are used in order to estimate µ0,
+and p (propensity score). The networks for fitting the outcome model consist of
+128 and 80 nodes within the two layers, respectively. For the propensity score,
+two layers contain 32 and 8 nodes, respectively.
+post-lasso Nuisance models are fitted in two steps, where all higher order terms up
+to order three are considered. First, a lasso variable selection step is performed
+using the R-package hdm (Chernozhukov et al. (2016)), and then maximum
+likelihood is used to fit the models with the selected covariates (those with co-
+efficients not shrinked to zero by lasso). Variable selection is performed using
+two different strategies: double selection which takes the union of variables se-
+lected by regressing outcome and treatment variables and uses that to refit both
+models (Belloni et al., 2014; Moosavi et al., 2021), and single selection, which
+utilizes two sets of variables selected by regressing outcome and treatment, and
+refits each of the models using their corresponding set (Farrell, 2015).
+We use the above nuisance function estimators in several different estimators of τt,
+as follows:
+AIPW is implemented using CNN, MLP, and post-lasso with both single and double
+selection (denoted respectively: DRcnn, DRmlp, DRss, DRds).
+OR estimator with post-lasso and double selection (denoted ORds).
+TMLE using the R-package tmle and the function SL.nnet to fit all nuisance func-
+tions (denoted TMLE).
+Full details with code for this simulation study are available at the Github page:
+https://github.com/stat4reg/Causal CNN/blob/main/README.md.
+We set sample sizes to n = 5000 and 10000, and run 1000 replicates.
+Bias,
+standard errors (both estimated and Monte Carlo), mean squared errors (MSE, bias
+squared plus variance) and empirical coverages for 95% confidence intervals are re-
+ported in Table 1 and 2.
+The DGP of Setting 1 (Table 1) is the least complex one and its polynomial form
+is favorable to the post-lasso methods which are based on higher order terms. It is
+11
+
+Table 1: Setting 1 DGP: Bias, standard errors (both estimated, Est sd, and Monte
+Carlo, MC sd), MSEs and empirical coverages for 95% confidence intervals over 1000
+replicates.
+bias
+coverage
+MC sd
+Est sd
+MSE
+n = 5000
+DRcnn
+0.185
+0.947
+1.379
+1.366
+1.935
+DRmlp
+0.426
+0.940
+1.353
+1.331
+2.013
+ORds
+0.001
+0.947
+1.404
+1.392
+1.970
+DRss
+0.002
+0.948
+1.404
+1.395
+1.971
+DRds
+0.003
+0.948
+1.404
+1.395
+1.972
+tmle
+0.044
+0.944
+1.396
+1.384
+1.951
+n = 10000
+DRcnn
+0.089
+0.958
+0.971
+0.972
+0.951
+DRmlp
+0.223
+0.949
+0.961
+0.958
+0.973
+ORds
+0.001
+0.955
+0.981
+0.982
+0.962
+DRss
+0.001
+0.955
+0.981
+0.984
+0.962
+DRds
+0.001
+0.955
+0.981
+0.984
+0.962
+tmle
+0.028
+0.957
+0.981
+0.978
+0.963
+therefore expected that the post-lasso methods ORds, DRss, DRds have low bias.
+DRcnn has larger bias, but decreasing with n. All methods have similar MSEs even
+though DRcnn has the lowest MSE for the largest sample size. Empirical coverages
+are close to the nominal 95% level for all methods.
+The DGP of Setting 2 (Table 2) is less smooth and we observe that the DRcnn and
+DRmlp methods have the lowest biases and MSEs, where DRcnn performs slightly
+better. TMLE and post-lasso both have such a large bias that the confidence interval
+coverages are far from being at the nominal level. It should be noted, however, that
+the TMLE uses the built-in feedforward neural network, which is not adjusted to the
+generated datasets, and so the comparison is not a fair one between TMLE and the
+DR estimators.
+4
+Effect of early retirement on health
+The existing theoretical and empirical evidence on the effects of early retirement on
+health are mixed, and both positive and negative results may be expected and have
+been reported in different situations; see Barban et al. (2020) and the references
+therein. In this context, we add to the empirical evidence by studying the effect of
+early retirement on health using a database linking at the individual level a collection
+12
+
+Table 2: Setting 2 DGP: Bias, standard errors (both estimated and Monte Carlo),
+MSEs and empirical coverages for 95% confidence intervals over 1000 replicates.
+bias
+coverage
+MC sd
+Est sd
+MSE
+n = 5000
+DRcnn
+0.047
+0.937
+0.094
+0.101
+0.011
+DRmlp
+0.076
+0.890
+0.095
+0.099
+0.015
+ORds
+0.233
+0.206
+0.083
+0.085
+0.061
+DRss
+0.233
+0.260
+0.083
+0.092
+0.061
+DRds
+0.235
+0.250
+0.083
+0.092
+0.062
+tmle
+0.233
+0.204
+0.083
+0.085
+0.061
+n = 10000
+DRcnn
+0.025
+0.945
+0.070
+0.072
+0.006
+DRmlp
+0.036
+0.928
+0.070
+0.072
+0.006
+ORds
+0.224
+0.040
+0.062
+0.061
+0.054
+DRss
+0.224
+0.054
+0.062
+0.065
+0.054
+DRds
+0.226
+0.045
+0.061
+0.065
+0.055
+tmle
+0.224
+0.041
+0.062
+0.061
+0.054
+of socio-economic and health registers on the whole Swedish population (Lindgren
+et al., 2016). This allows us to follow until 2017 two complete cohorts born in 1946
+and 1947 and residing in Sweden in 1990. As health outcome we observe the number
+of days in hospital per year after retirement. To study the effects of early retirement
+we consider those who were still alive at age 62, and either retire at age 62 (T = 1,
+treatment) or retire later (T = 0, control group). For the cohorts studied, retirement
+pensions are accessible from age 61, although the usual age of retirement is 65 years
+of age (see Figure 2 for descriptive statistics on age of retirement). Therefore, retiring
+at age of 62 is considered as early retirement since it decreases your annual pension
+transfers compared to later retirement; see, e.g.
+Barban et al. (2020) for details
+on the Swedish pension system. Barban et al. (2020) also contains a study on the
+effects of early retirement, although based on fewer measurements for time-dependent
+covariates and using a matching design. This study provides new evidence by taking
+into account richer histories of the time-dependent covariates, and by incorporating
+CNN to address complex dynamics in pre-treatment confounding covariates.
+More precisely, the design of the study is as follows. An individual alive at age
+62 is considered as taking early retirement at that age if hers/his pension transfers
+become larger than income from work at that age for the first time (i.e., they were
+never so earlier). For replicability, the exact definition using income and transfer
+variables from the Swedish registers are available at
+13
+
+https://github.com/stat4reg/Causal CNN/blob/main/population description.md.
+The health outcomes of interest are the number of hospitalization days per year dur-
+ing the next five years following early retirement. We, however, check first whether
+early retirement has an effect on survival during the five first years following early
+retirement. There were no (or hardly significant) such effects at the 5% level (results
+reported in the appendix, Table 4) and we therefore focus the analysis on the sur-
+vivors when looking at effects on hospitalization. The following covariates are used as
+input in the CNN architures to fit treatment and outcome nuisance models. Besides
+the birth year we include the the following (pre-treatment) covariates measured at 61
+years of age: marriage status, municipality, education level, Spouse education level,
+and the number of biological children. Moreover, we consider the measurements of
+the following covariates for each of the ten years preceding retirement: days of hos-
+pitalization and open health care, annual income from labour, annual income from
+pension, annual income from unemployment, annual income from early retirement
+and sickness benefit, annual compensation for illness, and spouse retirement status.
+Thus, covariates include eight time(-structured) series of ten observations each per
+individual. We do separate analyses for men and women which gives two samples of
+approximately 100000 individuals each.
+The code giving details on the CNN architectures and tuning parameters can be
+found at
+https://github.com/stat4reg/Causal CNN/blob/main/population description.md.
+We also apply the other methods evaluated in the simulation study above.
+Results are presented in Table 3. Based on the naive estimation not controlling for
+confounders, early retirement has a clear positive effect on health (varying between
+0.167 and 0.396 average number of days) and for the five years of follow up considered.
+These naive effects are larger for women in the beginning. These negative effects
+disappear when controlling for the considered covariates. This is seen most clearly
+with CNN which yields the effects smaller than 0.1 day (in absolute value) for all
+cases but one. Confidence intervals at the 95% level cover zero in most cases. Thus,
+while there appeared to be a positive effect on health of early retirement, this effect
+was probably due to confounding.
+5
+Discussion
+We have proposed and studied a semiparametric estimation strategy for average causal
+effects which combine convolutional neural network with AIPW.
+As long as the
+conditions given are met, this strategy yields locally efficient and uniformly valid
+inference. The use of CNN has the advantage over fully-connected feed forward neural
+networks that they are more efficient on the number of weights (free parameters) used
+14
+
+0
+250000
+500000
+750000
+50
+60
+70
+retirement age
+count
+0e+00
+2e+05
+4e+05
+6e+05
+8e+05
+count
+0
+250000
+500000
+750000
+1000000
+50
+60
+70
+retirement age
+count
+0
+250000
+500000
+750000
+1000000
+count
+Figure 2: Retirement age for men and women who belong to the cohorts born in
+Sweden 1946 and 1947. The last staple in each histogram displays those who retired
+at age 71 or later.
+15
+
+Table 3:
+Effect of early retirement on number of days in hospital during five years
+of follow up, for the early retirees, and 95% confidence intervals.
+Women
+Men
+The first year
+n=106437
+n=104459
+naive −0.396
+−0.167
+DRcnn
+0.062
+(−0.060,
+0.183) −0.037
+(−0.256, 0.183)
+DRmlp −0.062
+(−0.192,
+0.069)
+0.138
+(−0.079, 0.355)
+ORds −0.135
+(−0.231, −0.039)
+0.018
+(−0.156, 0.192)
+DRss −0.138
+(−0.232, −0.045)
+0.025
+(−0.148, 0.199)
+DRds −0.128
+(−0.222, −0.035)
+0.028
+(−0.145, 0.202)
+tmle −0.143
+(−0.237, −0.049)
+0.018
+(−0.156, 0.192)
+The second year
+n=105776
+n=103560
+naive −0.354
+−0.192
+DRcnn −0.082
+(−0.231,
+0.067)
+0.122
+(−0.042, 0.285)
+DRmlp −0.083
+(−0.224,
+0.058)
+0.214
+( 0.041, 0.387)
+ORds −0.104
+(−0.215,
+0.007) −0.007
+(−0.161, 0.146)
+DRss −0.098
+(−0.207,
+0.011)
+0.011
+(−0.142, 0.164)
+DRds −0.081
+(−0.190,
+0.027)
+0.014
+(−0.139, 0.167)
+tmle −0.111
+(−0.220, −0.002)
+0.011
+(−0.142, 0.163)
+The third year
+n=105039
+n=102547
+naive −0.277
+−0.173
+DRcnn
+0.011
+(−0.153,
+0.175)
+0.030
+(−0.140, 0.200)
+DRmlp
+0.200
+( 0.029,
+0.370)
+1.094
+(−0.894, 3.083)
+ORds −0.034
+(−0.192,
+0.124)
+0.014
+(−0.136, 0.165)
+DRss −0.018
+(−0.174,
+0.138)
+0.024
+(−0.126, 0.175)
+DRds −0.010
+(−0.167,
+0.146)
+0.027
+(−0.123, 0.178)
+tmle −0.018
+(−0.174,
+0.139)
+0.027
+(−0.123, 0.178)
+The fourth year
+n=104293
+n=101502
+naive −0.179
+−0.216
+DRcnn
+0.067
+(−0.181,
+0.315)
+0.085
+(−0.082, 0.252)
+DRmlp
+0.804
+(−0.432,
+2.040) −0.517
+(−1.723, 0.688)
+ORds
+0.032
+(−0.186,
+0.250) −0.032
+(−0.194, 0.131)
+DRss
+0.042
+(−0.176,
+0.259) −0.016
+(−0.178, 0.146)
+DRds
+0.056
+(−0.162,
+0.273) −0.008
+(−0.170, 0.154)
+tmle
+0.033
+(−0.185,
+0.251) −0.015
+(−0.177, 0.147)
+The fifth year
+n=103477
+n=100325
+naive −0.232
+−0.230
+DRcnn −0.012
+(−0.185,
+0.161) −0.095
+(−0.276, 0.085)
+DRmlp −0.979
+(−2.982,
+1.024) −0.153
+(−0.443, 0.138)
+ORds −0.019
+(−0.169,
+0.132) −0.050
+(−0.195, 0.095)
+DRss −0.013
+(−0.163,
+0.137) −0.035
+(−0.179, 0.109)
+DRds
+0.000
+(−0.150,
+0.151) −0.029
+(−0.173, 0.116)
+tmle −0.014
+(−0.164,
+0.136) −0.034
+(−0.178, 0.110)
+16
+
+for approximating the nuisance functions, and are geared to take into account time
+invariant local features in the data.
+A main contribution of the paper is to show under which conditions CNN fits of
+nuisance functions achieve n−1/4 convergence rate required to obtain uniformly valid
+inference on an ACE. Using the result in Farrell et al. (2021) in which convergence
+rate of a ReLU-based feed-forward network is shown to follow Equation (1), we show
+that the rate conditions in Assumption 3 are fulfilled.
+Specifically, we use CNN
+with a given complexity for which we show that approximation rate in (1) is small
+enough. A key component of result in Farrell et al. (2021) is the upper bound on
+empirical process terms found in Bartlett et al. (2005) based on Rademacher averages,
+which are measures of function space complexity. Global Rademacher averages do not
+provide fast rates as in (1). To derive this fast rate, localization analysis is employed
+which takes into consideration only intersection of the function space with a ball
+around the goal function; considering that in reality the algorithm only searches in
+a neighborhood around the true function. Moreover, the tight bound on Pseudo-
+dimension of deep nets found in Bartlett et al. (2019) is used.
+We also present numerical experiments showing the performance of the estimation
+strategy in finite sample and compare it to other machine learning based estimation
+strategies. The applicability of the method is illustrated through a population wide
+observational study of the effect of early retirement on hospitalization in Sweden.
+6
+Appendix
+In this section we also consider the Sobolev space Hβ(Ω) that is defined using the L2
+norm instead of the L∞ norm. Moreover, let ∥f∥L2(Ω) be the L2 norm of a function
+f defined on Ω.
+6.1
+Proof of Theorem 2
+Proof. Let Ω = [−1, 1]d. For any α that satisfies |α| ≤ β, by H¨older’s inequality, we
+have:
+∥Dαf∥L2(Ω) ≤ 2d/2 ∥Dαf∥L∞(Ω) .
+Therefore, f ∈ Hβ(Ω) and ∥f∥Hβ(Ω) ≤ 2d/2 hold by maxα,|α|≤β ∥Dαf∥L∞(Ω) ≤ 1.
+Moreover, Ω has a Lipschitz domain interior and measure zero boundary. Therefore,
+by Sobolev extension theorem (Stein, 1970, Theorem 5) there is an extension F ∈
+Hβ(Rd) of f for which
+∥F∥Hβ(Rd) ≤ C ∥f∥Hβ(Ω) ≤ C2d/2,
+17
+
+where C is a constant.
+By F ∈ Hβ(Rd) and the result in Klusowski and Barron (2018, Theorem 2), we
+have
+∥F − Fm∥C(Ω) ≤ c0vF,2 max{
+�
+log m,
+√
+d}m− 1
+2 − 1
+d,
+where m = E
+�
+(s−1)L
+d
+− 1
+�
+, Fm(x) = β0 + α0x + v
+m
+�m
+k=1 βk(αkx − tk)+, βk ∈ [−1, 1],
+∥αk∥1 = 1, tk ∈ [0, 1], β0 = F(0), α0 = ∇F(0) and |v| ≤ 2vF,2.
+Here, vF,2 :=
+�
+Rd ∥ω∥2
+1 | ˆF(ω)|dω ≤ cd,β ∥F∥Hβ(Rd), where cd,β is the finite constant
+��∥ω∥2
+1 (1 + |ω|2)−β/2��
+L2
+and ˆF(ω) is the Fourier transform of F.
+Let F i
+m := 1
+Eβ0 + 1
+Eα0x + v
+m
+�(i+1)m
+k=im+1 βk(αkx − tk)+. It is shown in Zhou (2020),
+using s ≥ 2 and L ≥ 2d/(s − 1), that F i
+m|Ω belongs to the following function space
+Cw,b
+L
+=
+� dL
+�
+k=1
+ckh(L)
+k (x) : c ∈ RdL
+�
+.
+Therefore, by the definition of C, we can directly conclude that Fm|Ω belongs to C.
+Let f w,b
+L,E := Fm|Ω. Then, we have
+���f − f w,b
+L,E
+���
+C(Ω) = ∥F − Fm∥C(Ω)
+≤ c0vF,2 max{
+�
+log m,
+√
+d}m− 1
+2 − 1
+d
+≤ c0cd,β ∥F∥Hβ(Rd) max{
+�
+log m,
+√
+d}m− 1
+2 − 1
+d.
+(6)
+The final result is obtained from the facts that 1
+2(s − 1)LE ≤ md ≤ (s − 1)LE, s ≥ 2
+and β > 2 + d/2 (cd,β is bounded).
+6.2
+Proof of Theorem 3
+Proof. The rate conditions (a) and (b) of Assumption 3 are fulfilled if both En[(ˆµt(xi)−
+µt(xi))2] and En[(ˆp(xi)−p(xi))2] are oP(n−1/2). For this, the three terms in the upper
+bound (5) in Theorem 1 must be oP(n−1/2). For the first term, based on 2, we have
+QL log Q
+n
+log n ≍ EL3 log(EL2)
+n
+log n ≍ log6(n) log(n
+d+1
+2d+4 log4 n)
+n
+2d+5
+4d+8
+log n = oP(n−1/2).
+The second term is also oP(n−1/2) if γ = o(n1/2). For the third term, using Theorem
+2 we have
+εW,C ≤
+�
+d+1
+2d+4 log n + log log2 n
+n
+d+1
+4d log
+d+2
+d n
+.
+18
+
+To prove condition (c) of Assumption 3, we use the proof of lemma Farrell et al.
+(2021, Lemma 10). In this proof it is shown that for an arbitrary class of feedforward
+neural networks with probability at least 1 − exp(−n
+d+3
+4d+8 log6 n)
+En
+�
+(ˆµt,C(xi) − µt(xi))
+�
+1 −
+1{ti = t}
+P[T = t|X = xi]
+��
+≤C
+�
+QL log Q
+n
+log n+
+log log n + n
+d+3
+4d+8 log6 n
+n
++ ε2
+W,C
+�
+.
+Therefore, we can conclude that under the stated conditions
+En
+�
+(ˆµt(xi) − µt(xi))
+�
+1 −
+1{ti = t}
+P[T = t|X = xi]
+��
+≤ CEn[(ˆµt(xi) − µt(xi))2] = oP(n−1/2).
+6.3
+Early retirement effects on survival
+Additional results on the effect of early retirement on survival can be found in Table
+4.
+References
+Barban, N., X. de Luna, E. Lundholm, I. Svensson, and F. C. Billari (2020). Causal
+effects of the timing of life-course events: Age at retirement and subsequent health.
+Sociological Methods & Research 49(1), 216–249.
+Bartlett, P. L., O. Bousquet, and S. Mendelson (2005). Local rademacher complexi-
+ties. The Annals of Statistics 33(4), 1497–1537.
+Bartlett, P. L., N. Harvey, C. Liaw, and A. Mehrabian (2019).
+Nearly-tight vc-
+dimension and pseudodimension bounds for piecewise linear neural networks. The
+Journal of Machine Learning Research 20(1), 2285–2301.
+Belloni, A., V. Chernozhukov, and C. Hansen (2014).
+Inference on treatment ef-
+fects after selection among high-dimensional controls. The Review of Economic
+Studies 81(2), 608–650.
+Chernozhukov, V., D. Chetverikov, M. Demirer, E. Duflo, C. Hansen, W. Newey, and
+J. Robins (2018). Double/debiased machine learning for treatment and structural
+parameters. The Econometrics Journal 21(1), C1–C68.
+19
+
+Chernozhukov, V., C. Hansen, and M. Spindler (2016). hdm: High-dimensional met-
+rics. R Journal 8(2), 185–199.
+Farrell, M. H. (2015, Nov). Robust inference on average treatment effects with pos-
+sibly more covariates than observations. Journal of Econometrics 189(1), 1–23.
+Farrell, M. H. (2018). Robust inference on average treatment effects with possibly
+more covariates than observations. arXiv:1309.4686v3.
+Farrell, M. H., T. Liang, and S. Misra (2021). Deep neural networks for estimation
+and inference. Econometrica 89(1), 181–213.
+Kennedy, E. H. (2016).
+Semiparametric theory and empirical processes in causal
+inference. In He H., Wu P., Chen D.-G. (eds) Statistical causal inferences and
+their applications in public health research, pp. 141–167. Springer.
+Klusowski, J. M. and A. R. Barron (2018). Approximation by combinations of relu
+and squared relu ridge functions with ℓ1 and ℓ0 controls. IEEE Transactions on
+Information Theory 64(12), 7649–7656.
+LeCun, Y., Y. Bengio, et al. (1995). Convolutional networks for images, speech, and
+time series. The handbook of brain theory and neural networks 3361(10), 1995.
+Lindgren, U., K. Nilsson, X. de Luna, and A. Ivarsson (2016). Data resource profile:
+Swedish microdata research from childhood into lifelong health and welfare (Ume˚a
+SIMSAM Lab). International Journal of Epidemiology 45, 1075–1075.
+Moosavi, N., J. H¨aggstr¨om, and X. de Luna (2021). The costs and benefits of uni-
+formly valid causal inference with high-dimensional nuisance parameters. To appear
+in Statistical Science. ArXiv preprint arXiv:2105.02071.
+Robins, J. M., A. Rotnitzky, and L. P. Zhao (1994). Estimation of regression coef-
+ficients when some regressors are not always observed. Journal of the American
+statistical Association 89(427), 846–866.
+Rubin, D. B. (1974).
+Estimating causal effects of treatments in randomized and
+nonrandomized studies. Journal of educational Psychology 66(5), 688.
+Scharfstein, D., A. Rotnitzky, and J. Robins (1999).
+Rejoinder to comments on
+“adjusting for non-ignorable drop-out using semiparametric non-response models?”.
+Journal of the American Statistical Association 94, 1121–1146.
+Stein, E. M. (1970). Singular Integrals and Differentiability Properties of Functions
+(PMS-30). Princeton University Press.
+20
+
+Tan, Z. (2007). Comment: Understanding OR, PS and DR. Statistical Science 22(4),
+560–568.
+Tsiatis, A. (2006).
+Semiparametric theory and missing data.
+Springer Science &
+Business Media.
+van der Laan, M. and S. Rose (2011). Targeted Learning: Causal Inference for Ob-
+servational and Experimental Data.
+Springer Series in Statistics. Springer New
+York.
+Zhou, D.-X. (2020). Universality of deep convolutional neural networks. Applied and
+computational harmonic analysis 48(2), 787–794.
+21
+
+Table 4:
+Effect of early retirement on survival (binary outcomes death=1), during
+five years of follow up for the early retirees, and 95% confidence intervals.
+Women n=106916
+Men n=105149
+The first year after treatment is considered for the outcome
+naive
+0.002
+−0.001
+DRcnn
+0.005
+( 0.002,
+0.008) −0.001
+(−0.005, 0.003)
+DRmlp
+0.004
+( 0.000,
+0.007) −0.003
+(−0.013, 0.007)
+ORds
+0.003
+(−0.194,
+0.200)
+0.000
+(−0.150, 0.151)
+DRss
+0.003
+( 0.000,
+0.006)
+0.000
+(−0.002, 0.003)
+DRds
+0.003
+( 0.000,
+0.007)
+0.000
+(−0.002, 0.003)
+tmle
+0.003
+( 0.000,
+0.006)
+0.000
+(−0.002, 0.003)
+The second year after treatment is considered for the outcome
+naive
+0.002
+−0.002
+DRcnn
+0.007
+( 0.003,
+0.012)
+0.001
+(−0.005, 0.007)
+DRmlp
+0.006
+( 0.002,
+0.011)
+0.001
+(−0.003, 0.006)
+ORds
+0.005
+(−0.164,
+0.174)
+0.002
+(−0.130, 0.133)
+DRss
+0.005
+( 0.001,
+0.009)
+0.002
+(−0.002, 0.005)
+DRds
+0.005
+( 0.001,
+0.009)
+0.002
+(−0.002, 0.005)
+tmle
+0.005
+( 0.001,
+0.009)
+0.002
+(−0.002, 0.005)
+The third year after treatment is considered for the outcome
+naive
+0.000
+−0.002
+DRcnn −0.052
+(−0.166,
+0.063)
+0.005
+( 0.000, 0.010)
+DRmlp
+0.004
+(−0.001,
+0.009)
+0.008
+( 0.003, 0.013)
+ORds
+0.004
+(−0.150,
+0.159)
+0.004
+(−0.114, 0.122)
+DRss
+0.005
+( 0.000,
+0.009)
+0.004
+(−0.000, 0.009)
+DRds
+0.005
+( 0.001,
+0.010)
+0.005
+( 0.000, 0.009)
+tmle
+0.005
+( 0.000,
+0.009)
+0.004
+(−0.000, 0.009)
+The fourth year after treatment is considered for the outcome
+naive −0.003
+−0.006
+DRcnn
+0.002
+(−0.005,
+0.008) −0.001
+(−0.007, 0.005)
+DRmlp
+0.001
+(−0.005,
+0.006)
+0.005
+(−0.000, 0.011)
+ORds
+0.003
+(−0.140,
+0.145)
+0.002
+(−0.107, 0.110)
+DRss
+0.003
+(−0.002,
+0.008)
+0.002
+(−0.003, 0.007)
+DRds
+0.004
+(−0.001,
+0.009)
+0.002
+(−0.003, 0.007)
+tmle
+0.003
+(−0.002,
+0.008)
+0.002
+(−0.003, 0.007)
+The fifth year after treatment is considered for the outcome
+naive −0.005
+−0.007
+DRcnn
+0.002
+(−0.004,
+0.009)
+0.001
+(−0.006, 0.008)
+DRmlp
+0.009
+( 0.002,
+0.016)
+0.009
+( 0.002, 0.016)
+ORds
+0.001
+(−0.131,
+0.134)
+0.002
+(−0.098, 0.102)
+DRss
+0.002
+(−0.004,
+0.008)
+0.002
+(−0.003, 0.008)
+DRds
+0.003
+(−0.003,
+0.008)
+0.003
+(−0.003, 0.009)
+tmle
+0.002
+(−0.004,
+0.008)
+0.002
+(−0.003, 0.008)
+22
+
diff --git a/qtFKT4oBgHgl3EQfIC2S/content/tmp_files/load_file.txt b/qtFKT4oBgHgl3EQfIC2S/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7e5c4697cdd1111b47a87fdbb2a0436ace282448
--- /dev/null
+++ b/qtFKT4oBgHgl3EQfIC2S/content/tmp_files/load_file.txt
@@ -0,0 +1,1022 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf,len=1021
+page_content='Convolutional neural networks for valid  and efficient causal inference  Mohammad Ghasempour∗, Niloofar Moosavi, Xavier de Luna  Department of Statistics, USBE, Ume˚a University, Ume˚a, Sweden  January 30, 2023  Abstract  Convolutional neural networks (CNN) have been successful in machine learn-  ing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Their success relies on their ability to consider space invariant  local features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We consider the use of CNN to fit nuisance models in semipara-  metric estimation of the average causal effect of a treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' In this setting,  nuisance models are functions of pre-treatment covariates that need to be con-  trolled for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' In an application where we want to estimate the effect of early  retirement on a health outcome, we propose to use CNN to control for time-  structured covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Thus, CNN is used when fitting nuisance models explain-  ing the treatment and the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' These fits are then combined into an aug-  mented inverse probability weighting estimator yielding efficient and uniformly  valid inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Theoretically, we contribute by providing rates of convergence  for CNN equipped with the rectified linear unit activation function and com-  pare it to an existing result for feedforward neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We also show  when those rates guarantee uniformly valid inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' A Monte Carlo study  is provided where the performance of the proposed estimator is evaluated and  compared with other strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Finally, we give results on a study of the effect  of early retirement on hospitalization using data covering the whole Swedish  population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  ∗We are grateful to Xijia Liu and Jenny H¨aggstr¨om for helpful comments that have improved  the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  We acknowledge funding from the Swedish Research Council and the Marianne and  Marcus Wallenberg Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The Ume˚a SIMSAM Lab data infrastructure used in this study  was developed with support from the Swedish Research Council, the Riksbanken Jubileumsfond and  by strategic funds from Ume˚a University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Correspondence: mohammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='ghasemour@umu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='se  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='11732v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='ML]  27 Jan 2023  1  Introduction  Convolutional Neural Networks (CNN) have been found successful in discovering  location-invariant patterns in speech, images, and time series data (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=',  1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' In particular, they have been shown to have a universal approximation prop-  erty and be more efficient in terms of number of hidden layers than fully-connected  multi-layer networks in high-dimensional situations (Zhou, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' In this paper we  show how CNN can be useful in controlling confounding information when using rich  observational databases in order to perform semiparametric inference on a low di-  mensional causal parameter: we focus on average causal effects of a binary treatment  on an outcome of interest, although our results are relevant for the semiparametric  estimation of other low dimensional parameters of interest (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Chernozhukov  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Augmented Inverse Probability Weighting (AIPW) estimators (also called Double  Robust (DR) estimators, Robins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 1994) attain the semiparametric efficiency  bound and yield uniformly valid inference as long as the nuisance functions of the  confounding covariates are fitted consistenlty with fast enough convergence rates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  all nuisance functions are estimated with order n−1/4 (Belloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Farrell,  2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Kennedy, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Moosavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  In this paper, we contribute to this  theory by showing that CNN fits of nuisance functions achieve the n−1/4 convergence  rate required to obtain uniformly valid inference on causal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  To show  this we use a result obtained by Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2021) for general Rectified Linear  Unit (ReLU)-based Feed-forward Neural Networks (FNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' They show that, for large  samples, the estimation error rate of FNN are bounded by the following term, with a  probability increasing exponentially with γ:  �  log n/n × complexity penalty +  �  (log log n + γ)/n + approximation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  (1)  We deduce the above approximation rate for CNN architectures inspired by earlier  work by Zhou (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' However, in contrast to the latter paper, we consider a larger  number of free parameters by considering multi-channel convolutional neural network,  so as to achieve a trade-off between complexity penalty and approximation rate in  (1), and thereby obtain the convergence rate n−1/4 for the CNN fit of the nuisance  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' In the next section we formally define the causal parameters of interest  using the potential outcome framework (Rubin, 1974), and introduce the assumptions  yielding identification, and locally efficient and uniformly valid inference when using  AIPW estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We also introduce the convolutional network architectures with  which we propose to fit the nuisance functions used by AIPW, followed by our main  theoretical results, including conditions to obtain uniformly valid inference when using  CNN based AIPW estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Section 3 presents numerical experiments illustrating  2  the finite sample behaviour of this estimation strategy under different data generat-  ing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The proposed estimator is compared to AIPW, Targeted Maximum  Likelihood Estimation (TMLE) (van der Laan and Rose, 2011) and Outcome Re-  gression (OR) estimation (Tan, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Moosavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 2021) using fully-connected  feed-forward ReLU based neural networks (Multilayer Perceptron (MLP) in Farrell  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 2021) and Lasso to fit the nuisance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' In Section 4, we study the effect  of early retirement (at 62 years old), compared to retiring later in life, on morbidity  and mortality outcomes (Barban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We use population wide Swedish reg-  ister data and follow cohorts born in 1946 and 1947 for which we have a rich reservoir  of potential pre-treatment confounders, including hospitalization and income histo-  ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' CNN allows us to consider that such life histories may contain location-invariant  patterns that confound the causal effects of the treatment (decision to retire early).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  2  Theory and method  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='1  Causal parameters and uniformly valid inference  The Average Causal Effect (ACE) and Average Causal Effect among the Treated  (ACET) of a binary treatment (T) are parameters defined using potential outcomes  (Rubin (1974)), respectively:  τ = E(Y (1)) − E(Y (0)),  τt = E(Y (1) − Y (0)|T = 1),  where Y (1) and Y (0) are the outcomes that would be observed if T = 1 and T = 0,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  For a given individual only one of these potential outcomes can be  observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' This intrinsic missing data problem implies that assumptions need to be  made to identify τ and τt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' For this purpose, and given a vector of observed pre-  treatment covariates X, we assume:  Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' No unmeasured confounders:  (i) Y (0)  T | X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  (ii) Y (1)  T | X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Overlap:  (i) P(T = 0 | X) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  (ii) P(T = 1 | X) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Consistency: The observed outcome is Y = Y (1)T + Y (0)(1 − T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  3  Note that ACET is identified if only 1a(i), 1b(i) and 1c hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Assumption 1a  requires that the observed vector X includes all confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Assumption 1b requires  that for any value X both treatment levels have non-zero probability to occur, and  by Assumption 1c one of the potential outcomes is observed for each individual, and  its value is not affected by the treatment received by other individuals in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  We aim at uniformly (over the class of Data Generating Process (DGP)s, for which  Assumption 2 hold) valid inference while using semiparametric estimation (Moosavi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Therefore, the following AIPW estimators of ACE are considered  (Robins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Scharfstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 1999):  ˆτ = En[ ˆψ1(zi) − ˆψ0(zi)]  where En[·] is the empirical mean operator,  ˆψt(zi) = 1{ti = t}(yi − ˆµt(xi))  ˆP[T = t|X = xi]  + ˆµt(xi),  and ˆµt(x) is an estimator of µt(x) = E(Y (t) | X = x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  For ACET, we use the estimator  ˆτt = En[ ˆψ1,1(zi) − ˆψ0,1(zi)],  where  ˆψt,t′(zi) =  ˆP[T = t′|X = xi]  ˆP[T = t′]  1{ti = t}(yi − ˆµt(xi))  ˆP[T = t|X = xi]  + 1{ti = t′}ˆµt(xi)  ˆP[T = t′]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  We use the following assumptions for the DGPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We have  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Let {(yi, ti, xi), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' , n} be an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' sample from (Y, T, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Let U = Y (t) − µt(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' There is some r > 0 for which E  �  |µt (xi) µt′ (xi)|1+r�  and E  �  |ui|4+r�  are bounded, for given values of t and t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  The estimators of the nuisance functions have yet to be introduced for these AIPW  estimation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' In order to get the desired results, the proposed estimators  should be well behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' More precisely, the following consistency and rate conditions  are considered for the nuisance function estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Let ˆp(x) be an estimator of P[T = 1|X = xi] which only depends on  {xi, ti}n  i=1 (assumption of “no additional randomness”, Farrell, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Moreover, for  a given t we have  4  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' En[(ˆp(xi) − p(xi))2] = oP(1) and En[(ˆµt(xi) − µt(xi))2] = oP(1),  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' En[(ˆµt(xi) − µt(xi))2]1/2En[(ˆp(xi) − p(xi))2]1/2 = oP(n−1/2),  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' En[(ˆµt(xi) − µt(xi))(1 − 1{ti = t}/P[T = t|X = xi])] = oP(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  The following proposition describes uniform validity results obtained by Farrell  (2018, Corollary 2 and 3) under the above regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' For each n, let Pn be the set of distributions obeying Assumptions  1a(i), 1b(i), 1c and 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Further, assume Assumption 2b holds for t = t′ = 0, and let  ˆp(x) and ˆµ0(x) fulfill Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Then, we have:  sup  P∈Pn  ����PP  �  τt ∈  �  ˆτt ± cα  �  ˆVt/n  ��  − (1 − α)  ���� → 0,  where  ˆVt = n2  n2  t  En  �  1(ti = 1) (yi − ˆµ0 (xi) − ˆτt)2�  +n2  n2  t  En  �  ˆp (xi)2  (1 − ˆp (xi))21(ti = 0) (yi − ˆµ0 (xi))2  �  ,  nt = Σn  i=11(T = t) and cα = Φ−1(1 − α/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Let also Assumptions 1a(ii), 1b(ii) be fulfilled and Assumption 2b hold for t, t′ ∈  {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Additionally, assume ˆµ1(x) fulfills Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Then, we have:  sup  P∈Pn  ����PP  �  τ ∈  �  ˆτ ± cα  �  ˆV /n  ��  − (1 − α)  ���� → 0,  where  ˆV = En  �  1(ti = 1) (yi − ˆµ1 (xi))2  ˆp (xi)2  �  + En  ��  ˆµ1 (xi) − En[ ˆψ1(zi)]  �2�  + En  �  1(ti = 0) (yi − ˆµ0 (xi))2  (1 − ˆp (xi))2  �  + En  ��  ˆµ0 (xi) − En[ ˆψ0(zi)]  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' A multiplicative rate condition as Assumption 3(b) is weaker than sepa-  rate conditions on the two nuisance model estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' It only requires that one of the  nuisance functions is estimated at faster rate if the other one is estimated at slower  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' However, using the regularity conditions in this paper and Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2021),  the rate oP(n−1/4) is obtained for each of the nuisance estimators separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' This is  to make sure Assumption 3(c) for ˆµ is also fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' This assumption can be, however,  dropped by considering sample splitting (Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Note, finally, that because the AIPW estimators is based on the efficient influence  function, its asymptotic variance is equal to the semiparametric efficiency bound  (Tsiatis, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='2  Convolutional neural networks  We consider a specific CNN architecture with parallel hidden layers structured as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Let the input column vector be denoted by h0 := (x1, · · · , xd)′ ∈ Ω, where  Ω ⊆ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We consider L to be the number of hidden layers in which we have E number  of parallel vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Let σ be the ReLU function defined on the space of real numbers  as σ(z) = max(0, z) for z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The vectors in each hidden layer l ∈ {1, · · · , �L} have  size dl and are defined by hl  e = σ(W l  ehl−1  e  − be  l ), where e ∈ {1, · · · , E}, h0  e = h0, be  l  is a vector called bias vector in the neural network literature, W l  e is a weight matrix  defined as:  W l  e =  �  ���������������������  we  0,l  0  0  0  · ·  0  we  1,l  we  0,l  0  0  · ·  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  · ·  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  we  S,l  we  S−1,l  · ·  we  0,l  0 · · ·  0  0  we  S,l  we  S−1,l  · ·  we  0,l · · ·  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='  · ·  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  · ·  0  we  S,l  · ·  we  0,l  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  · ·  · ·  0  we  S,l · · ·  we  1,l  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  0  · ·  · ·  · ·  0  we  S,l  we  S−1,l  0  · ·  · ·  · ·  · · 0  we  S,l  �  ���������������������  ,  and the weight parameter we  i,l is the i-the element of the filter mask (we  0,l, · · · , we  S,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  S is considered fixed through all values of l and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We have dl = d0 + Sl, where dl is  the size of vectors in the l-th layer and d0 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  The structure of a CNN as defined above provides us with the following function  space on Ω:  C = {  E  �  e=1  c′  ehL  e : ce ∈ RdL}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Similar to Zhou (2020) (who, however, uses E = 1) we assume that for all values  of e in {1, · · · E} and l in {1, · · · , L−1} the vector be  l has the form (be  l,1, · · · , be  l,S, be  l,S+1,  · · , be  l,S+1, be  l,dl−S+1, · · · , be  l,dl)′ with the dl − 2S repeated components in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Therefore, the total number of parameters (duplicate/shared weights are counted  more than once), Q for the neural network fulfills the following inequality:  Q = E(d(1 + s)(L − 1) + s(1 + s)L(L − 1)/2 + (s + 3)(d + sL)) ≍ EL2,  (2)  where the notation xn ≍ yn for two sequences of random variables xn and yn means  xn = OP(yn) and yn = OP(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='3  Main results  In this section, we provide results that allow us to use the CNN architectures described  above to fit the nuisance models needed for AIPW estimations of ACE and ACET,  and obtain uniformly valid inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' In particular, we need to show that the rate  conditions in Assumption 3 are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Assume that zi = (ui, v′  i)′ , 1 ≤ i ≤ n are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  copies of Z =  (U, V ) ∈ [−1, 1]d × V, where U is continuously distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' For an absolute constant  M > 0 and a target f ∗ (a nuisance function), assume V ∈ [−M, M] and ∥f ∗∥∞ ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Let ℓ be a loss function that, for any arbitrary functions f and g  and z a realization of Z, fulfills:  |ℓ(f, z) − ℓ(g, z)| ≤ Cℓ|f(x) − g(x)|,  c1E((f − ˜f)2) ≤ E(ℓ(f, Z)) − E(ℓ( ˜f, Z)) ≤ c2E((f − ˜f)2),  where ˜f = arg min E(ℓ(f, Z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Further, let F be an arbitrary set of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We define:  ˆfF := arg min  f∈F  ∥f∥∞≤M′  n  �  i=1  ℓ(f, z),  (3)  and  εF,F′ := sup  f∈F  inf  f′∈F′  ∥f′∥∞≤M′  ∥f − f ′∥∞ ,  (4)  where ∥·∥∞ is the supremum norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The following result is a special case of Theorem  2(b) given in Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2021), and gives the rate of convergence for our estimate  ˆfC of a target f ∗ (a nuisance function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Prior to presenting the theorem, the target  space in which we seek to find the nuisance functions need to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We use the  notation W β,∞(Ω) for the Sobolev space with smoothness β ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We consider W to  be the unit ball on the Sobolev space defined on [−1, 1]d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' W := B1(W β,∞([−1, 1]d))  such as:  W := {f : max  α,|α|≤β ∥Dαf(x)∥L∞([−1,1]d) ≤ 1},  where ∥f∥L∞(Ω) is the essential supremum norm of the absolute value of a function f  defined on Ω, α = (α1, · · · , αd)′, |α| = α1 + · · · + αd, and Dαf is the weak derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  7  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Let f ∗ ∈ W, and assume that Assumptions 4-5 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' For an absolute  constant M > 0 and M ′ = 2M, let the approximation error εW,C be defined as in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  With probability at least 1 − e−γ, and for large enough n,  En[( ˆfC − f ∗)2] ≤ C  �  QL log Q  n  log n + log log n + γ  n  + ε2  W,C  �  ,  (5)  where the constant C > 0 is independent of n, but may depend on d, M, and other  fixed constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Our next step is to bound the term ε2  W,C in the above inequality on the estimation  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The following result is similar to Theorem 1 in Zhou (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' However, we allow  here the number of parallel layers to grow with n, which translates into faster growing  function space for our specific CNN architectures, thereby a larger decay rate for the  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Let 2 ≤ s ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' If L ≥ 2d/(s − 1) and f ∈ W, ∥f∥∞ ≤ M and an  integer index β > 2 + d/2, then there exist w, b and f w,b  L,E ∈ C such that  ���f − f w,b  L,E  ���  C([−1,1]d) ≤ c2d/2�  log(LE) 1  LE  1  2 + 1  d,  where c is an absolute constant and therefore  εW,C ≤ C  �  log(LE) 1  LE  1  2 + 1  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Finally, our main result gives conditions for the CNN nuisance function fit to fulfill  Assumption 3, and therefore conditions for AIPW estimation based on CNN fit to  yield uniformly valid inference for ACE and ACET (Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Let the conditions in Theorem 1 hold for ui = xi, vi = yi, and f ∗ = µt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Moreover the same conditions are fulfilled for ui = xi, vi = ti, and f ∗ = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Let  also E ≍ n  d  2d+4, L ≍ n  1  4d+8 log2(n), β = βp ∧ βµ, and s and d fulfill the assumptions  of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Then, the rate conditions in Assumption 3 are fulfilled for nuisance  functions estimators (3) with F = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The rate result for CNN based estimators in Theorem 3 corresponds to  similar results for MLP in Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2021, Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' However, the conditions  on smoothness is less strict in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The rate of growth for the number of pa-  rameters considered for MLP based estimators in Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2021, Theorem 1) is  n  d  β+d log5 n, while we have considered the growth rate n  d+1  2d+4 log4(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' These rates are  similar up to a log factor if d is large and β is close to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Notice, however, that these  are not necessarily minimal rates required for valid inference on ACE(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  8  3  Simulation Study  We perform a simulation study to evaluate the use of the CNN together with AIPW  estimation strategy of average causal effects proposed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We focus here on τt  (ACET) but similar results are obtained for τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Comparisons are made with other  algorithms to fit nuisance functions, including, MLP, (Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 2021) and post-  lasso estimators (Farrell, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Moreover, we also include alternative strategies to  AIPW, including OR estimation (Tan, 2007) and targeted learning (van der Laan  and Rose, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  The simulation results are based on 1000 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The neural networks initial  weights are different for each replication, which makes the results more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' A se-  quence of ten covariates are independently generated having normal distributions with  increasing means and varying variances: X1 ∼ N(100, 20), X2 ∼ N(102, 15), X3 ∼  N(105, 13), X4 ∼ N(107, 11), X5 ∼ N(109, 8), X6 ∼ N(110, 20), X7 ∼ N(112, 15),  X8 ∼ N(115, 13), X9 ∼ N(117, 11), and X10 ∼ N(119, 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' While these variables  are independent, their ordering is going to be important in how the outcomes and  treatment assignment are generated as described below in two different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Setting 1 Potential outcomes and treatment assignment are generated as follows:  Y (0) = 1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='001((X2−X1)2+(X4−X3)3+(X6−X5)2+(X8−X7)3+(X10−X9)2)+e0,  Y (1) = 2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='001((X2−X1)2+(X4−X3)3+(X6−X5)2+(X8−X7)3+(X10−X9)2)+e1,  P(T = 1|X) =1/(1 + exp(5 × 10−6((X2 − X1)2+  (X4 − X3)3 + (X6 − X5)2 + (X8 − X7)3 + (X10 − X9)2))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  In this setting, polynomial functions of the difference of some neighboring vari-  ables in the sequence are used to generate the nuisance models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Setting 2 Potential outcomes and treatment assignment are generated as follows:  Y (0) = 1 + (l1(X1, X2, X3, X4) + l2(X4, X5, X6, X7) + l3(X6, X7, X8, X9)) + e0,  Y (1) = 2 − (l1(X1, X2, X3, X4) + l2(X4, X5, X6, X7) + l3(X6, X7, X8, X9)) + e1,  P(T = 1|X) =1/(1 + exp(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='05X5−  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='1(l1(X1, X2, X3, X4) + l2(X4, X5, X6, X7) + l3(X6, X7, X8, X9)))),  where  l1(x, y, z, w) =  �  10,  if y/x, z/y, and w/z > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='15,  0,  otherwise,  9  E1  E2  E2  El-2  El-1  El-1  Flattened  Fully connected  Figure 1: Illustration of the convolutional neural network utilized in the simulations  and the real data study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Each layer l contains different number of channels El, where  E0 (not shown in the Figure) is the number of time series in the set of input covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Each channel (feature map) in layer l (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' the blue vector in the second layer) is  formed by first applying El−1 number of kernels (convolution filters) on the vectors  in layer l − 1 (as demonstrated by the blue arrow in the set of kernels applied on the  first layer) and then summing the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' There exist some covariates which are not  time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Inclusion of those vectors in the neural network is not illustrated in the  figure for simplification purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' A fully connected feed-forward structure has been  used for those covariates and the results are then combined with the results from the  time-series data in the flattening layer demonstrated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  l2(x, y, z, w) =  �  5,  if y/x, z/y, and w/z < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='05,  0,  otherwise,  l3(x, y, z, w) =  �  3,  if (y − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='1x)(z − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='1y)(w − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='1z) < 0,  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Here, the output of l1 and l2 are non-zero only if an increase by a factor bigger  than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='15 and a decrease by a factor bigger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='05 is found in all consecutive  pair of covariates in the input of size four, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The value of l3 is nonzero  if an increase by a factor bigger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='1 is followed by a decrease and followed  by another increase by a factor of bigger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='1 or if the oscillation has the  opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  The following algorithms to estimate the nuisance functions are evaluated:  CNN A convolutional neural network using the ReLU activation function, with ar-  chitecture illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Note that this architecture is slightly more  flexible than the one considered in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='2, hence, we expect it to perform  at least as well in terms of approximation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Both for the potential out-  come model µ0 and the propensity score p, the number of channels in the input  10  layer is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' For the outcome models 128 channels in the first hidden layer  and 16 channels in the second hidden layer have been used, while 32 channels  in the first hidden layer and 8 channels in the second hidden layer have been  used for fitting the propensity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We have implemented the CNN together  with AIPW estimation strategy in the package DNNCausal which is available at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='com/stat4reg/DNNCausal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  MLP A ReLU activation function based feedforward neural network with two hidden  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Two different feedforward architectures are used in order to estimate µ0,  and p (propensity score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The networks for fitting the outcome model consist of  128 and 80 nodes within the two layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' For the propensity score,  two layers contain 32 and 8 nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  post-lasso Nuisance models are fitted in two steps, where all higher order terms up  to order three are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' First, a lasso variable selection step is performed  using the R-package hdm (Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2016)), and then maximum  likelihood is used to fit the models with the selected covariates (those with co-  efficients not shrinked to zero by lasso).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Variable selection is performed using  two different strategies: double selection which takes the union of variables se-  lected by regressing outcome and treatment variables and uses that to refit both  models (Belloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Moosavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 2021), and single selection, which  utilizes two sets of variables selected by regressing outcome and treatment, and  refits each of the models using their corresponding set (Farrell, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  We use the above nuisance function estimators in several different estimators of τt,  as follows:  AIPW is implemented using CNN, MLP, and post-lasso with both single and double  selection (denoted respectively: DRcnn, DRmlp, DRss, DRds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  OR estimator with post-lasso and double selection (denoted ORds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  TMLE using the R-package tmle and the function SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='nnet to fit all nuisance func-  tions (denoted TMLE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Full details with code for this simulation study are available at the Github page:  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='com/stat4reg/Causal CNN/blob/main/README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  We set sample sizes to n = 5000 and 10000, and run 1000 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Bias,  standard errors (both estimated and Monte Carlo), mean squared errors (MSE, bias  squared plus variance) and empirical coverages for 95% confidence intervals are re-  ported in Table 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  The DGP of Setting 1 (Table 1) is the least complex one and its polynomial form  is favorable to the post-lasso methods which are based on higher order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' It is  11  Table 1: Setting 1 DGP: Bias, standard errors (both estimated, Est sd, and Monte  Carlo, MC sd), MSEs and empirical coverages for 95% confidence intervals over 1000  replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  bias  coverage  MC sd  Est sd  MSE  n = 5000  DRcnn  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='185  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='935  DRmlp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='970  DRss  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='971  DRds  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='384  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='951  n = 10000  DRcnn  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='951  DRmlp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='973  ORds  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='962  DRss  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='962  DRds  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='962  tmle  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='981  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='978  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='963  therefore expected that the post-lasso methods ORds, DRss, DRds have low bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  DRcnn has larger bias, but decreasing with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' All methods have similar MSEs even  though DRcnn has the lowest MSE for the largest sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Empirical coverages  are close to the nominal 95% level for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  The DGP of Setting 2 (Table 2) is less smooth and we observe that the DRcnn and  DRmlp methods have the lowest biases and MSEs, where DRcnn performs slightly  better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' TMLE and post-lasso both have such a large bias that the confidence interval  coverages are far from being at the nominal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' It should be noted, however, that  the TMLE uses the built-in feedforward neural network, which is not adjusted to the  generated datasets, and so the comparison is not a fair one between TMLE and the  DR estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  4  Effect of early retirement on health  The existing theoretical and empirical evidence on the effects of early retirement on  health are mixed, and both positive and negative results may be expected and have  been reported in different situations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' see Barban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2020) and the references  therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' In this context, we add to the empirical evidence by studying the effect of  early retirement on health using a database linking at the individual level a collection  12  Table 2: Setting 2 DGP: Bias, standard errors (both estimated and Monte Carlo),  MSEs and empirical coverages for 95% confidence intervals over 1000 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  bias  coverage  MC sd  Est sd  MSE  n = 5000  DRcnn  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='054  of socio-economic and health registers on the whole Swedish population (Lindgren  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' This allows us to follow until 2017 two complete cohorts born in 1946  and 1947 and residing in Sweden in 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' As health outcome we observe the number  of days in hospital per year after retirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' To study the effects of early retirement  we consider those who were still alive at age 62, and either retire at age 62 (T = 1,  treatment) or retire later (T = 0, control group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' For the cohorts studied, retirement  pensions are accessible from age 61, although the usual age of retirement is 65 years  of age (see Figure 2 for descriptive statistics on age of retirement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Therefore, retiring  at age of 62 is considered as early retirement since it decreases your annual pension  transfers compared to later retirement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Barban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2020) for details  on the Swedish pension system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Barban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2020) also contains a study on the  effects of early retirement, although based on fewer measurements for time-dependent  covariates and using a matching design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' This study provides new evidence by taking  into account richer histories of the time-dependent covariates, and by incorporating  CNN to address complex dynamics in pre-treatment confounding covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  More precisely, the design of the study is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' An individual alive at age  62 is considered as taking early retirement at that age if hers/his pension transfers  become larger than income from work at that age for the first time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=', they were  never so earlier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' For replicability, the exact definition using income and transfer  variables from the Swedish registers are available at  13  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='com/stat4reg/Causal CNN/blob/main/population description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  The health outcomes of interest are the number of hospitalization days per year dur-  ing the next five years following early retirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We, however, check first whether  early retirement has an effect on survival during the five first years following early  retirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' There were no (or hardly significant) such effects at the 5% level (results  reported in the appendix, Table 4) and we therefore focus the analysis on the sur-  vivors when looking at effects on hospitalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The following covariates are used as  input in the CNN architures to fit treatment and outcome nuisance models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Besides  the birth year we include the the following (pre-treatment) covariates measured at 61  years of age: marriage status, municipality, education level, Spouse education level,  and the number of biological children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Moreover, we consider the measurements of  the following covariates for each of the ten years preceding retirement: days of hos-  pitalization and open health care, annual income from labour, annual income from  pension, annual income from unemployment, annual income from early retirement  and sickness benefit, annual compensation for illness, and spouse retirement status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Thus, covariates include eight time(-structured) series of ten observations each per  individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' We do separate analyses for men and women which gives two samples of  approximately 100000 individuals each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  The code giving details on the CNN architectures and tuning parameters can be  found at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='com/stat4reg/Causal CNN/blob/main/population description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  We also apply the other methods evaluated in the simulation study above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Results are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Based on the naive estimation not controlling for  confounders, early retirement has a clear positive effect on health (varying between  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='167 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='396 average number of days) and for the five years of follow up considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  These naive effects are larger for women in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' These negative effects  disappear when controlling for the considered covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' This is seen most clearly  with CNN which yields the effects smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='1 day (in absolute value) for all  cases but one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Confidence intervals at the 95% level cover zero in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Thus,  while there appeared to be a positive effect on health of early retirement, this effect  was probably due to confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  5  Discussion  We have proposed and studied a semiparametric estimation strategy for average causal  effects which combine convolutional neural network with AIPW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  As long as the  conditions given are met, this strategy yields locally efficient and uniformly valid  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The use of CNN has the advantage over fully-connected feed forward neural  networks that they are more efficient on the number of weights (free parameters) used  14  0  250000  500000  750000  50  60  70  retirement age  count  0e+00  2e+05  4e+05  6e+05  8e+05  count  0  250000  500000  750000  1000000  50  60  70  retirement age  count  0  250000  500000  750000  1000000  count  Figure 2: Retirement age for men and women who belong to the cohorts born in  Sweden 1946 and 1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The last staple in each histogram displays those who retired  at age 71 or later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  15  Table 3:  Effect of early retirement on number of days in hospital during five years  of follow up, for the early retirees, and 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Women  Men  The first year  n=106437  n=104459  naive −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='167  DRcnn  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='110)  16  for approximating the nuisance functions, and are geared to take into account time  invariant local features in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  A main contribution of the paper is to show under which conditions CNN fits of  nuisance functions achieve n−1/4 convergence rate required to obtain uniformly valid  inference on an ACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Using the result in Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2021) in which convergence  rate of a ReLU-based feed-forward network is shown to follow Equation (1), we show  that the rate conditions in Assumption 3 are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Specifically, we use CNN  with a given complexity for which we show that approximation rate in (1) is small  enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' A key component of result in Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2021) is the upper bound on  empirical process terms found in Bartlett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2005) based on Rademacher averages,  which are measures of function space complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Global Rademacher averages do not  provide fast rates as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' To derive this fast rate, localization analysis is employed  which takes into consideration only intersection of the function space with a ball  around the goal function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' considering that in reality the algorithm only searches in  a neighborhood around the true function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Moreover, the tight bound on Pseudo-  dimension of deep nets found in Bartlett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' (2019) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  We also present numerical experiments showing the performance of the estimation  strategy in finite sample and compare it to other machine learning based estimation  strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The applicability of the method is illustrated through a population wide  observational study of the effect of early retirement on hospitalization in Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  6  Appendix  In this section we also consider the Sobolev space Hβ(Ω) that is defined using the L2  norm instead of the L∞ norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Moreover, let ∥f∥L2(Ω) be the L2 norm of a function  f defined on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='1  Proof of Theorem 2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Let Ω = [−1, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' For any α that satisfies |α| ≤ β, by H¨older’s inequality, we  have:  ∥Dαf∥L2(Ω) ≤ 2d/2 ∥Dαf∥L∞(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Therefore, f ∈ Hβ(Ω) and ∥f∥Hβ(Ω) ≤ 2d/2 hold by maxα,|α|≤β ∥Dαf∥L∞(Ω) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Moreover, Ω has a Lipschitz domain interior and measure zero boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Therefore,  by Sobolev extension theorem (Stein, 1970, Theorem 5) there is an extension F ∈  Hβ(Rd) of f for which  ∥F∥Hβ(Rd) ≤ C ∥f∥Hβ(Ω) ≤ C2d/2,  17  where C is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  By F ∈ Hβ(Rd) and the result in Klusowski and Barron (2018, Theorem 2), we  have  ∥F − Fm∥C(Ω) ≤ c0vF,2 max{  �  log m,  √  d}m− 1  2 − 1  d,  where m = E  �  (s−1)L  d  − 1  �  , Fm(x) = β0 + α0x + v  m  �m  k=1 βk(αkx − tk)+, βk ∈ [−1, 1],  ∥αk∥1 = 1, tk ∈ [0, 1], β0 = F(0), α0 = ∇F(0) and |v| ≤ 2vF,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Here, vF,2 :=  �  Rd ∥ω∥2  1 | ˆF(ω)|dω ≤ cd,β ∥F∥Hβ(Rd), where cd,β is the finite constant  ��∥ω∥2  1 (1 + |ω|2)−β/2��  L2  and ˆF(ω) is the Fourier transform of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Let F i  m := 1  Eβ0 + 1  Eα0x + v  m  �(i+1)m  k=im+1 βk(αkx − tk)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' It is shown in Zhou (2020),  using s ≥ 2 and L ≥ 2d/(s − 1), that F i  m|Ω belongs to the following function space  Cw,b  L  =  � dL  �  k=1  ckh(L)  k (x) : c ∈ RdL  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Therefore, by the definition of C, we can directly conclude that Fm|Ω belongs to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Let f w,b  L,E := Fm|Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Then, we have  ���f − f w,b  L,E  ���  C(Ω) = ∥F − Fm∥C(Ω)  ≤ c0vF,2 max{  �  log m,  √  d}m− 1  2 − 1  d  ≤ c0cd,β ∥F∥Hβ(Rd) max{  �  log m,  √  d}m− 1  2 − 1  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  (6)  The final result is obtained from the facts that 1  2(s − 1)LE ≤ md ≤ (s − 1)LE, s ≥ 2  and β > 2 + d/2 (cd,β is bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='2  Proof of Theorem 3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' The rate conditions (a) and (b) of Assumption 3 are fulfilled if both En[(ˆµt(xi)−  µt(xi))2] and En[(ˆp(xi)−p(xi))2] are oP(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' For this, the three terms in the upper  bound (5) in Theorem 1 must be oP(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' For the first term, based on 2, we have  QL log Q  n  log n ≍ EL3 log(EL2)  n  log n ≍ log6(n) log(n  d+1  2d+4 log4 n)  n  2d+5  4d+8  log n = oP(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  The second term is also oP(n−1/2) if γ = o(n1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' For the third term, using Theorem  2 we have  εW,C ≤  �  d+1  2d+4 log n + log log2 n  n  d+1  4d log  d+2  d n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  18  To prove condition (c) of Assumption 3, we use the proof of lemma Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  (2021, Lemma 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' In this proof it is shown that for an arbitrary class of feedforward  neural networks with probability at least 1 − exp(−n  d+3  4d+8 log6 n)  En  �  (ˆµt,C(xi) − µt(xi))  �  1 −  1{ti = t}  P[T = t|X = xi]  ��  ≤C  �  QL log Q  n  log n+  log log n + n  d+3  4d+8 log6 n  n  + ε2  W,C  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Therefore, we can conclude that under the stated conditions  En  �  (ˆµt(xi) − µt(xi))  �  1 −  1{ti = t}  P[T = t|X = xi]  ��  ≤ CEn[(ˆµt(xi) − µt(xi))2] = oP(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='3  Early retirement effects on survival  Additional results on the effect of early retirement on survival can be found in Table  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content=' (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  Estimating causal effects of treatments in randomized and  nonrandomized studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content='  Rejoinder to comments on  “adjusting for non-ignorable drop-out using semiparametric non-response models?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+page_content=' Singular Integrals and Differentiability Properties of Functions  (PMS-30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content=' Princeton University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
+page_content='  20  Tan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtFKT4oBgHgl3EQfIC2S/content/2301.11732v1.pdf'}
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+arXiv:2301.08597v1  [math.DS]  20 Jan 2023
+Dynamics of the fifth Painlev´e foliation
+E. Paul ∗
+J.P. Ramis†
+January 23, 2023
+Abstract. The leaves of the Painlev´e foliations appear as the isomonodromic deformations
+of a rank 2 linear connection on a moduli space of connections. Therefore they are the fibers of
+the Riemann-Hilbert correspondence that sends each connection on its monodromy data, and
+this correspondence induces a conjugation between the dynamics of the foliation and a dynamic
+on a space of representations of some fundamental groupoid (a character variety). This one
+can be identified to a family of cubic surfaces through trace coordinates. We describe here the
+dynamics on the character variety related to the Painlev´e V equation. We have here to consider
+irregular connections, and the representations of wild groupoids. We describe and compare all
+the dynamics which appear on this wild character variety: the tame dynamics, the confluent
+dynamics, the canonical symplectic dynamics and the wild dynamics.
+Contents
+Introduction
+3
+1
+The moduli space MV
+10
+1.1
+The local classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+1.2
+The global gauge moduli space
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+1.3
+The moduli space of connections MV
+. . . . . . . . . . . . . . . . . . . . . . . .
+12
+2
+The wild fundamental groupoid πV
+1 (X, S)
+13
+2.1
+Stokes operators, formal monodromy and exponential tori . . . . . . . . . . . . .
+13
+2.2
+The local wild fundamental groupoid
+. . . . . . . . . . . . . . . . . . . . . . . .
+16
+2.3
+The global wild fundamental groupoid . . . . . . . . . . . . . . . . . . . . . . . .
+18
+2.4
+A confluent morphism of groupoid
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+3
+The character variety χV
+20
+3.1
+Definition of χV
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+20
+3.2
+The character variety χV and cubic surfaces . . . . . . . . . . . . . . . . . . . . .
+21
+3.3
+Lines and reducibility locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+24
+3.4
+Log-canonical coordinates on CV (θ) and cluster sequences . . . . . . . . . . . . .
+28
+3.5
+The Laurent property
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+31
+3.6
+Families of confluent and diffluent morphisms . . . . . . . . . . . . . . . . . . . .
+32
+∗Institut of Mathematics of Toulouse, 118 route de Narbonne, 31062 Toulouse Cedex, France
+emmanuel.paul math.univ-toulouse.fr
+†Institut of Mathematics of Toulouse, 118 route de Narbonne, 31062 Toulouse Cedex, France
+Institut de France (Acad´emie des Sciences), France
+ramis.jean-pierre@wanadoo.fr
+1
+
+4
+The Painlev´e V vector field on MV .
+35
+4.1
+The Riemann-Hilbert map RHV
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+35
+4.2
+Isomonodromic families on MV . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+36
+4.3
+Singularities of the Painlev´e V vector field in the Okamoto’s compactification of
+MV (α)
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+37
+5
+Dynamics on χV
+39
+5.1
+The tame dynamics
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+39
+5.2
+The confluent dynamic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+40
+5.3
+The canonical dynamics on CV (θ) . . . . . . . . . . . . . . . . . . . . . . . . . . .
+43
+5.4
+Comparison between the confluent and the canonical dynamics . . . . . . . . . .
+48
+5.5
+Comparison with the wild dynamics
+. . . . . . . . . . . . . . . . . . . . . . . . .
+52
+6
+Conclusion and open questions
+55
+Appendix. Structure of the symplectic Cremona group
+56
+Bibliography
+58
+2
+
+Introduction
+The transverse dynamics of an holomorphic foliation is usually defined by its holonomy groupoid.
+In general we use a specialization of this groupoid along a particular leaf L: the holonomy group
+of L. This one is a representation of the fundamental group of the leaf on a germ of transversal,
+defined by the analytic continuation of the solutions. It only describes the transverse structure
+of the foliation in a neigbourhood of L.
+The Painlev´e foliations are holomorphic foliations defined by the Painlev´e equations. These
+ones are order two non linear ordinary differential equations. Their writings are available for
+example in [27]. We only recall here the Painlev´e VI and Painlev´e V families:
+(PVI)
+d2w
+dt2 = 1
+2
+� 1
+w +
+1
+w − 1 +
+1
+w − t
+� �dw
+dt
+�2
+−
+� 1
+w +
+1
+t − 1 +
+1
+w − t
+� dw
+dt
++ w(w − 1)(w − t)
+2t2(t − 1)2
+�
+(α4 − 1)2 − α2
+3
+t
+w2 + α2
+1
+t − 1
+(w − 1)2 + (1 − α2
+2) t(t − 1)
+(w − t)2
+�
+,
+(1)
+(PV ) d2w
+dt2 =
+� 1
+2w +
+1
+w − 1
+� �dw
+dt
+�2
+− 1
+t
+dw
+dt + (w − 1)2
+t2
+�α2
+1
+2 w − α2
+3
+2w
+�
++ (1 − α1 − 2α2 − α3)w
+t − 1
+2
+w(w + 1)
+w − 1
+.
+(2)
+They define families of one dimensional foliation on {(t, w, w′) ∈ C3}. They share with all the
+Painlev´e differential equations type three properties, which will allow us to give a more global
+and explicit description of their dynamics. Let us detail these properties for the Painlev´e VI
+foliation:
+1- The Painlev´e property : all the solutions (w(t), w′(t)) have a meromorphic continuation
+over the universal covering the time space T = P1(C) punctured at the fixed singularities 0, 1 and
+∞. This property generalizes the property of holomorphic extension of the solutions of linear
+differential equations. The search of non linear differential equations satisfying this property was
+the initial motivation of Painlev´e and then Gambier in order to construct new transcentendal
+functions. It gives rise to the six well known families of Painlev´e equations.
+According to this property, the movable singularities, i.e. the singularities of the solutions
+outside the fixed singularities defined by the equation itself, are only poles. K. Okamoto has
+introduced a semi-compactification process in order to get a foliation transverse to a fibration
+over the time space. After this process, one can define the dynamic of the Painlev´e foliation by
+its non linear monodromy, which is defined by an action of the fundamental group of the time
+space T, over the Okamoto fiber (the space of initial values). We call it the tame dynamics.
+2- The isomonodromic property.
+In order to study the non linear monodromy of this
+foliation, it is useful to introduce a conjugacy between this dynamical system and a simpler one:
+the Riemann-Hilbert map. Let us recall the different ingredients involved by this property for
+the Painlev´e VI equation.
+• The moduli space of connections MV I. We consider the family of linear connections on the
+trivial bundle over P1 defined the family SV I of rank 2 trace free differential systems
+(A) : dY
+dx =
+� 4
+�
+i=1
+Ai
+x − pi
+�
+· Y, pi ̸= pj for i ̸= j, Ai ∈ sl2(C),
+4
+�
+i=1
+Ai = 0, x ∈ P1,
+3
+
+up to the global gauge action: Y �→ Y · P, P ∈ SL2(C), and to a change of the independent
+variable x �→ ϕ(x), ϕ ∈ Aut(P1). The extension of the system to x = ∞ ∈ P1 is defined by the
+change of variable z = x−1. The family SV I is a 13 dimensional space (including the variables
+pi), and the above quotient is a 7 dimensional space. The local parameter space is defined by
+α(A) = (det(Ai) = α2
+i /4, i = 1, . . . 4),
+where ±αi/2’s are the eigenvalues of each residue matrix Ai. The time parameter is the cross
+ratio t ∈ T = P1 \{0, 1, ∞} of the configurations of the singular points S = {p1, p2, p3, p4}. Over
+a value α, the fiber MV I(α) is a 3 dimension space, and t(A) defines a fibration of MV I(α) over
+T.
+• The character variety χV I as representations of a fundamental groupoid. For a system A in
+SV I we can define its monodromy representation by using the analytic continuation of a local
+matrix solution Y0 along any element γ of the fundamental group π1(P1\S, x0). We consider here
+another (equivalent) description of this monodromy, introduced by the authors in [53], essential
+for what will follows.
+Instead of considering as usually a fundamental group, we consider a
+fundamental groupoid, which allows to make use of several base points. A groupoid is a small
+category whose morphisms are invertible. For details on groupoids see [13]. The fundamental
+groupoid π1(X) of a variety X is the groupoid whose objects are the points of X and the
+morphisms the paths between two points up to homotopy, with the obvious composition. In
+what follows, “paths” always means “path up to an homotopy”. We consider the variety X
+obtained by 4 real blowing up of the singular points pi in P1, and we choose a base point si on
+each divisor (i.e. a direction out of pi). Let S = {s1, s2, s3, s4}. The groupoid πV I
+1 (X, S) is the
+restriction of the fundamental groupoid π1(X) to this finite set of objects S. πV I
+1 (X, S) has a
+presentation given by the following picture:
+ 
+s1
+s2
+s4
+s3
+γ1,1
+γ2,2
+γ3,3
+γ4,4
+γ1,2
+γ2,3
+γ4,1
+γ3,4
+Figure 1: A presentation of the groupoid πV I
+1 (X, S).
+The morphisms are generated by the local loops γi,i based in si homotopic to each divisor and
+the paths γi,i+1 from si to si+1 satisfying the following relations:
+Rext : γ1,2γ2,3γ3,4γ4,1 = ⋆1 (the trivial loop based in s1)
+Rint : γ1,1γ1,2γ2,2γ2,3γ3,3γ3,4γ4,4γ4,1 = ⋆1.
+A linear representation ρ of πV I
+1 (X, S) is a morphism of groupoid from πV I
+1 (X, S) into the
+category of vector spaces. In the Painlev´e context, we only consider linear representations ρ
+4
+
+such that ρ(s) = Vs is a 2-dimensional vector space and ρ(γs,t) belongs to the set of linear
+morphisms which preserve the area.
+A choice of a basis in each Vs defines a map ρ(γs,t) ≃ Mγs,t in SL2(C). We obtain a representation
+of the groupoid πV I
+1 (X, S) in the group SL2(C). If we change the basis in each Vs, we obtain
+another representation ρ′ of πV I
+1 (X, S) in the group SL2(C) which is equivalent, i.e. for any s,
+there exist matrices Ps such that
+M′
+γs,t = P −1
+s
+· Mγs,t · Pt.
+Therefore a rank two trace free linear representation of πV I
+1 (X, S) is also a class of equivalent
+representations of πV I
+1 (X, S) in SL2(C).
+Definition 0.1 The character variety χV I is the categorical quotient 1 of the set of rank 2 trace
+free linear representations up to the above equivalence.
+The local data of an element ρ in χV I is the restriction of ρ to the groupoid πV I,loc
+1
+(X, S)
+obtained by forgetting the generators γi,i+1. It is characterized by the class of each ρ(γi,i), or
+by a = (ai = tr(ρ(γi,i), i = 1, ..., 4). We denote by χV I(a) the corresponding fiber in χV I.
+• The character variety as a family of cubic surfaces CV I. Given a presentation of the groupoid
+πV I
+1 (X, S) by figure (1), we define the trace coordinates (a, x) by
+ai(ρ) = tr(ρ(γi,i)), i = 1, · · · 4, xk(ρ) = tr(ρ(γi,iγi,jγj,j)), {i, j, k} = {1, 2, 3}.
+The trace coordinates do not change in a class of equivalent representations in SL2(C). Therefore
+it can be convenient to make use of normalized representations: A representation ρ of the
+groupoid πV I
+1 (X, S) in the group SL2(C) is normalized if for any indexes i, j, i ̸= j we have:
+ρ(γi,j) = I. One can always choose the representation of the objects in order to get a normalized
+representation by choosing arbitrarily the representation of a first one and by representing the
+further consecutive ones such that ρ(γi,i+1) = I. A normalized representation is given by 4
+matrices Mi = ρ(γi,i) unique up to a common conjugacy which satisfy, according to the relation
+Rint, M1M2M3M4 = I. We recover here the usual monodromy data. The trace coordinates of
+a normalized representation ρ are defined by
+ai = tr(Mi), i = 1, · · · 4, x1 = tr(M2M3), x2 = tr(M3M1), x3 = tr(M1M2).
+Proposition 0.2 The trace map TrV I defined by (a, x) sends each χV I(a) on the cubic surface
+CV I(θ) in C3 defined by the equation FV I(x, θ) = 0 with
+FV I(x, θ) = x1x2x3 + x2
+1 + x2
+2 + x2
+3 − θ1x1 − θ2x2 − θ3x3 + θ4,
+and θi = aia4 + ajak, {i, j, k} = {1, 2, 3}, θ4 = a1a2a3a4 + �
+i a2
+i − 4.
+From [31], this trace map is an homeomorphism only on an open set χ0
+V I(a) in the character
+variety. For a generic value of the local data a, we have χ0
+V I(a) = χV I(a). This condition implies
+that Mi ̸= ±I for i = 1, 2, 3.
+• The Riemann-Hilbert correspondence RHV I : MV I → χV I. Let A in MV I(α). We choose
+a representative of each base point si by a fundamental system of solutions Ys of A.
+The
+1This means that we consider the affine variety defined by the ring of invariants functions over the set of linear
+representations: see [12] or [61].
+5
+
+representation of a path γ from s to t is obtained by comparing the analytic continuation �Ys
+γ
+of Ys along γ with Yt:
+ρA(γ) = Mγ ⇔ Yt = �Ys
+γ · Mγ.
+A change of representation of the objects, or a gauge action on A gives an equivalent represen-
+tation. The Riemann-Hilbert map RHV I is defined by RHV I(A) = [ρA]. The map induced by
+RHV I on the local parameters is defined by
+RHloc
+V I(αi) = 2 cos(παi) = ai.
+We do not discuss here about the surjectivity of RHV I (the so called Riemann-Hilbert prob-
+lem) or its properness. For a survey on this wide subject, see [29] for the Painlev´e VI case, and
+[54] for the other cases.
+• Isomonodromic families and Painlev´e VI equation. An isomonodromic family on MV I is a
+fiber of the map RHV I, parametrized by a variable t, i.e. a family of connections defined by
+dY
+dx = A(x, t) · Y such that the monodromy representation is locally constant in χV I along this
+family. The relationship with the Painlev´e VI equation was discovered by R. Fuchs in 1907.
+Theorem 0.3 There exists coordinates w, w′, t over a Zariski open set in MV (α) such that the
+isomonodromic families are the solutions the Painlev´e VI equation (1).
+A sketch of proof [30].
+Given a fundamental system of solutions Y (x, t), by isomonodromy,
+d
+dtY (x, t) and Y (x, t) have the same behaviour along any continuation. Therefore the quotient
+B(x, t) = d
+dtY (x, t) · Y (x, t)−1
+is univalued outside the fixed singular points. Since the singular points are regular singular
+points (see section 2 for the definition) B extends to the singular set in a meromorphic way.
+Therefore Y (x, t) satisfies two rational linear systems dY
+dx = A(x, t)·Y and dY
+dt = B(x, t)·Y . The
+compatibility condition requires that the two operators d/dx − A and d/dt − B must commute :
+dA
+dt − dB
+dz + [B, A] = 0.
+(3)
+The pair (A, B) is called an isomonodromic Lax pair, and the above equation the Schelsinger
+equation. If A is irreducible, B is unique. Now, one can find coordinates such that the Schelsinger
+equation (3) is equivalent to the Painlev´e VI equation: see [30]. ✷
+3- The Hamiltonian property : Any Painlev´e differential equation is equivalent to a (non
+autonomous) hamiltonian system. This fact was first remarked by J. Malmquist [46], and ex-
+tended by K. Okamoto in [50]. For example the family of Painlev´e VI equations is equivalent to
+˙p = −∂HV I
+∂q
+, ˙q = ∂HV I
+∂p
+,
+(4)
+with
+HV I(α) = q(q − 1)(q − t)
+t(t − 1)
+�
+p2 − (α3
+q +
+α1
+q − 1 + α2 − 1
+q − t )p +
+β
+4q(q − 1)
+�
+.
+with β = (α1 + α2 + α3 + α4 − 2)(α1 + α2 + α3 − α4), and where the numbers ±αi/2 are the
+eigenvalues of the residues matrices Ai2.
+2The numbering is chosen here such that the two first indices correspond to the singularities 1 and t which are
+confluent in the usual confluent process of the Painlev´e equations.
+6
+
+We denote by PV I(α) the corresponding family of Painlev´e vector fields, and by FV I(α) the
+family of dimension one holomorphic foliations on C3
+(p,q,t) ⊂ C2 × P1 (where P1 is the complex
+one dimensional projective space). The fibers t = 0, 1, ∞ are singular and the foliation has 4
+singular points on each of this fibers.
+Remark 0.4 Painlev´e sixth equation was not discovered by Painlev´e and Gambier, using the
+Painlev´e property, but by Richard Fuchs, using an isomonodromic deformation. R. Fuchs consid-
+ered the monodromy preserving deformation of a second order Fuchsian differential equation with
+four regular singular points and an apparent singularity3 : Dy = y′′ + a1(x, t)y′ + a2(x, t) = 0,
+where a1 and a2 are rational functions on x depending holomorphically of t.
+We write the
+Riemann scheme of Dy = 0 :
+
+
+
+
+
+
+
+
+
+
+
+
+
+x = 0
+x = 1
+x = t
+x = q
+x = ∞
+0
+0
+0
+0
+κ0
+κ1
+κ2
+κ3
+2
+κ0 + κ4
+
+
+
+
+
+
+
+
+
+
+
+
+
+.
+In this scheme the exponents κi, i = 0, 1, 2, 3, 4 are complex parameters subject to the Fuchs
+relation :
+κ1 + κ2 + κ3 + κ4 + 2κ0 = −1.
+The κi, i = 1, 2, 3, 4 are related to the eigenvalues ±αi/2 of the Fuchsian systems (A) by
+κ1 = α1, κ2 = α2, κ3 = α3 and κ4 = α4 − 1.
+For a proof, see [29]. It is easy to derive the Hamiltonian formulation (cf. [48]). The variable q
+is the position of the apparent singularity, the variable p is defined by p = Resx=qa2(x, t)dx and
+the hamiltonian by (4).
+The family of Painlev´e V equations (2) is equivalent to
+˙p = −∂HV
+∂q , ˙q = ∂HV
+∂p ,
+(5)
+with HV (α) = t−1 (p(p + t)q(q − 1) − α1p(q − 1) + α2qt − α3pq) .
+We have used here the notations of [60]. The values
+alphai/2 are not here eigenvalues of the residues matrices of the connection but are directly
+related to them by an affine invertible map whose expression in available in the appendix C of
+[35].
+We denote by PV (α) the corresponding families of Painlev´e vector fields, and by FV (α) the
+family of dimension one holomorphic foliations on C3
+(p,q,t) ⊂ C2 × P1. There is here only two
+singular fibers over t = 0 and t = ∞, with 2 singular points over 0, and 5 singular points over
+∞, 3 of them are of saddle-node type i.e. with a vanishing eigenvalue.
+This construction provides a symplectic structure ΩV I(α) on MV I(α) given by dq ∧dp. This
+Poisson structure on MV I also comes from a general construction of Atiyah and Bott [1]: the
+symplectic structure on the infinite dimensional space of all the connections induces a Poisson
+structure by a symplectic reduction under the gauge action. There also exists several direct
+3Respectively x = 0, 1, t, ∞ and x = q.
+7
+
+finite dimensional constructions, by using either ciliated fat graphs [24], [2], or quasi-hamiltonian
+geometry and quivers varieties [10], or symplectic reduction of multi-Poisson structures [18].
+On the right-hand side, the space of linear representations χV I has also a Poisson structure
+which induces a symplectic form ωχV I(a) on each fiber χV I(a). This fact was first proved by
+Goldman in [26], and makes use of the Poincar´e-Lefschetz duality on the tangent space. It can
+be extended to irregular cases by using the concept of decorated character variety : [15], or
+quasi-hamiltonian geometry [9].
+There also exists a Poisson structure on the family of cubic surfaces CV I(θ) given by:
+ωV I(θ) = dx1 ∧ dx2
+FV I,x3
+= dx2 ∧ dx3
+FV I,x1
+= dx3 ∧ dx1
+FV I,x2
+,
+where FV I,xi = ∂FV I/∂xi = xjxk + 2xi − θi = ∂FV I/∂xi for i=1,2,3. This structure obtained
+by the Poincar´e residue of the volume form coincide with the Goldman structure: see [31].
+Finally K.Iwasaki has proved that the Riemann-Hilbert map RHV I is a Poisson morphism
+from (MV I, ΩV I) to (CV I, 2iπωV I): [32]. One can also find a proof of this fact in [40] obtained
+from the Jimbo’s asymptotic formula [34]. This result has been extended for the Riemann-Hilbert
+map in a local irregular case by P. Boalch: see [8].
+One can summarize these three properties for the Painlev´e foliation by the following state-
+ment: the dynamics of the Painlev´e VI foliation FV I(κ) is conjugated through the Riemann-
+Hilbert map to the dynamics on CV I(θ) induces by the automorphisms of πV I
+1 (X, S).
+Since the inner automorphisms γi,j �→ αi,iγi,jα−1
+j,j acts trivially on the character variety, this
+action factorizes to the quotient
+Out(πV I
+1 (X, S)) = Aut(πV I
+1 (X, S))/Inn(πV I
+1 (X, S))
+which is defined by pure braids over S in X. This braid group is generated by three elementary
+braids b1,2, b2,3 and b3,1 such that b1,2 · b2,3 · b3,1 = 1. The action of each generator on χV I has
+been first computed by Dubrovin and Mazzocco [23]. See also Iwasaki [31], or Cantat and Loray
+[14] or the authors in [53] for a simple computation using groupoids instead of groups. We have
+Proposition 0.5 The action of bi,j is given by the automorphism:
+hi,j :
+
+
+
+
+
+xi → −xi + xjxk + xix2
+k − θjxk + θi
+xj → −xj − xixk + θj
+xk → xk.
+This dynamics has the remarkable “tame property”: any element and its inverse is given by
+polynomials.
+Contents of this article.
+Our aim is to present the main tools that will allow us to
+extend this description for all the Painlev´e equations, by considering here the first case of PV .
+In this case, which can be obtained from PV I, by a confluent process, new difficulties arise:
+– The monodromy around the two singular fibers 0 and ∞ do not describe the whole trans-
+verse structure of the foliation. This monodromy will only give a “tame” dynamics generated
+by just one element. This is a consequence of the “irregular” type of the singularities of the
+foliation over ∞ which are now saddle-nodes, with non linear Stokes phenomenon.
+– On the other side, the class of connections on which PV naturally arises is now a class of
+irregular connections. The representation of such a connection must take into account beyond
+the usual monodromy, new operators related to such singular points: the linear Stokes operators
+and exponential tori.
+8
+
+We solve these problems by constructing a wild fundamental groupoid πV
+1 (X, S) with its
+linear representations (modulo equivalence):
+the wild character variety χV .
+The action of
+Out(πV
+1 (X, S)) allows us to recover the tame dynamics.
+In order to complete this dynamics, one can construct a family of (non invertible) confluent
+morphisms from πV
+1 (X, S) to πV I
+1 (X, S). The key point is that these morphisms induce families
+of birational maps between the corresponding character varieties. This will allow us to define
+and compute a confluent dynamics Conf(PV ) on χV . This one was previously found by Martin
+Klimes in [40]. Our technique based here on wild fundamental groupoids allows us to expect a
+generalization for all the other Painlev´e equations. Notice hat the dynamics that we obtain is
+no longer polynomial but only a rational one.
+We will also discuss about a canonical dynamic which exists on the cubic surfaces CV , with-
+out any reference to a Painlev´e equation.
+The central idea is that here the cubic surface is
+symplectically birationally equivalent to
+�
+C2, du
+u ∧ dv
+v
+�
+by a sequence of log-canonical coordi-
+nates which satisfies some exchange relations, which appear in cluster algebra. The pull-back
+of the symplectic Cremona group Symp
+�
+C2, du
+u ∧ dv
+v
+�
+delivers a very rich structure denoted by
+Dyn(CV ). We will compare Conf(PV ) and Dyn(CV ).
+The wild dynamics of the Painlev´e V foliation is defined by the non linear monodromy, non
+linear Stokes operators and non linear tori in a neighborhood of a saddle-node singular point.
+Following M. Klimes [40], we will recall the definition of this dynamics and its image through the
+Riemann-Hilbert map. We also recover here the canonical symplectic dynamics, thus confirming
+in this case the rationality conjecture of the second author.
+The last part is dedicated to open problems.
+Acknowledgments.
+We would like to thank Martin Klimes for the discussions we had
+together on this subject. His work [40] is a primary source of inspiration for this article.
+9
+
+1
+The moduli space MV
+According to [54], the moduli space of connections on which lives the Painlev´e V foliation is
+defined by:
+SV =
+�dY
+dx = A(x) · Y, A(x) =
+A0
+x − s0
++
+A1
+x − s1
++ A∞, A0, A1, A∞ ∈ sl2(C)
+�
+such that A∞ is a non trivial semi-simple element. The singularities, i.e. the points s such that
+A is not holomorphic around s are s0, s1 and ∞. We suppose that the eigenvalues ±t of A∞ are
+distinct (t ̸= 0). One can extend the basis to P1(C) by setting z = x−1:
+dY
+dz = −z−2A(z−1) · Y.
+As explained in the introduction, we consider this family up to a gauge action Y = PZ, which
+acts on the system by A → AP = P −1AP − P −1 dP
+dx , and up to a change of independent variable
+ϕ(x), ϕ in the M¨obius group Aut(P1).
+1.1
+The local classification
+Suppose that x = 0 is a singular point of a germ of connection defined by
+xp+1 dY
+dx = A(x) · Y, A ∈ sl2(C{x}), A0 = A(0) ̸= 0, p ≥ 0.
+The local formal classification is the classification of such a system up to a gauge transformation
+Y = PZ, with P in SL2 (C((x))). The integer p is called the Poincar´e rank of the system. It is
+not a local gauge invariant. If p = 0 the singular point is called a fuchsian singularity. Given
+an element of SV , the points s0 and s1 are fuchsian singularities. The local formal classification
+around such a point is given by the following general result (cf [64]):
+Proposition 1.1 We consider the fuchsian system xY ′ = A(x) · Y, A0 = A(0) ̸= 0.
+1. Suppose that the eigenvalues of A0 do not differ from a positive integer (the non resonant
+case). There exists a local meromorphic gauge transformation P in SL2 (C((x))) which
+conjugates the fuchsian singular system to xZ′ = A0 · Z
+2. In this case, the system (1) admits a local fundamental solution Y = P(x)xA0, P in
+SL2 (C((x))). Furthermore if A is a convergent data, then P is also a convergent matrix.
+The general case reduces to the non resonant case by using shearing transformations, which can
+eventually modify the Poincar´e rank. We also obtain a local fundamental system Y = P(x)xL
+for some constant matrix L. In any cases, the sectoral local solutions admits a moderate growth :
+fuchsian singularities are regular singular points. The classification of non fuchsian singularities
+is given by the Hukuhara Turritin Theorem [3] :
+Theorem 1.2 We consider a sl2-system zp+1Y ′ = A(z).Y, A0 = A(0) ̸= 0, p ≥ 1. There exists
+a formal meromorphic gauge transformation Y = PZ and a ramification z = yk such that the
+differential system has the canonical form
+yZ′ = (L + D1y−1 + · · · + Dmy−m) · Z,
+where the Di’s are diagonal matrices, and L commutes with the Di’s.
+10
+
+If the irregular part D1y−1 + · · · + Dmy−m vanishes, we still get a regular singular point. Other-
+wise, the singular point is irregular, and the degree r = m/k is called the Katz invariant of the
+irregular singular point.
+Remark 1.3 If the eigenvalues of A0 are distincts, we do not need to use a ramification z = yk.
+Otherwise, for example if A0 is a nilpotent element of sl2(C), we first have to use shearing
+transformations in order to modify the leading term A0, which may change the Poincar´e rank
+and requires a ramification. This may occur for some Painlev´e families.
+Given an element A of SV , the singular point x = ∞ (z = 0) is an irregular singular point
+with Katz rank 1. Since the eigenvalues ±t of A∞ are distinct, we do not need here a ramification
+of z. The explicit computation of the residue matrix L∞ gives:
+L∞ =
+
+
+
+a0 + a1
+0
+0
+−a0 − a1
+
+
+ .
+Therefore the system (A) has a formal fundamental matrix solution around z = 0 given by
+�Y (z) = P(z)zL∞ exp Q(z), Q(z) = diag(αz−1, −αz−1).
+1.2
+The global gauge moduli space
+We want to compute the categorical quotient SV //Sl2(C), i.e the affine variety defined by the
+ring of invariant functions under the action of the constant gauge action of Sl2(C). One can
+first use a constant gauge transformation in order to diagonalize A∞:
+A∞ =
+
+
+
+t/2
+0
+0
+−t/2
+
+
+ , Ai =
+
+
+
+ai
+bi
+ci
+−ai
+
+
+ , i = 0, 1.
+Let S′
+V be the subset of SV defined be the above writing. For a fixed singular set s0, s1, ∞, it is
+a 7-1=6 dimensional variety. The remaining gauge action which normalizes the diagonal matrix
+A∞ is the group N =< T, P > generated by
+T = {Tm =
+
+
+
+m
+0
+0
+m−1
+
+
+ , m ∈ C∗} and P =
+
+
+
+0
+1
+−1
+0
+
+
+ .
+This group acts on sl2(C) by
+Tm ·
+
+
+
+a
+b
+c
+−a
+
+
+ =
+
+
+
+a
+bm2
+cm−2
+−a
+
+
+ , P ·
+
+
+
+a
+b
+c
+−a
+
+
+ · P −1 =
+
+
+
+−a
+−c
+−b
+a
+
+
+ = −AT .
+After this first reduction, SV //SL2(C) = S′
+V //N. The following functions are invariant by the
+action of N:
+α0 = a2
+0 + b0c0
+α1 = a2
+1 + b1c1
+α∞ = (a0 + a1)2
+τ = a0t
+β0 = b0c1 + b1c0
+β1 = t(b0c1 − b1c0)
+(6)
+11
+
+Notice that the three coordinates α = (αi), i = 0, 1, ∞ are local invariants around each singular
+point. As mentioned in the introduction the αi parameters used in the expressions of the Painlev´e
+V equation and in its hamiltonien HV are not exactly equal to the present parameters but only
+related to the eigenvalues of the Ai by an affine invertible map : see [35].
+Proposition 1.4 SV //SL2(C) = Spec(α0, α1, α∞, τ, β0, β1).
+Proof.
+It suffices to prove that the map (α0, α1, α∞, τ, β0, β1) :
+S′V //SL2(C) → C6 is
+invertible over the Zariski open set a0 ̸= 0, b0c0 ̸= 0. Given a value (α0, α1, α∞, τ, β0, β1), we
+first choose arbitrarily t ̸= 0. Now a0 is uniquely determined by ta0 = τ, and b0c0 is uniquely
+determined by α0−a2
+0. The values a2
+0 and b0c0 determine a unique class [A0] modulo N. It suffices
+to prove that the choice of A0 in this class determines in a unique way the triple (A0, A1, A∞).
+A∞ is uniquely defined by a0t = τ. The two last equations define a linear system in (b1, c1)
+which admits a unique solution for b0c0 ̸= 0. The relation
+2a0a1 = α2
+∞ − α2
+0 − α2
+1 + b0c0 + b1c1
+determines a unique value of a1 for a0 ̸= 0. Another choice of A0 in [A0] will give an equivalent
+triple (A0, A1, A∞) modulo N. ✷
+This quotient is endowed with a Poisson structure whose Casimir functions are α0, α1, α∞
+and τ.
+Therefore, each fiber has a canonical symplectic structure. See the introduction for
+references.
+1.3
+The moduli space of connections MV
+We now consider the action of a global change of independent variable. We have previously
+fixed one singular point at ∞. The positions s0 and s1 are now free parameters. One can use a
+translation x → x+λ in order to get s0 = 0. This action do not modify A0 A1 and A∞. Now we
+want to normalize the singular point s1 to the value 1, by using the last change of independent
+variable that we can use: x �→ µx. Since s1 ̸= s0 = 0, we can introduce the parameter s−1
+1 . The
+linear system
+dY = (A0 +
+A1
+1 − s1x−1 + xA∞)dx
+x ) · Y
+is changed in
+dY = (A0 +
+A1
+1 − s1µ−1x−1 + xµA∞)dx
+x ) · Y.
+This action do not modify A0 and A1 but modify A∞ in µA∞. Therefore it keeps invariant the
+local variables αi and modify the others variables by:
+(τ, β0, β1, s−1
+1 ) → (µβ0, β1, µβ2, µs−1
+1 ).
+The quotient of this action is the weighted projective space P3
+(1,0,1,1). The usual choice of µ such
+that s1 = 1, corresponds to a choice of chart in P3
+(1,0,1,1). Therefore
+Proposition 1.5 The moduli space of connections MV (α) induced by SV for fixed local invari-
+ants is isomorphic to P3
+(1,0,1,1).
+Remark 1.6
+1. This space is identical to that obtained by H. Chiba in [17] when he seeks a
+natural compactification on which lives the vector field PV by using its Newton polyhedra.
+12
+
+2. Since one weight vanishes, the quotient space is isomorphic to P2(C) × C, and it is not a
+compact space. This fact is shared with the quotient spaces corresponding to the equations
+PV I and PIII,D6, while we obtain a compact weighted projective space for the other families
+(I), (II), (IV ), (III, D7), (III, D8): [16]. To deal with this problem, we will use the
+trick introduced by H. Chiba, who introduces two copies of the previous space, glued by a
+convenient B¨acklund transformation.
+2
+The wild fundamental groupoid πV
+1 (X, S)
+The fundamental group of π1(P1 \{s0, s1, s∞}, b0) acts on a local fundamental system Y0 around
+b0 by analytic continuation of Y0 along a path in P1 \ {s0, s1, s∞}. We first enlarge this usual
+monodromy representation by considering new operators around the irregular point s∞: the
+formal monodromy, the Stokes operators and the exponential torus. We first describe theses
+operators in the present context, by using a point of view very similar to that introduced by
+Stokes himself in [58], starting from a formal solution and using the summability theory.
+2.1
+Stokes operators, formal monodromy and exponential tori
+Definition 2.1
+1. Let k > 0. A formal power series �f = �
+i≥0 aixi is 1/k-Gevrey if there
+exists M > 0 and A > 0 such that for all i |ai| ≤ MAi(i!)1/k.
+2. Let V be a sector at the origin and f an holomorphic function on V . f is 1/k-Gevrey
+asymptotic to �f if for every strict subsector W in V , there exists MW > 0 and AW > 0
+such that for all n ≥ 1
+| (f(x) −
+n−1
+�
+i=0
+aixi |≤ MW An
+W (n!)1/k|x|n.
+We denote by C[[x]]1/k the algebra of series which are 1/k-Gevrey, A1/k(V ), the algebra
+of holomorphic functions on V which are 1/k-Gevrey asymptotic to some formal series. We
+mention here the following facts (for the proofs see [44]):
+- If f is 1/k-Gevrey asymptotic to �f, then �f is a 1/k-Gevrey series. Therefore the Taylor
+map T : A1/k(V ) → C[[x]]1/k is well defined.
+- For V “narrow” i.e. with an opening ≤ π/k, the Taylor map T is surjective: this is a
+Gevrey extension of the Borel-Ritt theorem.
+- For V “large” i.e. with an opening > π/k, the Taylor map T is injective: this is a Gevrey
+extension of the Watson Lemma.
+- Any formal solution of any linear or non linear analytic differential equation is 1/k-Gevrey
+for some k. This a theorem of Maillet. In the linear case, the second author gives the
+optimal Gevrey index by using the Newton polyhedra. In particular, for a linear system
+xk+1dY/dx = A(x) · Y , this optimal order is the Katz rank of the system.
+Definition 2.2
+1. Let d ∈ S1 a direction.
+A formal power series �f = �
+k≥0 aixi is k-
+summable in the direction d if there exists a sector Vd bisected by d, whose opening is
+greater than π/k and an holomorphic function f on V which is 1/k-Gevrey asymptotic to
+�f.
+13
+
+2. A formal power series �f is k-summable if �f is k-summable in any direction d excepted a
+finite number of directions (the singular directions).
+Let C{x}1/k,d ⊂ C[[x]]1/k be the algebra of k-summable series in the direction d, and C{x}1/k ⊂
+C[[x]]1/k the algebra of k-summable series. Since Vd is a large sector, the summation operator
+Σd : C{x}1/k,d → A(Vd) is an injective morphism of C-differential algebra.
+Theorem 2.3 Let x2dY/dx = A(x)Y , A(x) in sl2(C{x}), be a germ at x = 0 of meromorphic
+differential system. We suppose that the eigenvalues of A(0) are distincts (which avoids a rami-
+fication of the independent variable), and that the singularity is irregular, with a Katz rank equal
+to 1.
+1. The formal series appearing in a formal fundamental system of solutions �Y are 1-summable.
+2. For any non singular direction d, the operator Σd extends to a unique differential morphism
+from the algebra C{x}1[xλ, x−λ, exp t/x, exp(−t/x)] into O(Vd), such that this morphism
+induces the identity map on C[xλ, x−λ, exp t/x, exp(−t/x)]. This allows us to define Σd(�Y )
+for a formal matrix solution �Y .
+3. Given a formal fundamental system of solutions
+�Y (x) = P(x)xL exp Q(x), Q(x) = diag(tx, −tx),
+the singular directions d on which Σd(�Y ) is not defined are characterized by arg(x) = d if
+and only if exp(2tx) has a maximal decay, i.e. by tx ∈ R−.
+Let Σ be the set of singular directions. The Stokes operators {Sd, d ∈ Σ} are defined in the
+following way. We choose a regular direction d0. For any singular direction d, we choose two
+regular directions d− and d+ such that the arc (d−, d+) contains the only one singular direction
+d. We choose a determination �Y0 of �Y around a d0. This choice induces a determination �Y − of �Y
+around the direction d− (resp. a determination �Y + of �Y around d+) by analytic continuation of
+the logarithm along the positive arc (d0, d−) (resp. (d0, d+)). The summations Y − = Σd−(�Y −)
+and Y + = Σd+(�Y +) are defined on large sectors V −
+d and V +
+d which intersects non trivially on an
+open sector around d. Therefore there exists some constant matrix Sd such that Y + = Y −Sd.
+One can easily check that Sd does not depend on the choices of d+ and d− around d. A change
+of choice for �Y or for its determination �Y0 around a d0 will modify the family {Sd, d ∈ Σ} by a
+common conjugation.
+Remark 2.4 Starting from a formal solution �Y0, the Stokes operator Sd are defined by the
+successive operations: analytic continuation of the formal solution �Y0 along an arc from d0 to
+d−, summation at d−, analytic continuation of Y − until d+ on V + ∩ V −, anti-summation at d+
+(i.e. Taylor expansion of the actual solution Y +) and finally analytic continuation of the formal
+solution �Y + from d+ to d0.
+Definition 2.5
+1. The Stokes operators at each singular direction are defined by the above
+composition of operators. Given a formal solution �Y0, they are characterized by matrices
+Ud, up to a common conjugation.
+2. The formal monodromy is the operator generated by a loop �γ around 0 acting on a de-
+termination of �Y .
+For a given determination �Y0, it is defined by a matrix denoted by
+�
+M.
+14
+
+3. The geometric monodromy is the operator generated by a loop γ around 0 acting on a
+determination of an actual solution Y . For the actual solution Y0 obtained by summation
+of �Y0, it is defined by a matrix denoted by M.
+4. The (formal) exponential torus is an action of the algebraic group C∗ on �Y0 induced by
+a rescaling of the exponential map: exp q → τ exp q. If �Y0 = PxL exp Q is chosen such
+that Q is diagonal, the action of τ is given by a diagonal matrix Tτ = diag(τ, τ −1). In the
+general case Tτ is a Cartan component of SL2(C)
+The motivation for the introduction of the exponential torus action is given by the following
+heuristic interpretation in a one dimensional context: we consider the equation x2y′ + y = 0.
+We introduce an unfolding of the irregular singularity at the origin : we replace D = x2d/dx + 1
+by Dε := x(x − ε)d/dx + 1 (ε ∈ C∗). Then the irregular singularity is replaced by two regular
+singularities at 0 and ε, with respective exponents 1/ε and −1/ε. An important point to notice
+is the coupling between the relative positions of the singularity and the exponents. The general
+solution of Dεy = 0 is y = Cx1/ε(x − ǫ)−1/ε. It is invariant by the monodromy action of a
+simple loop around the two points 0 and ε but the monodromy action of a simple loop around
+0 passing between the two points will transform f into e−2iπ/εf. When ε → 0, this monodromy
+action does not have a limit. However we can replace the continuous limit by a discrete limit.
+We fix ε0 ∈ C∗ and we define a sequence (εn)n∈N by
+1
+εn :=
+1
+ε0 + n, n ∈ N. Then the sequence
+(e−2iπ/εn)n∈N is constant and we can interpret f �→ fτ0, with τ0 := e−2iπ/ε0 as the action of a
+simple loop passing between two infinitely near points4. As the choice of ε0 is arbitrary, then τ0
+is arbitrary in C∗ and we can consider the exponential torus as a generalized monodromy group.
+This group is no longer discrete, it is an algebraic group of dimension one. The “monodromy of
+a loop between the two infinitely near singularities” can be interpreted as a “random point” in
+this group. In the unfolding bifurcation Dε there is a breaking of symmetry, the choice of ε fix
+a point e−2iπ/ε ∈ C∗ and the exponential torus is replaced by the ordinary monodromy group
+generated by f �→ e−2iπ/εf. The random point is replaced by a true point.
+Proposition 2.6
+1. Let d1, d2 = d1+π be the 2 singular directions of the differential system.
+We have M = �
+M · Ud2 · Ud1.
+2. If �Y0 = PxL exp Q is chosen such that Q is diagonal, Ud1 is a lower triangular unipotent
+matrix and Ud2 is a upper triangular unipotent matrix. In the general case, Ud1 and Ud2
+are in the two Borel components of the Cartan component of the exponential torus.
+3. In the present case (a non ramified case), the formal monodromy commutes with the expo-
+nential torus (and belongs to it). This fact is no longer true in the ramified case.
+A Borel-Cartan configuration is a triple of subgroups (B−, C, B+) in SL2(C) such that C is a
+maximal torus, isomorphic to the algebraic group C∗, and B− and B+ are two maximal solvable
+subgroups which contains C. Any Borel-Cartan configuration is conjugated to the canonical
+one given by (T −, D, T +) where D is the subgroup of diagonal matrices, T − (resp. T +) the
+subgroup of lower (resp. upper) triangular matrices. The unipotent subgroups of B− and B+
+are respectively denoted by U − and U +.
+The triple (Ud1, �
+M, Ud2) and the triples (Ud1, Tτ, Ud2) take their values in a unipotent Borel-
+Cartan configuration (U −, C, U +).
+The local wild dynamics of the irregular connection is the subgroup (well defined up to
+a conjugation) of SL2(C) generated by the Stokes operators, the formal monodromy and the
+4The idea of considering an irregular singularity as a pack of infinitely near regular singularities is due to Ren´e
+Garnier in 1919 [25]. It will be used as a guideline in our work.
+15
+
+exponential torus. According to a result of the second author [55], the Zariski closure of the
+local dynamics in SL2(C) is the differential Galois group of the local linear differential system.
+For the study of isomonodromic deformations, we need a parametric version of Theorem
+(2.3). They are parametric variants of Gevrey power series and of Gevrey expansions (with
+analytic parameters). One supposes that the estimates in definition (2.1) are uniform on the
+parameter space U ⊂ Cp, or more generally on every compact K of U : one replaces MW and
+AW by MW,K and AW,K (resp. uniform on the parameter space). For an open set U ⊂ Cp,
+we will denote O(U)[[x]]1/k the algebra of Gevrey series of order 1/k uniformly on U. They
+are also a similar parametric variant of k-summability using the uniformly parametrized 1/k-
+Gevrey expansions : the definition is the same mutatis mutandis. We will speak of uniform
+k-summability. Then the sums of the uniformly k-summable series in a direction d are analytic
+in the variable and in the parameters.5 For more information on these subjects cf. [57].
+Theorem 2.7 Let U ⊂ Cp be an open subset.
+Let D ⊂ C be an open disc centered at 0.
+Let [A] :
+x2dY/dx = A(x, t)Y , where A ∈ sl2 (O(D × U)), be a parametrized meromorphic
+differential system. We suppose that, for all t = (t1, . . . tp) ∈ U, the eigenvalues of A(0, t) are
+distinct and that the singularity is irregular, with a Katz rank equal to 1.
+(i) Locally on U, the system [A] admits a formal fundamental solution analytic in the param-
+eters :
+ˆF = P �HxLeQ, where P ∈ SL2 (O(V )), �H ∈ SL2 (O(V )[[x]]1), �H(0, t) = I (for
+every t ∈ V ), L ∈ sl2(C), Q = (q, −q), with q ∈ x−1O(V )[x−1].
+(ii) Let t0 ∈ U and V ⊂ U a neighborhood of t0 such that the system admits on V a formal
+fundamental solution analytic in the parameters as above. Let t1 ∈ U and d a nonsingular
+direction for the system x2dY/dx = A(x, t1)Y . Then there exists a neighborhood W ⊂ V
+of t1 such that P �H is uniformly 1-summable in the direction d on W.
+Proof. (i) Let t0 ∈ U. If an open polydisc V centered at t0 is sufficiently small, then there
+exists P ∈ GL (O(V )) conjugating the restriction of A0 = A(0, t) to V to a diagonal matrix B0.
+Therefore we are reduced to the case where A0 is diagonal. Then we can use a parametrized
+variant of the splitting lemma of [3]. It reduces the system to two parametrized formal one
+dimensional equations. The result follows easily.
+(ii) A first proof. We use a parametrized generalization of the Improved Splitting Lemma of
+[3] (8.1, Lemma 11, page 124). The proof is the same mutatis mutandis6.
+A second proof. We imitate a proof of the unparamatrized version based on Gevrey asymp-
+totics (cf. [44]), replacing the Gevrey Main Asymptotic Existence Theorem by a parametrized
+version. ✷
+2.2
+The local wild fundamental groupoid
+Following an idea which appears in a correspondence between P. Deligne B. Malgrange and the
+second author [19], the Stokes operators, formal and geometric monodromy around an irregular
+point appear as a representation of a “local wild fundamental groupoid”. We want to encode
+all these operators by loops, or paths if we allow several base points, in a “halo” (see also [44])
+around the singular point whose internal boundary describes the formal world (represented by
+5when these sums exist for all values of the parameters.
+In general the singular directions move with the
+parameters and it is necessary to reduce the domain.
+6It uses the interpretation of 1-summability in terms of the Borel-Laplace method and delicate estimates in
+the Borel plane.
+16
+
+determinations of formal solutions) and its exterior the analytic world (represented by sectoral
+holomorphic solutions).
+We consider a small disc ∆0 around x = 0, and we perform a real blowing up E : X0 → ∆0
+of x = 0. Let D = E−1(0) ⊂ X0. We draw a second divisor �D inside the disc bounded by D.
+We mark two opposite directions by drawing two points p1 and p2 in the annulus. Let A be the
+punctured annulus. We consider the groupoid π1(X0, D ∪ �D) whose objects are the points of
+D ∪ �D and whose morphisms are the paths or loops up to homotopy between two objects inside
+the punctured annulus A. Let S0 = {s1, s2} two opposite directions distinct from p1 and p2.
+Definition 2.8 The local wild groupoid π♭
+1(X0, S0) is the restriction of π1(X0, D ∪ �D) over S0.
+We define a presentation of π♭
+1(X0, S0) in the following way:
+ 
+s1
+s2
+p2
+p1
+r+
+1
+r−
+1
+α1
+�γ−
+1
+�γ+
+1
+r1
+r2
+�γr
+1,2
+γr
+1,2
+�γl
+1,2
+γl
+1,2
+Figure 2: A presentation of π♭
+1(X0, S0).
+We choose rays r−
+i , r+
+i on the left and right side of pi for i = 1, 2. We choose two opposite
+base points s1 and s2 on D on the orthogonal direction to (p1, p2), endpoints of 2 rays r1 and
+r2. Let �γ−
+i (resp.�γ+
+i ) be the arc from si to the origin of r−
+i (resp. r+
+i ) in �D, and αi the arc on D
+from the end of r−
+i to the end of r+
+i on D. The two Stokes loops are the loops based in si (see
+figure above):
+σi = r−1
+i
+· �γ−
+i · r−
+i · αi · (r+
+i )−1 · (�γ+
+i )−1ri, i = 1, 2.
+Let γr
+1,2 and γl
+1,2 the lower and upper arcs from s1 to s2 on D, and �γr
+1,2 and �γl
+1,2 their analog on
+�D by using r1, an arc on �D, and r−1
+2 . The local wild groupoid is generated by σ1, σ2, γr
+1,2, γl
+1,2,
+�γr
+1,2 and �γl
+1,2. Let γ1,1 = γr
+1,2 · γl
+1,2, �γ1,1 = �γr
+1,2 · �γl
+1,2. From the above picture we have the relation
+γ1,1 = σ1 · �γr
+1,2 · σ2 · �γl
+2,1.
+In order to include the exponential torus in a linear representation of this groupoid, we introduce
+a (non discrete) extension π1(X0, S0) of π♭
+1(X0, S0): we glue a representation of (C∗, ×) by adding
+two collections of loops �t1,1(κ), κ ∈ C∗ based in �s1, and �t2,2(κ), κ ∈ C∗ based in �s2. We denote:
+t1,1(κ) = r−1
+1
+· �t1,1(κ) · r1( based in s1), t2,2(κ) = r−1
+2
+· �t2,2(κ) · r2( based in s2).
+17
+
+We require the relations Rloc:
+(i) ∀κ ∈ C∗, ∀κ′ ∈ C∗, ti,i(κκ′) = ti,i(κ) · ti,i(κ′), i = 1, 2;
+(ii) ∀κ ∈ C∗, [[σi, ti,i(κ)], σi] = ⋆i, i = 1, 2;
+(iii) ∀κ ∈ C∗, �γi,i · ti,i(κ) · �γ−1
+i,i ti,i(κ)−1 = ⋆i, i = 1, 2;
+(iv) t1,1(κ) · �γl
+1,2 · t2,2(κ) · �γr
+2,1 = ⋆1.
+Definition 2.9 The (complete) wild local groupoid is the groupoid π1(X0, S0) generated by
+π♭
+1(X0, S0) and the families T1,1 = {t1,1(κ), κ ∈ C∗} and T2,2 = {t2,2(κ), κ ∈ C∗}, with the
+above relations Rloc.
+ 
+s1
+s2
+p2
+p1
+t1,1(κ)
+t2,2(κ)
+Figure 3: A presentation of π1(X0, S0).
+2.3
+The global wild fundamental groupoid
+The local wild fundamental groupoid suffices to deal with the Euler equation, or the Kummer
+equations by adding one base point [47]. Nevertheless for the Painlev´e equations, we have to
+consider several regular and irregular singularities, with connecting data between them (and in
+some cases an order two ramification).
+In order to represent the global monodromy data together with the wild local dynamic of a
+connection in MV , we consider the following groupoid. We begin with B = P1\{p0, p1, p∞}. We
+perform real blowing up at each of these points, and we obtain a variety X with divisors D0, D1
+and D∞. We choose two opposite base points s1 and s2 on D∞, a base point s3 on D0, and s4 on
+D1. Let S = {s1, s2, s3, s4}. We consider the groupoid whose morphisms are the paths between
+the base points up to homotopy. Inside (D∞, s1, s2), we glue the local wild groupoid π♭
+1(X0, S0)
+(resp. the complete local wild groupoid π1(X0, S0)). We obtain a new groupoid denoted by
+πV,♭
+1 (X, S) (resp. πV
+1 (X, S) for the complete version). A presentation of this groupoid is given
+by the figure:
+18
+
+ 
+ 
+s3
+s4
+γ2,3
+γ4,1
+γ3,4
+p1
+p2
+s1
+s2
+γ3,3
+γ4,4
+Figure 4: A presentation of πV
+1 (X, S).
+The relations of this presentation are generated by :
+– the local relations Rloc previously defined;
+– Rext: γl
+1,2 · γ2,3 · γ3,4 · γ4,1 = ⋆1;
+– Rint: γr
+1,2 · γ2,3 · γ3,3 · γ3,4 · γ4,4 · γ4,1 = ⋆1.
+We denote by πV,κ
+1
+(X, S) the fiber of the morphism K :
+πV
+1 (X, S) → C∗ induced by
+K(t1,1(κ)) = κ.
+2.4
+A confluent morphism of groupoid
+The comparison of πV,κ
+1
+(X, S) with πV I
+1 (X, S) is a central point in order to define the confluent
+process. For this purpose, we introduce another presentation of this groupoid, setting:
+t1,2(κ) = t1,1(κ−1) · �γr
+1,2 = �γl
+1,2 · t2,2(κ−1),
+γ2,3(κ) = t2,2(κ−1) · σ−1
+2
+· γ2,3.
+We obtain the following figure
+19
+
+ 
+ 
+≃
+t1,2(κ)
+t1,1(κ)
+t2,2(κ)
+γ2,3(κ)
+σ1
+σ2
+ 
+t1,2(κ)
+t1,1(κ)
+σ1
+t2,2(κ)
+σ2
+γ2,3(κ)
+Figure 5: Another presentation for πV,κ
+1
+(X, S).
+This last figure looks like the figure (1) for the groupoid πV I
+1 (X, S). More precisely, we denote
+by s′
+i the objects of πV I
+1 (X, S), and we consider a presentation {γ′
+i,j, R′
+int, R′
+ext} of πV I
+1 (X, S)
+given by figure (1). We consider the groupoid morphism ϕκ from πV I
+1 (X, S) to πV,κ
+1
+(X, S) which
+sends s′
+i on si and which is defined by:
+• ϕκ(γ′
+3,3) = γ3,3, ϕκ(γ′
+4,4) = γ4,4, ϕκ(γ′
+3,4) = γ3,4, ϕκ(γ′
+4,1) = γ4,1,
+• ϕκ(γ′
+1,1) = σ1 · t1,1(κ), ϕκ(γ′
+2,2) = σ2 · t2,2(κ),
+• ϕκ(γ′
+1,2) = t1,2(κ), ϕκ(γ′
+2,3) = γ2,3(κ).
+This morphism ϕκ is an injective morphim but not a surjective one: the pre-image of a
+Stokes loop or a loop of an exponential tori are not defined.
+Definition 2.10 The confluent morphism from ϕ :
+πV I
+1 (X, S) → πV
+1 (X, S) is defined by the
+family ϕ = (ϕκ)κ∈C∗ .
+There exists a similar result in the context of the hypergeometric equation and the confluent
+hypergeometric equation in Kummer form.
+3
+The character variety χV
+3.1
+Definition of χV
+The class of rank 2 linear representations of a fundamental groupoid in SL2(C) has been defined
+in the introduction. Here we are only interested in a subclass of linear representations ρ which
+satisfy the following property
+Definition 3.1 A representation ρ of πV
+1 (X, S) satisfies the property (⋆) if there exists a Borel-
+Cartan configuration (B−, C, B+) such that:
+1. ρ(t1,1(κ), κ ∈ C∗) = C,
+2. ρ(σ1) ∈ U −, ρ(�γl
+1,2σ2�γl
+2,1) ∈ U +.
+If this property holds for ρ it still holds for ρ′ equivalent to ρ.
+20
+
+Definition 3.2 The character variety χV is the categorical quotient of the set of linear repre-
+sentations πV
+1 (X, S) which satisfy the property (⋆) through the equivalence of representations.
+A local representation is a representation over the sub-groupoid restricted to the local loops,
+generated by �γ1,1, γ3,3, γ4,4.
+It is characterized by a = (a0, a3, a4) with a0 = tr(ρ(�γ1,1)) =
+tr(ρ(�γ2,2)), a3 = tr(ρ(γ3,3)), a4 = tr(ρ(γ4,4)). Any representation ρ in χV determines a unique
+local representation. The fiber of this map is denoted by χV (a).
+The inclusion of πV,♭
+1 (X, S) in πV
+1 (X, S) induces by restriction a map r whose image is denoted
+by χ♭
+V .
+Proposition 3.3 For a such that a0 ̸= ±2, r is a (2:1) map over the open set of χ♭
+V defined by
+ρ(σ1) or ρ(σ2) is a non trivial element.
+Proof. If we suppose that σ1 is non trivial, one can choose �
+Y0 such that
+ρ(σ1) =
+
+
+
+1
+0
+u1
+1
+
+
+ .
+In this case, ρ(t1,1(κ), κ ∈ C∗) = C is the diagonal subgroup D of SL2(C), and we have
+ρ(t1,1(κ)) =
+
+
+
+f(κ)
+0
+0
+f(κ−1)
+
+
+ .
+From the relation t1,1(κ)t1,1(κ′) = t1,1(κκ′) we deduce that f belongs to Aut(C∗, ×), and there-
+fore f(κ) = κ or f(κ) = κ−1 . ✷
+Corollary 3.4 The character variety χV (a) = χ+
+V (a) ∪ χ−
+V (a) with
+χ+
+V (a) = {(U1, M0, U2, M3, M4, Dκ)}//SL2(C),
+χ−
+V (a) = {(U1, M0, U2, M3, M4, D−1
+κ )}//SL2(C).
+The two copies χ+
+V (a) and χ−
+V (a) coincide over a such that e0 = e−1
+0
+i.e. such that a0 = ±2.
+3.2
+The character variety χV and cubic surfaces
+Definition 3.5 A representation of πV,♭
+1 (X, S) in SL2(C) is normalized if,
+ρ(γl
+1,2) = ρ(γ2,3) = ρ(γ3,4) = I,
+where the γi,j are generators of the presentation introduced in figure 4.
+There always exists a normalized representation in each class of equivalent representations in
+SL2(C), obtained by choosing a representation of an initial object –say s2– and by representing
+the successive others objects by analytic continuation along γ2,3, γ3,4 and γl
+2,1. From the relation
+Rext, for a normalized representation we also have ρ(γ4,1) = I.
+Proposition 3.6 A normalized representation of πV,♭
+1 (X, S) is characterized by the data
+M0 = ρ(�γ1,1), U1 = ρ(σ1), U2 = ρ(�γl
+1,2 · σ2 · �γl
+2,1), M3 = ρ(γ3,3), M4 = ρ(γ4,4)
+such that U1M0U2M3M4 = I, up to a common conjugacy.
+21
+
+Proof.
+We have to prove that, for a normalized representation ρ, the images of all the
+generators of the groupoid by ρ are well defined. Since ρ(γl
+1,2) = I, we have
+ρ(�γl
+1,2) = U2, ρ(�γr
+1,2) = M0U2, ρ(γr
+1,2) = U1M0U2.
+The relation Rint gives U1M0U2M3M4 = I. A change of representation of the initial object will
+change this data by a common conjugacy. ✷
+One can suppose, by using a conjugacy, that the Cartan subgroup C = {ρ(t1,1(κ)), κ ∈ C∗} ⊂
+SL2(C) is the diagonal one D. From the relation (iii) of the local groupoid presentation, �
+M0
+commutes with each element of C. Therefore from the relations (i) (ii) and (iii), we have:
+U1 =
+
+
+
+1
+0
+u1
+1
+
+
+ , M0 =
+
+
+
+e0
+0
+0
+e−1
+0
+
+
+ , U2 =
+
+
+
+1
+u2
+0
+1
+
+
+ ,
+M3 =
+
+
+
+α3
+β3
+γ3
+δ3
+
+
+ , M4 =
+
+
+
+α4
+β4
+γ4
+δ4
+
+
+ .
+This data is now defined up to a conjugacy by an element of D. Finally we have
+χ♭
+V = {(U1, M0, U2, M3, M4) ∈ (U −, D, U +) × SL2(C)2, U1M0U2M3M4 = I}//D.
+This algebraic quotient is a 5 dimensional space. The function (a+, x+) = (e0, a3, a4, x+
+1 , x+
+2 , x+
+3 ):
+e0 = M0[1, 1], a3 = tr(M3), a4 = tr(M4),
+x+
+1 = M3[2, 2] = δ3, x+
+2 = M4[2, 2] = δ4, x+
+3 = tr(U1M0U2) = e0 + e−1
+0
++ e0u1u2,
+is invariant under the action of D.
+Proposition 3.7 The coordinates (a+, x+) define a map Tr+
+V , invertible for a generic a, from
+χ♭
+V (a) to the family affine cubic surface CV (θ+) defined by
+FV (θ+, x) = x1x2x3 + x2
+1 + x2
+2 − θ+
+1 x1 − θ+
+2 x2 − θ+
+3 x3 + θ+
+4 = 0,
+where θ+
+1 = a3 + e0a4, θ+
+2 = a4 + e0a3, θ+
+3 = e0, θ+
+4 = e2
+0 + e0a3a4 + 1.
+Lemma 3.8 We have:
+e0u1u2 = x3 − e0 − e−1
+0
+e0γ3s2 = x2 − e0a3 + e0x1
+e0β3s1 = −x1x3 + e0x1 − x2 + a4
+Proof. We have x3 = e0 + e−1
+0
++ e0u1u2. From U1M0U2M3 = M−1
+4
+we obtain
+δ4 = α3e0 + γ3e0u2, α4 = β3e0s1 + δ3e0u1u2 + δ3e−1
+0 .
+Using αi = ai − δi, we obtain the two other equalities. ✷
+Proof of Proposition (3.7). Since α3δ3 − β3γ3 = 1, we have
+(e2
+0u1u2)(α3δ3) − (e0β3u1)(e0γ3u2) − e2
+0u1u2 = 0.
+22
+
+If we replace e0s1s2, e0γ3s2 and e0β3s1 by the above expressions, we obtain the equation
+FV (x, θ+) = 0. This proves that the map (a+, x+) takes its values in the cubic surface. Given
+a point in the cubic surface, we recover α3 = a3 − x1, α4 = a4 − x2, β3γ3 = 1 − x1(a3 − x1),
+β4γ4 = 1 − x2(a4 − x2), γ3s2 = e−1
+0 x2 − a3 + x1, and therefore, for generic values, a unique point
+of χ♭
+V (a). ✷
+Remark 3.9 It might seen more natural to consider the trace coordinates:
+y1(ρ) = tr(ρ(�γ1,1γ1,3γ3,3)), y2(ρ) = tr(ρ(�γ2,2γ2,4γ4,4)), y3(ρ) = tr(ρ(γ3,3γ3,4γ4,4))
+which, for a normalized representation, give:
+y1 = tr(M0M3), y2 = tr(M0M4), y3 = tr(M3M4).
+But clearly, the coordinates y1 and y2 degenerate if M0 = ±I, since we have y1 = a3 and y2 = a4
+in this case. The coordinates
+x+
+1 = y1 − e0a3
+e−1
+0
+− e0
+= δ3, x+
+2 = y2 − e0a4
+e−1
+0
+− e0
+= δ4, x+
+3 = y3
+give an extension to this exceptional case M0 = ±I. This choice is also the usual one in the
+litterature: see [54] or [40]. Since the coordinates (a+, x+) are directly related to the above trace
+coordinates through an affine map, we still denote the corresponding map by Tr+
+V , and we call it
+a trace map.
+The normalization of the representation by (U1, M0, U2, M3, M4) used in order to construct Tr+
+V
+requires that U1 is a lower triangular matrix. If we require that U1 must be an upper triangular
+matrix, we obtain the data (U −
+1 , M−
+0 , U −
+2 , M−
+3 , M−
+4 ) which is equivalent to the first data by the
+conjugacy with the matrix
+P =
+
+
+
+0
+1
+−1
+0
+
+
+ .
+We define the trace map Tr−(ρ) = (a−, x−) = (e−1
+0 , a3, a4, x−
+1 , x−
+2 , x−
+3 ) with
+e−1
+0
+= M−
+0 [1, 1], x−
+1 = M−
+3 [1, 1] = α3, x−
+2 = M−
+4 [1, 1] = α4, x−
+3 = tr(U −
+1 M−
+0 U −
+2 ) = x+
+3 .
+This map takes its values in the cubic surface CV (θ−):
+FV (x, θ−) = x1x2x3 + x2
+1 + x2
+2 − θ−
+1 x1 − θ−
+2 x2 − θ−
+3 x3 + θ−
+4 = 0,
+where θ−
+1 = a3 + e−1
+0 a4, θ−
+2 = a4 + e−1
+0 a3, θ−
+3 = e−1
+0 , θ−
+4 = e−2
+0
++ e−1
+0 a3a4 + 1 which can also
+be identified to the categorical quotient χ♭
+V (a).
+The points p+ ∈ CV (θ+) and p− = Tr−
+V ◦
+(Tr+
+V )−1(p+) correspond to the same representation in χ♭
+V (a).
+From (3.4) we know that χV (a) = χ+
+V (a) ∪ χ−
+V (a) where
+χ+
+V (a) = {(U1, M0, U2, M3, M4, Dκ)//SL2(C)},
+χ−
+V (a) = {(U1, M0, U2, M3, M4, D−1
+κ )//SL2(C)}.
+Using a conjugacy with P this second data is equivalent to (U −
+1 , M−
+0 , U −
+2 , M−
+3 , M−
+4 , ρ(t1,1(κ)) =
+Dκ), which define an element in CV (θ−). Therefore χV (a) is identified through the trace coor-
+dinates to the union of the two affine surfaces CV (θ−) ∪ CV (θ+)
+We denote by Tr±,κ the restriction of Tr± over πV,κ
+1
+(X, S).
+23
+
+3.3
+Lines and reducibility locus
+For what follows in this article, excepted (5.3), we will suppose that the parameters satisfy the
+generic conditions:
+e0 ̸= ±1, e3 ̸= ±1, e4 ̸= ±1, e0eε3
+3 eε4
+4 ̸= 1 for ε3 = ±1, ε4 = ±1.
+The description of the lines in cubic surfaces can be found in [22]. In particular the compacti-
+fication of a generic element CV I(θ) in P3(C) contains 27 lines. In the figure below the central
+triangle is the union of the three lines at infinity:
+Le1,e2 Le−1
+1 ,e−1
+2
+Le3,e4 Le−1
+3 ,e−1
+4
+Le1,e−1
+2 Le−1
+1 ,e2 Le3,e−1
+4 Le−1
+3 ,e4
+Le1,e3
+Le2,e3
+Le−1
+1 ,e−1
+3
+Le−1
+2 ,e−1
+3
+Le1,e3
+Le−1
+1 ,e−1
+3
+Le1,e−1
+3
+Le−1
+1 ,e3
+Le2,e4
+Le−1
+2 ,e−1
+4
+Le2,e−1
+4
+Le−1
+2 ,e4
+Le2,e−1
+3
+Le−1
+2 ,e3
+Le1,e4
+Le−1
+1 ,e−1
+4
+Le1,e−1
+4
+Le−1
+1 ,e4
+Figure 6: The 24 lines in CV I(θ).
+The equations of the lines are obtained from the following decomposition which appears in [40]:
+FV I(x, θ) = (xk − cei,ej)(FV I,xk − xk + cei,ej) + lei,ejle−1
+i
+e−1
+j ,
+where FV I,xk is the partial derivative of FV I(x, θ) with respect to the variable xk, cα,β = αβ−1 +
+α−1β and
+lei,ej = eixi + ejxj − akeiej − a4, le−1
+i
+,e−1
+j
+= e−1
+i xi + e−1
+j xj − ake−1
+i e−1
+j
+− a4.
+Therefore the plane xk = cei,ej intersects the cubic surface along a degenerated conic, union of
+the two lines
+Lei,ej : xk = cei,ej = eie−1
+j
++ e−1
+i ej and lei,ej = 0, Le−1
+i
+,e−1
+j
+: xk = ce−1
+i
+,e−1
+j
+and le−1
+i
+,e−1
+j
+= 0.
+The compactification of CV (θ) in P3(C) has a singular point of type A1 at infinity, and admits
+21 lines, of which 18 are in the affine part:
+24
+
+ 
+∆e4
+∆e−1
+4
+∆e3
+∆e−1
+3
+Dl
+e−1
+4
+Dr
+e3
+Dl
+e4
+Dr
+e−1
+3
+Dl
+e3
+Dr
+e−1
+4
+Dl
+e−1
+3
+Dr
+e4
+De3,e4
+De−1
+3 ,e−1
+4
+De3,e−1
+4 De−1
+3 ,e4
+Df0,f−1
+0
+Df−1
+0
+,f0
+Figure 7: The 18 lines in CV (θ).
+Remark 3.10 The equations of these lines are given by
+1. z0 = −x2
+1x3 − x1x2 + θ2x1 − e0 = 0 for Dl
+e3 ∪ Dl
+e−1
+3
+∪ Dl
+e−1
+4
+∪ Dl
+e4;
+2. z1 = x1x2 − e0 = 0 for ∆e3 ∪ ∆e−1
+3
+∪ ∆e4 ∪ ∆e−1
+4 ;
+3. z2 = −x2
+2x3 − x1x2 + θ1x2 − e0 = 0 for Dr
+e−1
+3
+∪ Dr
+e3 ∪ Dr
+e4 ∪ Dr
+e−1
+4 ;
+4. x3 = ce3,e4 or ce3,e−1
+4
+or ce0,e−1
+0
+for the pairs of lines which cut the basis of the triangle.
+In this section, we want to characterize the representations whose image is a point on a line.7
+The local loops in πV I
+1 (X, S) are the loops γi,i based in si. The local loops in πV
+1 (X, S) are γ3,3
+in s3, γ4,4 in s4, and a generic element of the torus t1,1(κ) in s1 and t2,2(κ) in s2. In any cases,
+the local loops at si are denoted below by γi.
+Definition 3.11 Let γ be a morphism of the groupoid πV I
+1 (X, S) or πV
+1 (X, S), defined by some
+path from si to sj, j ̸= i, γi the local loop in si and γj the local loop in si. A linear representation ρ
+of the groupoid in SL2(C) is reducible along γ if ρ(γ−1γiγ) and ρ(γj) have a common eigenvector.
+If ρ is the linear representation associated with a regular connection ∇, and γ is represented
+by analytic continuation, it means that there exists some local solution at si which is an eigen-
+vector of the local monodromy around si and whose analytic continuation along γ is also an
+eigenvector of the local monodromy around sj. This semi-local solution in a neighbourhood of
+γ has an abelian monodromy.
+Remark 3.12
+1. If ρ is reducible along the path γ, any equivalent representation is reducible
+along γ.
+2. The representation ρ is reducible along γ if and only ρ is reducible along γ−1.
+3. If ρ(γi) = ±I, ρ is reducible along any path joining si to another base point.
+7We thanks Martin Klimes for discussions on this topics (see also [40]).
+25
+
+4. Let ρ be a representation in SL2(C) given by a choice of basis of solutions Xi on each Vi.
+We set ρ(γi) = Mi, ρ(γ) = Mγ. ρ is reducible on γ if and only if the matrices M−1
+γ MiMγ
+and Mj have a common eigenvector.
+5. In particular, if ρ is a normalized representation associated to the presentation given by
+figure (1) of the groupoid πV I
+1 (X, S), with ρ(γi) = Mi and ρ(γi,j) = I for i ̸= j, ρ is
+reducible on the generator path γi,j if and only if Mi and Mj have a common eigenvector.
+Definition 3.13 A pair of matrices in SL2(C) is reducible if and only if they have a common
+eigenvector.
+Definition 3.14
+1. The reducibility locus associated to a path γ is the set R(γ) of linear
+representations ρ in χV I which are reducible on γ.
+2. Given a presentation of πV I
+1 (X, S), the generating reducibility locus is the union of the sets
+R(γi,j) for all the generators γi,j, i ̸= j.
+3. The total reducibility locus is the set R of linear representations ρ in χV I such that there
+exists a path between two distinct objects on which ρ is reducible.
+Theorem 3.15 Let {γi,j), Ri} be the presentation of πV I
+1 (X, S) given by the figure (1), and
+TrV I : χV I(a) → CV I(θ) the trace map associated to this presentation. The 24 lines in CV I(θ)
+are the reducibility locus of the 6 paths: γi,i+1, i = 1, .., 4 and γ1,3 = γ1,2 · γ2,3, γ2,4 = γ2,3 · γ3,4.
+Lemma 3.16 Let M1 and M2 be two matrices in SL2(C) distinct from ±I, e1, e−1
+1 , and e2,
+e−1
+2
+their eigenvalues (we may have e1 = e−1
+1
+or e2 = e−1
+2 ). Let
+cα,β = αβ−1 + α−1β.
+The pair (M1, M2) is reducible if and only if
+tr(M1M2) = ce1,e2, or tr(M1M2) = ce1,e−1
+2 .
+Proof. We choose a ”mixed basis” taking an eigenvector of M2, and an eigenvector of M1.
+In such a basis, we have:
+M1 =
+
+
+
+e1
+0
+f1
+e−1
+1
+
+
+ ,
+M2 =
+
+
+
+e2
+f2
+0
+e−1
+2
+
+
+
+or any similar writing obtained by changing e1 with e−1
+1
+or e2 with e−1
+2 . Let Di = diag(ei, e−1
+i ).
+We have
+tr(M1M2) = tr(D1D2) + f1f2.
+Such a pair has a common eigenvector if and only if f1 = 0 or f2 = 0, i.e.
+if and only if
+tr(M1M2) = tr(D1D2). We have tr(D1D2) = ce1,e2 or tr(D1D2) = ce1,e−1
+2 . ✷
+Proof of Theorem (3.15). We consider a normalized representation ρ and the six (non ori-
+ented) generating paths γi,j, i ̸= j.
+For a normalized representation ρ, these paths satisfy:
+ρ(γi,j) = I. Therefore ρ is reducible over γi,j if and only if the pair of matrices (Mi, Mj) is
+reducible. For {i, j} in {1,2,3}, according to Lemma (3.16), ρ is reducible on the path γi,j if and
+only if xk = cei,ej or xk = cei,e−1
+j . Therefore the reducibility locus R(γi,j) is given by the four
+lines Le±1
+i
+,e±1
+j . ✷
+26
+
+Definition 3.17 The 12 Kaneko points are the intersections
+pei,ej = Lei,ej ∩ Le−1
+i
+,e−1
+j , pei,e−1
+j
+= Lei,e−1
+j
+∩ Le−1
+i
+,ej, {i, j} ⊂ {1, 2, 3, 4}.
+Proposition 3.18 The Kaneko points pei,ej and pei,e−1
+j
+correspond to a normalized monodromy
+representation such that Mi and Mj commute.
+Proof. In a ”mixed basis”,
+Mi =
+
+
+
+ei
+0
+fi
+e−1
+i
+
+
+ ,
+Mj =
+
+
+
+ej
+fj
+0
+e−1
+j
+
+
+
+(7)
+or the same writing changing ej in e−1
+j .
+The reducibility locus is given by fifj = 0, which
+defines two components Lei,ej and Le−1
+i
+,e−1
+j
+(or Lei,e−1
+j
+and Le−1
+i
+,ej). Therefore pei,ej (or pei,e−1
+j )
+is defined by fi = 0 and fj = 0.✷
+The Painlev´e VI foliation admits 12 singular points, 4 over each singular fiber. The local
+description of the foliation in a neighborhood of these non linear singularities of “regular type”
+can be found in the chapter 4 of [30]. The Painlev´e V foliation also admits 3 non linear regular
+singular points over 0. The germ of Painlev´e equation admits a unique meromorphic solution
+around such a singular point, defining an analytic leave for the germ of foliation: we call them
+the central solutions. These solutions have been studied by K. Kaneko [36]. Furthermore, they
+appears as the intersection of two codim 1 germs of invariant analytic surfaces [30].
+Theorem 3.19
+1. The Riemann-Hilbert correspondence RHV I sends the 12 central solutions
+to the 12 Kaneko points.
+2. The Riemann-Hilbert correspondence RHV I sends the local invariant varieties near a sin-
+gular point to the 2 germs of lines intersecting at the corresponding Kaneko point.
+Proof.
+We consider the 4 singular points over 0.
+The non linear monodromy generated
+by a simple loop around 0 will keep invariant each meromorphic solutions and the analytic
+invariant surfaces which intersect on it, around this singular point. Through the Riemann-Hilbert
+correspondence this non linear monodromy is conjugated to one of the polynomial dynamics hi,j
+given by (0.5). The central solutions are sent on fixed points of hi,j. Since each hi,j fixes 4
+central points, and has no more than 4 fixed points, this proves the first point.
+Now we search for invariant curves under the action of hi,j through the central point pei,ej.
+Clearly the two lines Lei,ej and Le−1
+i
+,e−1
+j
+are invariant since the union of the two lines is (xk =
+cei,ej) ∩ (FV I(x, θ) = 0) which is invariant. Suppose that there exists another local analytic
+invariant curve transverse to dxk = 0. It cut the plane (xk = c) on a finite set of points, for any
+value c near from cei,ej. This would create a periodic orbit on each (xk = c). The restriction of
+hi,j on each (xk = c) is a family of affine maps, and for a generic affine map, there doesn’t exist
+such a periodic orbit. Therefore the local invariant surfaces are sent on the germ of two lines
+around each central point. ✷
+Remark 3.20 Using Proposition (3.18) and Theorem (3.19), we recover here a result of K.
+Kaneko [36]: the linear monodromy data preserved along a Kaneko solution is generated by
+3 matrices with a pair of commuting matrices.
+In particular, it generates a solvable group.
+Nevertheless, the theorem of K. Kaneko is much more precise: it describes explicitely this linear
+solvable monodromy data.
+27
+
+We have a result similar to Theorem (3.15) for the Painlev´e V foliation.
+Theorem 3.21 Let {γi,j, R} be the presentation of πV
+1 (X, S) given by figure (4), and Tr+
+V :
+χ+
+V (a) → CV (θ+) the trace map associated to this presentation. Any line in CV (θ+) is the image
+of a reducibility locus in χV (a) for some path in πV
+1 (X, S).
+The proof is similar to the one of Theorem (3.15, but one needs to investigate different cases.
+We just mention here an example:
+Lemma 3.22 The reducibility locus R(γr
+1,2) is given by a pair of lines defined by (x3 = a0)∩CV .
+Proof. For a normalized representation ρ we have
+ρ(γr
+1,2) = U1M0U2, ρ(�γ1,1) = ρ(�γ2,2) = M0.
+Therefore, according to Remark (3.12), we have:
+ρ ∈ R(γr
+1,2) ⇔ ((U1M0U2)−1M0(U1M0U2), M0) is a reducible pair
+⇔ (M−1
+0 U −1
+1 M0U1, U2M0U −1
+2 M−1
+0 ) is a reducible pair
+⇔ u1u2 = 0
+⇔ tr(U1M0U2) = tr(M0),
+⇔ x3 = a0 = f 2
+0 + f −2
+0
+where f 2
+0 = e0.
+The intersection (x3 = a0) ∩ CV is given by one pair of lines. Indeed we have:
+FV (x, θ) = (x3 − e0 − e−1
+0 )(x1x2 − e0) + df0,f−1
+0 df−1
+0
+,f0,
+with
+df0,f−1
+0
+= f0x1 + f −1
+0 x2 − f0a3, df−1
+0
+,f0 = f −1
+0 x1 + f0x2 − f0a4.
+Therefore the reducibility locus R(γr
+1,2) is given by the pair of lines:
+Df0,f−1
+0
+: x3 = e0 + e−1
+0
+and df0,f−1
+0
+= 0,
+Df−1
+0
+,f0 :
+x3 = e0 + e−1
+0
+and df−1
+0
+,f0 = 0. ✷
+The others cases are treated with similar computations. Here we also have a correspondence
+between the 3 Kaneko solutions described by K. Kaneko and Y. Ohyama in and [37], and the 3
+Kaneko points pe3,e4, pe3,e−1
+4
+and pe0,e−1
+0
+through RHV .
+The complete reducibility locus, defined by all the paths γ of the fondamental groupoid,
+contains these lines and their images by the tame dynamics. We don’t know if the lines and the
+tame action generate the complete reducibility locus (maybe we have to add some parabolas).
+3.4
+Log-canonical coordinates on CV (θ) and cluster sequences
+Let CV (θ) be the family of symplectic cubic affine surfaces in C3 defined by FV (x, θ) = 0. The
+cubic surfaces CV (θ) are birational to C2. Indeed, since the equation FV (θ) is affine in x3, the
+map π3 : CV (θ) → C2 induced by (x1, x2, x3) → (x1, x2) is rationally invertible and therefore
+birational. The polar locus of π−1
+3
+is given by FV,x3 = x1x2 − e0 = 0. Recall that CV (θ) has a
+symplectic structure defined by
+ωV (θ) = dxi ∧ dxj
+∂FV,θ/∂xk
+,
+for any i, j, k = 1, 2, 3. We want here to introduce symplectic birational maps:
+Notations (see Appendix):
+28
+
+- ωlog = du
+u ∧ dv
+v : the log-canonical symplectic form on C2;
+- Bir(C2) : the group of the birational automorphisms of the complex plane;
+- Symp(C2) = Symp+(C2) : the subgroup of Bir(C2) of the automorphisms which preserve
+ωlog;
+- Symp−(C2) : the subset of Bir(C2) of the automorphisms ϕ such that ϕ∗ωlog = −ωlog;
+- Symp±(C2) : the subgroup of Bir(C2) generated by Symp+(C2) ∪ Symp−(C2).
+Definition 3.23
+1. A log-canonical system of coordinates on CV (θ) is a birational symplectic
+morphism from (CV (θ), ωV (θ)) to (C2, ωlog).
+2. A log-canonical function on CV (θ) is a component of some log-canonical system.
+3. A log-canonical sequence of coordinates is a sequence of coordinates such that two consec-
+utive elements define a log-canonical system of coordinates.
+4. A log-anti-canonical system of coordinates on CV (θ) is a birational symplectic morphism
+from (CV (θ), ωV (θ)) to (C2, −ωlog).
+Remark 3.24
+1. If (x, y) is a log-anti-canonical system of coordinates, (y, x) and (x−1, y)
+are log-canonical systems of coordinates.
+2. If (x, y) is a log-canonical system of coordinates, for any λ, µ in C∗, (λx, µy) is a log-
+canonical system of coordinates.
+The cubic surface CV (θ) is symplectic birationally equivalent to (C2, ωlog). Indeed we have two
+first examples of log-canonical systems of coordinates on CV (θ):
+Proposition 3.25 Let y1 = x1, y2 = x2 and z1 = FV,x3 = x1x2 − e0. The pairs (y1, z1), and
+(z1, y2) are log-canonical systems of coordinates on CV (θ). A map z1 such that both (y1, z1), and
+(z1, y2) are log-canonical systems of coordinates is unique up to a multiplicative constant.
+Proof. We have:
+dy1
+y1
+∧ dz1
+z1
+= dx1
+x1
+∧ x1dx2 + x2dx1
+z1
+= dx1 ∧ dx2
+x1x2 − e0
+= ωV (θ),
+and we have a similar computation for (z2, y2). All the maps such that (y1, z) is a log-canonical
+system of coordinates write z = c(y1)z1. Therefore the only maps such that (y1, z), and (z, y2)
+are log-canonical systems of coordinates are z = cz1 for some c in C∗. ✷
+Therefore (y1, z1, y2) is a log-canonical triple, which satisfies: z1 = y1y2−e0. In order to construct
+new log-canonical or log-anti-canonical systems of coordinates we consider the two maps:
+σ1(x1, x2, x3) = (x1 − FV,x1, x2, x3) = (−x1 − x2x3 + θ1, x2, x3)
+σ2(x1, x2, x3) = (x1, x2 − FV,x2, x3) = (x1, −x2 − x1x3 + θ2, x3).
+Lemma 3.26 σ1 and σ2 define polynomial involutive automorphims of CV (θ). Furthermore they
+are anti-symplectic automorphisms, i.e. σ∗
+i ωV (θ) = −ωV (θ).
+29
+
+Proof. The involutive property is a direct computation. We have FV ◦σi = FV , and therefore
+the σi’s are polynomial automorphims. Finally, by using ωV (θ) =
+dx2∧dx3
+x2x3+2x1−θ1 (resp. ωV (θ) =
+dx3∧dx1
+x3x1+2x2−θ2), we obtain σ∗
+1ωV (θ) = −ωV (θ) (resp. σ∗
+2ωV (θ) = −ωV (θ)). ✷
+We set:
+g = σ1 ◦ σ2,
+g−1 = σ2 ◦ σ1.
+From Lemma (3.26), g is a polynomial symplectic automorphisms of (CV (θ), ωV (θ)). Therefore,
+starting from the log-canonical triple (y1, z1, y2), we obtain two other log-canonical triples by
+applying σ∗
+i , i = 1, 2 and reversing the order of the triple. We set:
+(y2, z2, y3) := (σ∗
+1y2, σ∗
+1z1, σ∗
+1y1);
+(y0, z0, y1) := (σ∗
+2y2, σ∗
+2z1, σ∗
+2y1).
+Therefore we obtain a log-canonical sequence of length seven:
+H : (y0, z0, y1, z1, y2, z2, y3).
+We call it the fundamental log-canonical heptuple. Note that we have:
+(y2, z2, y3) = g−1∗(y0, z0, y1).
+We extend the fundamental log-canonical sequence to an infinite log-canonical sequence by
+setting for any k in Z:
+(y2k, z2k, y2k+1, z2k+1, y2k+2, z2k+3, y2k+3) := (g−k∗)(y0, z0, y1, z1, y2, z2, y3).
+This sequence satisfies the following relations:
+Proposition 3.27 [exchange relations] let AV = Z[e±
+0 , e±
+3 , e±
+4 ]. Let P, Q1 and Q2 in AV [t]
+defined by
+P(t) = t + e0,
+Q1(t) = (t − e0e−1
+4 )(t − e0e4)(t − e−1
+3 )(t − e3),
+Q2(t) = (t − e0e−1
+3 )(t − e0e3)(t − e−1
+4 )(t − e4).
+For all k in Z we have
+ykyk+1 = P(zk)
+z2kz2k+1 = Q1(y2k+1)
+z2k+1z2k+2 = Q2(y2k+2).
+Lemma 3.28 We have :
+σ∗
+1x1 = x−1
+2
+�
+e0 + z−1
+1 Q2(x2)
+�
+,
+and σ∗
+2x2 = x−1
+1
+�
+e0 + z−1
+1 Q1(x1)
+�
+.
+(8)
+with :
+Q2(t) = t4 − θ2t3 + θ4t2 − e0θ1t + e2
+0 = (x − e0e−1
+3 )(x − e0e3)(x − e−1
+4 )(x − e4),
+(9)
+Q1(t) = t4 − θ1t3 + θ4t2 − e0θ2t + e2
+0 = (x − e0e−1
+4 )(x − e0e4)(x − e−1
+3 )(x − e3).
+(10)
+30
+
+Proof.
+1. We have :
+x2σ∗
+1x1 = −x1x2 − x2
+2x3 + θ1x2
+= −e0 − z1 + x2
+2z−1
+1 (x2
+1 + x2
+2 − θ1x1 − θ2x2 + θ4) + θ1x2
+= −e0 − z1 + z−1
+1
+�
+(e0 + z1)2 + x4
+2 − θ1x2(e0 + z1) − θ2x3
+2 + θ4x2
+2
+�
++ θ1x2
+= e0 + z−1
+1
+�
+e2
+0 + x4
+2 − θ1x2 − θ2x3
+2 + θ4x2
+2
+�
+= e0 + z−1
+1 Q2(x2).
+Hence σ∗
+1x1 = e0x−1
+2
++ x−1
+2 z−1
+1 Q2(x2).
+The proof of the other equality is similar.
+2. If we develop the right term of (9), we get :
+t4 − (a4 + e0a3)t3 + (e0a3a4 + e2
+0 + 1)t2 − e0(a3 + e0a4)t + e2
+0
+= t4 − θ2t3 + θ4t2 − e0θ1t + e2
+0.
+If we develop the right term of (10), we get :
+t4 − (a3 + e0a4)t3 + (e0a3a4 + e2
+0 + 1)t2 − e0(a4 + e0a3)t + e2
+0
+= t4 − θ1t3 + θ4t2 − e0θ2t + e2
+0. ✷
+Proof of Proposition (3.27). From lemma 3.28, we get : z2 = x2σ∗
+1x1 − e0 = z−1
+1 Q2(x2).
+Hence z1z2 = Q2(x2). Similarly : z0 = σ∗
+2x2 − e0 = z−1
+1 Q1(x1). Hence z0z1 = Q1(x1). The
+others relations are obtained by applying g−k∗ to the previous one. ✷
+Remark 3.29
+1. We have Q1 = e−2
+0 t4Q2(e0t−1).
+2. According to Remark (3.10), z2k = 0 are the equations of the lines Dr
+· , Dl
+·, or their images
+by the dynamic < g >, and z2k+1 = 0 are the equations of the lines ∆· or their images by
+the dynamic < g >.
+3.5
+The Laurent property
+The exchange relations obtained at Proposition (3.27) suggests that the sequence of log-canonical
+systems of coordinates is related to a structure of generalized cluster algebra. Without going
+into the theory of cluster algebras (for details see [38]), we just recall here the fact that under
+some hypothesis they satisfy the “Laurent property” (see also [15]):
+Definition 3.30
+1. A rational map r ∈ C(x, y) satifies the Laurent property if its polar set
+is included in xy = 0.
+2. A birational map r satifies the Laurent property if both r and r−1 have the Laurent property.
+3. Let X be an affine surface, and let (yn, zn) be a sequence of algebraic morphisms from X
+to C2. This sequence satisfies the Laurent property, if given an element (yn, zn), any other
+regular function ym (or zm) = r(xn, yn) satisfies the Laurent property.
+We must remark that the Laurent property is not stable by composition or inversion. It
+turns out that in a cluster sequence some simplications arise from the exchange relations and
+give this property. A direct proof of this fact needs heavy computations. We can prove here the
+Laurent property for the canonical sequence by an argument which avoid these computations
+with the exchange relations.
+31
+
+Proposition 3.31 The log-canonical sequence satisfies the Laurent property.
+Proof. We first prove that all the log-canonical variables are polynomials in (x1, x2, x3). This
+is true for (y1, z1, y2) = (x1, x1x2 − e0, x2). Recall that σ1, σ2 (and therefore g) are polynomials
+in (x1, x2, x3). All the other canonical variables are obtained by the action of a word in σ∗
+1, σ∗
+2
+from a variable in this triple. Therefore they are still polynomials in (x1, x2, x3). We set:
+pn : CV (θ) → C2, u = yn(x), v = zn(x),
+qn : CV (θ) → C2, u = zn(x), v = yn+1(x).
+We claim that pn and qn satisfy the Laurent property. Since pn is polynomial, we only have to
+prove that the polar set of p−1
+n
+is included in uv = 0. This is true for p1 = (y1, z1):
+x1 = y1
+x2 = (z1 + e0)y−1
+1
+x3 = z−1
+1 (−y2
+1 − (z1 + e0)y−2
+1
++ θ1y1 + θ2(z1 + e0)y−1
+1
+− θ4.
+This is also true for q1 = (z1, y2) by using a similar computation. Let
+ι : (u, v) → (v, u).
+We have :
+q0 = ι ◦ p1 ◦ σ2, p2 = ι ◦ q1 ◦ σ1.
+Therefore, q0 and p2 also satisfy the Laurent property. Since the property is true for p1 and
+p2, and since the tame dynamic g is a polynomial automorphism, it remains true for any pn =
+g−1∗pn−2. Since the property is true for q0 and q1 it remains true for any qn = g−1∗qn−2. Finally,
+since pn satisfies the Laurent property and pm is polynomial, the map
+pm ◦ p−1
+n
+: (ym, zm) = (r(yn, zn), s(ym, zm))
+satisfies the Laurent property.✷
+From the above proof, one can remark that the Laurent property comes from the fact that the
+antisymplectic tame dynamics (σ1 and σ2) and the symplectic dynamics (g) are automorphisms
+of the cubic surface.
+3.6
+Families of confluent and diffluent morphisms
+These families were first discovered by M. Klimes in [40] through an analytic confluent process,
+both on Painlev´e equations and on character varieties. We recover them by using a groupoid
+point of view. In definition (2.10) we have introduced a family of morphisms ϕκ : πV I
+1 (X, S) →
+πV,κ
+1
+(X, S). Therefore we obtain a family
+ϕ∗
+κ : χκ
+V (a) → χV I(aκ)
+ρ → ρ ◦ ϕκ.
+On the restriction over χ±
+V we have 2 families ϕ±∗
+κ .
+Proposition 3.32 The morphisms ϕ±∗
+κ
+are invertible on the open set in χV I(aκ) defined by the
+elements (M1, M2, M3, M4) such that (M1, M2) is a non reducible pair.
+32
+
+Proof. The matrix expression of ϕ+∗
+κ
+is
+[U1, M0, U2, M3, M4]D → [U1Dκ, Dκ−1M0U2, M3, M4)]SL2(C).
+Given (M1, M2, M3, M4), and a parameter κ, we search for matrices M0, U1, U2, M′
+3, M′
+4,
+(U1, M0, U2) in U − × D × U + and a matrix Q in SL2(C) such that
+
+
+
+
+
+
+
+
+
+
+
+U1Dκ = Q−1M1Q
+D−1
+κ M0U2 = Q−1M2Q
+M′
+3 = Q−1M3Q
+M′
+4 = Q−1M4Q
+(11)
+For a given local data (a1, a2, a3, a4), and a value κ, we have only two choices for the eigenvalues
+(e0, e−1
+0 ) of M0, solutions of κ−1e0 + κe−1
+0
+= a2, one centered around a2κ and the other around
+a2κ−1, induced by the change κ → κ−1. Let e±
+1 1, e±
+2 be the eigenvalues of M1 and M2. The two
+solutions for M0 are: M0 = diag(e1e2, e−1
+1 e−1
+2 ), or M0 = diag(e−1
+1 e−1
+2 , e1e2). We choose the first
+one (the second will give a pre-image for ϕ−∗
+κ ).
+We recall the LDU decomposition in SL2(C):
+Lemma 3.33 Let M be an element of SL2(C) such that M[1, 1] ̸= 0. There exists a unique
+triple (L, D, U) in U − × D × U + such that M = L × D × U.
+Proof. We have
+
+
+
+a
+b
+c
+d
+
+
+ =
+
+
+
+1
+0
+l
+1
+
+
+ ×
+
+
+
+e
+0
+0
+e−1
+
+
+ ×
+
+
+
+1
+u
+0
+1
+
+
+ ⇔
+
+
+
+
+
+
+
+
+
+
+
+a = e
+b = eu
+c = el
+d = e−1 + elu
+which system has the unique solution (l, e, u) = (c/a, a, b/a) if a ̸= 0. ✷
+We call the above decomposition the LDU-decomposition of M and we denote:
+LDU(M) = (L, D, U).
+Lemma 3.34 Let M1 and M2 be two non vanishing matrices in SL2(C), M1 ̸= ±I, M2 ̸= ±I,
+with eigenvalues (e1, e−1
+1 ) and (e2, e−1
+2 ). Suppose that the eigenvectors related to e−1
+2
+and to e1
+are independent. There exists a matrix Q in SL2(C) such that the diagonal component of the
+decomposition LDU of Q−1M1M2Q is diag(e1e2, e−1
+1 e−1
+2 ).
+Proof. The hypothesis of the Lemma gives the existence of a ”mixed basis” (u, v) given by
+an eigenvector u of M2 related to the eigenvalue e−1
+2 , and an eigenvector v of M1 related to the
+eigenvalue e1. Let Q be the matrix of this change of basis. We have:
+Q−1M1Q =
+
+
+
+e1
+0
+c1
+e−1
+1
+
+
+ , Q−1M2Q =
+
+
+
+e2
+b2
+0
+e−1
+2
+
+
+ .
+33
+
+Therefore
+Q−1M1M2Q =
+
+
+
+e1e2
+e1b2
+e2c1
+e−1
+1 e−1
+2
++ b2c1
+
+
+ =
+
+
+
+1
+0
+c1
+1
+
+
+ ·
+
+
+
+e1e2
+0
+0
+e−1
+1 e−1
+2
+
+
+ ·
+
+
+
+1
+b2
+0
+1
+
+
+ .
+✷
+End of the proof of Proposition (3.32). Suppose that the pair of matrices (M1,κ, M2,κ) satisfies
+the hypothesis of Lemma (3.34). We choose the matrix Q given by this Lemma and we obtain
+a solution of (11):
+(U1, M0, U2) = LDU(Q−1M1,κM2,κQ), M′
+3 = Q−1M3Q, M′
+4 = Q−1M4Q.
+A change of mixed basis will modify Q by a multiplication on the righta side with a diagonal
+matrix. Therefore the class [(U1, M0, U2, M′
+3, M′
+4)]D is unique. The representations ρ for which
+the hypothesis of Lemma (3.34) is not satisfied are the representations for which the eigen-
+directions related to e1,κ and e−1
+2,κ are the same. This defines a component –here the line Le1,e2–
+of the reducibility locus along the path γ1,2 in πV I
+1 (X, S). Therefore Φκ is invertible outside
+Le1,e2. Note that for the other choices of κ or e0 we shall find one of the other components of
+R(γ1,2). ✷
+Using the trace maps, ϕ±∗
+κ
+induces two families of morphisms of cubic surfaces:
+Φ±
+κ = (Tr±κ
+V )−1 ◦ ϕ±∗
+κ
+◦ TrV I : CV (θ±) → CV I(κ(θ±))
+where κ(θ+) = κ(e0, a3, a4) = (κ + κ−1, e0κ−1 + e−1
+0 κ, a3, a4) and κ(a−) = κ(e−1
+0 , a3, a4) =
+(κ + κ−1, e−1
+0 κ−1 + e0κ, a3, a4).
+Theorem 3.35 (Confluent and diffluent morphisms) The morphisms Φ±
+κ are symplectic
+birational morphisms from CV (θ±) to CV I(κ(θ±).
+Proof. We have to prove that Φ+
+κ has rational expression in trace coordinates and is invertible
+in the class of birational transformations.
+Lemma 3.36 The map Φ+
+κ : CV (θ+) to CV I(κ(θ+) is defined by
+
+
+
+
+
+x1,κ = e−1
+0 κx1 + κ−1x2
+x2,κ = −e−1
+0 κx1x3 + κ−1x1 − e−1
+0 κx2 + a3κ + a4e−1
+0 κ
+x3,κ = x3.
+Proof. We set M1,κ = U1Dκ, M2,κ = Dκ−1M0U2. We have
+
+
+
+
+
+x1,κ = tr(M2,κM3) = α3e0κ−1 + e0γ3s2κ−1 + δ3e−1
+0 κ
+x2,κ = tr(M3M1,κ) = α3κ + β3s1κ + δ3κ−1
+x3,κ = tr(M1,κM2,κ) = e0s1s2 + e0 + e−1
+0 .
+Using Lemma (3.8), we obtain the expressions of Φκ in trace coordinates. ✷
+Lemma 3.37 This map is invertible outside the lines Le1,κ,e2,κ ∪ Le−1
+1,κ,e−1
+2,κ in χV I(aκ) and Φ−1
+κ
+is given by
+
+
+
+
+
+x1 = (−κx1,κ − e0κ−1x2,κ + a3e0 + a4)(x3,κ − ce1,κ,e2,κ)−1
+x2 = (κx1,κx3,κ − e0κ−1x1,κ + κx2,κ − a3κ2 − a4κ2e−1
+0 )(x3,κ − ce1,κ,e2,κ)−1
+x3 = x3,κ.
+34
+
+Proof. In order to find Φ−1
+κ
+we solve the system
+�
+e−1
+0 κx1 + κ−1x2 = x1,κ
+(−e−1
+0 κx3,κ) + κ−1)x1 − e−1
+0 κx2 = x2,κ − a3κ − a4e−1
+0 κ
+For a fixed value of x3 = x3,κ this system is linear and invertible outside the set
+e−1
+0 (x3,κ − e−1
+0 κ2 − e0κ−2) = 0.
+This set is the pair of lines x3,κ = e1,κe−1
+2,κ + e1,κe−1
+2,κ = ce1,κ,e2,κ, i.e. the pair of lines Le1,κ,e2,κ ∪
+Le−1
+1,κ,e−1
+2,κ in χV I(aκ). Solving this system, we obtain the expression of Φ−1
+κ
+given in the statement
+of the Lemma.✷
+Finally one can prove by using the above expressions that Φ+
+κ is a symplectic morphism (see
+also [40]) with respects to the symplectcic forms ωκ
+V (θ+) and ωV I(θκ). We have a similar result
+for Φ−
+κ .
+4
+The Painlev´e V vector field on MV .
+4.1
+The Riemann-Hilbert map RHV
+Proposition 4.1
+1. Any local connection ∇, defined by a germ of irregular sl2-system of
+Katz rank 1 induces a representation ρ∇ of the local wild groupoid π1(X0, S0) in SL2(C).
+2. Any connection A in CV induces a representation ρA of πV
+1 (X, S) in SL2(C).
+Proof. The representation of the local wild fundamental groupoid induced by a local irregular
+system is defined in the following way. We first consider an extension of the set of base points to
+any point of D or �D which is an extremity of the rays ri, r±
+i , i = 1, 2. The points p1 and p2 are
+defined by the singular directions of the connection. Any point of �D is arbitrarily represented
+by a determination in the direction d of a formal fundamental system of solutions �Yd of the
+connection and any point of D is represented by an actual holomorphic fundamental system
+of solutions Yd on a small sector around d. Any arc γ on D with origin a and end point b is
+represented, as usually by the comparison between the analytic continuation �
+Ya
+γ of Ya along γ
+with of Yb, i.e. by the matrix Mγ such that
+Yb = �
+Ya
+γ · Mγ.
+Any arc on �D is represented in the same way by using the formal representations of its extrem-
+ities, and the analytic continuation of the logarithm. Let rd be a regular ray whose origin and
+end-point are represented by �
+Yd and Yd. The morphism rd is represented by the comparison
+between the summation of �
+Yd (the summation replace here he analytic continuation) with Yd :
+Yd = Sd(�
+Yd) · Mrd.
+We represent the loops ti,i(κ) in the following way. One can suppose that the matrix �Ysi is a
+diagonal matrix. Then the action of �ti,i(κ) on �Ysi is given by the product with the diagonal
+matrix diag(κ−1, κ).Therefore ρA (ti,i(κ)) = Dκ. We have defined the representation ρ∇ on all
+the generators of π1(X0, S0) and therefore on π1(X0, S0).
+Given a global system A in SV , we complete the previous local wild representation to a
+representation of πV
+1 (X, S) by defining ρ∇(γi,j), (i, j) = (2, 3), (3, 4), (4, 1), (3, 3), (4, 4) in
+35
+
+the usual monodromic way, by analytic continuation along these paths or loops.
+A change
+of representations of the objects, or a gauge action on the linear system, induce equivalent
+representations, which end the proof. ✷
+We have defined a Riemann-Hilbert map
+RHV : MV (α) → χV (a).
+The general Riemann-Hilbert problem discusses about the surjectivity of RH: in the present
+context, for any irreducible representation ρ in χV , there exists a unique connection in MV on
+the trivial bundle whose linear representation is ρ: for details see [54], [11].
+4.2
+Isomonodromic families on MV
+Any isomonodromic family on MV (α) is a fiber of the map RHV , parametrized by a variable t,
+i.e. a family of connections defined by
+dY
+dx = A(x, t) · Y
+such that the monodromy and the Stokes operators are locally constant in χV (a).
+Theorem 4.2 There exists an open Zariski set in MV (α) and coordinates p, q on the fiber
+MV (α) such that the isomonodromic families are solutions the Painlev´e V hamiltonian differ-
+ential system (5).
+A sketch of proof.
+We follow here an argument of [54] wich extends the argument of
+Schlesinger for fuchsian connections. Given a fundamental system of solutions Y (x, t), by isomon-
+odromy, d
+dtY (x, t) and Y (x, t) have the same behaviour by analytic continuation or under a Stokes
+operator. Therefore the quotient
+B(x, t) = d
+dtY (x, t) · Y (x, t)−1
+is univalued outside the fixed singular points.
+Lemma 4.3 B(x, t) extends to the singular set in a meromorphic way.
+Proof.
+Let U ⊂ C be an open subset, and let D be an open disc centered at ∞.
+Let
+dY/dx = A(x−1, t)Y , where A ∈ sl2(O(D × U), be a parametrized meromorphic differential
+system. We suppose that, for all t ∈ U, the eigenvalues of A(0, t) are distinct and that the
+singularity is irregular, with a Katz rank equal to 1. From Theorem (2.7), the system dY/dx =
+AY admits, in a small neighborhood of each t0 ∈ U a formal fundamental solution analytic in
+t : �Y = �ΦxLeQ. The polynomials Q and the matrix L are analytic in t and the entries of �Φ
+belongs to O(V )[[x−1]], where V is a small open disc centered at t0. Moreover these coefficients
+are 1-summable uniformly in t. For a direction d non-singular at t0, if |t − t0| is small, then d
+remains non-singular and the 1-sum is analytic in t. Multiplying on the right by an invertible
+diagonal matrix analytic in t, we get a fundamental solution with similar properties.
+If the system is isomonodromic, then the matrix L is constant and it is possible to choose �Y
+in such a way that the Stokes multipliers does not depend on t.
+By a simple calculation, we see that �B(x, t) =
+d
+dt �Y (x, t) · �Y (x, t)−1 does not contain expo-
+nentials, and is invariant by the formal monodromy and by the (constant) Stokes multipliers.
+Therefore its entries belongs to O(D)((x−1)). These entries are 1-summable uniformly in t and
+36
+
+are invariant by the Stokes multipliers, therefore they are meromorphic in x (with a pole at ∞)
+and analytic in t. ✷
+Therefore Y (x, t) satisfies two rational linear systems dY
+dx = A(x, t) · Y and dY
+dt = B(x, t) · Y .
+The compatibility condition requires that the two operators d/dx − A and d/dt − B have to
+commute :
+dA
+dt − dB
+dz + [B, A] = 0.
+(12)
+The pair (A, B) is called an isomonodromic Lax pair. If A is irreducible, B is unique. In [54], the
+computation of B and the equivalence of the compatibility condition (12) with the hamiltonian
+system of PV have been performed. ✷
+4.3
+Singularities of the Painlev´e V vector field in the Okamoto’s compactifi-
+cation of MV (α)
+The Okamoto’s compactification [49], [29]. All the Painlev´e foliations satisfy the Painlev´e
+property: any solution admits an extension along any path is the basis given by the time variable
+punctured by the fixed singular points, as a meromorphic function.
+In a first step, we search for a compactification on which appear the “polar” singular set,
+which corresponds to the poles of the meromorphic solutions. Here, following H. Chiba [16] and
+[17], we have to distinguish the equations PIV , PII and PI from PV I, PV and PIII. For the first
+triple we obtain this compactification by a convenient weighted projective space. For the second
+one, the natural weighted projective space presents a vanishing weight and we can’t catch all
+the polar set. In this case we need to glue two copies of this weighted projective space by a
+convenient B¨acklund transformation.
+In a second step one can remove the polar singular set by a convenient sequence of blowing
+up’s. In the version of H. Chiba, it suffices to perform only one weighted blowing up whose
+weights are given by the eigenvalues of the polar singular point, which are positive integers.
+The first step creates new singular points outside the polar singular set, over z = ∞, which
+are saddle-nodes. We describe here this set for the PV foliation. In this case, the weights are
+ω = (1, 0, 1, 1).
+Since the second weight vanishes, this space P3
+(1,0,1,1) is not compact and is
+isomorphic to P2 × C. This space is here a manifold (an orbifold for other Painlev´e equations:
+[Chiba1-2-4]), covered by 3 charts:
+U1 = {(X1 : X2 : X3 : X4), X1 ̸= 0}, (X1 : X2 : X3 : X4) = (1 : u1 : u2 : x);
+U3 = {(X1 : X2 : X3 : X4), X3 ̸= 0}, (X1 : X2 : X3 : X4) = (v1 : v2 : 1 : y);
+U4 = {(X1 : X2 : X3 : X4), X4 ̸= 0}, (X1 : X2 : X3 : X4) = (w1 : w2 : z : 1).
+The change of charts are given by
+w1 = v1y−1 = x−1, w2 = v2 = u1, z = y−1 = u2x−1.
+Singular points.
+The chart U4 is the ”initial chart”, before compactification. In this chart
+the vector field corresponding to the hamiltonian zHV is defined by
+
+
+
+
+
+z ˙w1 = −2w2
+1w2 + w2
+1 + w1z − 2w1w2z + (α1 + α3)w1 − α2z
+z ˙w2 = 2w1w2
+2 − 2w1w2 − w2z + w2
+2z − (α1 + α3)w2 + α1
+˙z = z
+37
+
+For each vector field, the plane z = 0 is invariant and the singular points of the restriction of
+the vector field to this plane are given by
+
+
+
+
+
+(−2w1w2 + w1 + (α1 + α3))w1 = 0
+2w1w2
+2 − 2w1w2 − (α1 + α3)w2 + α1 = 0
+z = 0
+For α1 ̸= ±α3, we obtain 2 singular points r1 and r2:
+(w1, w2, z) =
+�
+0,
+α1
+α1 + α3
+, 0
+�
+,
+(w1, w2, z) =
+�
+α1 − α3,
+α1
+α1 − α3
+, 0
+�
+.
+Now we search for singular points over z = ∞. In the first chart the Painlev´e vector field is (see
+[17]):
+
+
+
+
+
+˙u1 = −2u1 + 2u2
+1 − u1u2 + u2
+1u2 + α1x − (α1 + α3)u1x
+˙u2 = −u2 + 2u1u2 − u2
+2 + u2x + 2u1u2
+2 − (α1 + α3)u2x + α2u2
+2x
+˙x = x(−1 + 2u1 − u2 + 2u1u2 − (α1 + α3)x + α2u2x)
+The plane (x = 0) is invariant and PV |(x=0) is the autonomous hamiltonian vector field
+�
+˙u1 = −2u1 + 2u2
+1 − u1u2 + u2
+1u2 = u1(u1 − 1)(u2 + 2)
+˙u2 = −u2 + 2u1u2 − u2
+2 + 2u1u2
+2 = u2(2u1 − 1)(u2 + 1)
+We have 5 singularities p1, p2, s1, s2, s3 in this first chart, which do not depend on the parameters
+αi:
+(u1, u2, x) = (0, 0, 0), (1, 0, 0), (0, −1, 0), (1
+2, −2, 0), (1, −1, 0).
+The singularities p1 and p2 are polar singular points: H. Chiba has proved in [17] that they
+correspond to solutions given by Laurent series at a movable pole z = z0. One can remove these
+singularities by a weighted blowing up. The eigenvalues of linear part of PV are (−2, −1, −1)
+and (2, 1, 1) respectively. They are analytically linearizable.
+In the third chart –the Boutroux chart–, the Painlev´e vector field is given by
+
+
+
+
+
+˙v1 = v1(1 + v1 − 2v2 − 2v1v2 + (α1 + α3 − 1)y) − α2y
+˙v2 = v2(−1 + v2 − 2v1 + 2v1v2 − (α1 + α3)y) + α1y
+˙y = −y2
+The plane y = 0 is invariant and the restriction of the vector field at infinity is
+�
+˙v1 = v1(1 + v1 − 2v2 − 2v1v2) = v1(v1 + 1)(1 − 2v2)
+˙v2 = v2(−1 + v2 − 2v1 + 2v1v2) = v2(v2 − 1)(2v1 + 1)
+There are five singular points s1, s2, s3, s4 and s5 in y = 0 given by
+(v1, v2, y) = (−1, 0, 0), (−1
+2, 1
+2, 0), (−1, 1, 0), (0, 0, 0), (0, 1, 0).
+38
+
+The polar singular set of PV contains 4 points. In order to catch the two missing polar singular
+points, H. Chiba introduces a second copy of �
+P3(1,0,1,1) glued to the first one by the B¨acklund
+transformation π, given in each chart by
+π :( ˜w1, ˜w2, ˜z, ˜α0, ˜α1, ˜α2, ˜α3) = (z(w2 − 1), −w1z−1, z, α1, α2, α3, α0)
+(˜v1, ˜v2, ˜y, ˜α0, ˜α1, ˜α2, ˜α3) = (v2 − 1, −v1, y, α1, α2, α3, α0)
+(˜u1, ˜u2, ˜x, ˜α0, ˜α1, ˜α2, ˜α3) = (−u−1
+2 , (u1 − 1)−1, (u1 − 1)−1u−1
+2 x, α1, α2, α3, α0)
+By computing the singularities of the vector field in the chart ( ˜w1, ˜w2, ˜z), we find two singular
+points r3 and r4 given by:
+( ˜w1, ˜w2, ˜z) = (˜α1 − ˜α3 + 1,
+˜α1
+˜α1 − ˜α3 + 1, 0), ( ˜w1, ˜w2, ˜z) = (0, −
+˜α1
+˜α0 + ˜α2
+, 0).
+We have r4 = r1, and therefore we obtain 3 singular points of regular type.
+In the chart
+(˜u1, ˜u2, ˜x), we find five singular points:
+p3 : (˜u1, ˜u2, ˜x) = (0, 0, 0),
+p4 : (˜u1, ˜u2, ˜x) = (1, 0, 0),
+s1 : (˜u1, ˜u2, ˜x) = (1, −1, 0),
+s2 : (˜u1, ˜u2, ˜x) = (1
+2, −2, 0),
+s4 : (˜u1, ˜u2, ˜x) = (0, −1, 0).
+According to [17], the singular points p3 and p4 are the two missing polar singular points. The
+singularities s1, s2 and s4 was yet detected in the first copy. The singular points in the Boutroux
+chart (˜v1, ˜v2, y) = (v2 −1, −v1, y) are s1, s2, s3, s4 and s5 without new singular point (the change
+of chart is polynomial). We have obtained:
+Proposition 4.4 The Painlev´e vector field admits 3 singular points r1, r2, r3 of regular type
+over z = 0, 4 polar singular points pi, i = 1...4 over z = ∞, which can be removed by a weighted
+blowing-up, and 5 singular points si over z = ∞, of saddle-node type.
+By a local study around each saddle-node singularity, one can recover the 5 solutions of A.
+Parusnikova in Laurent series around z = ∞ for the Painlev´e V equation: [52].
+5
+Dynamics on χV
+5.1
+The tame dynamics
+As mentionned in the introduction, the dynamic of the Painlev´e VI equation on χV I is induced
+by the automorphisms of the groupoid, given by braids over 3 points. Here since the singular
+set reduces to 3 points, the braid group over two points has only one generator, which can
+be represented by the pure braid b over s3 and s4. Furthermore one can consider the “half-
+monodromy” defined the (non pure) braid b3,4 which permutes the positions of s3 and s4 and
+induces an involution on the local parameters. These actions induced by the trace coordinates
+on CV (θ+) has been computed by the authors in [53] and M. Klimes in [40].
+Proposition 5.1 The action of the braid b3,4 : CV (θ+
+1 , θ+
+2 , e0) → CV (e−1
+0 θ+
+2 , e−1
+0 θ1+, e−1
+0 ) is
+defined by:
+
+
+
+
+
+x′
+1 = e−1
+0 (−x2 − x1x3 + θ+
+2 )
+x′
+2 = e−1
+0 x1
+x′
+3 = x3
+and
+
+
+
+
+
+e′
+0 = e−1
+0
+θ+
+1
+′ = e−1
+0 θ+
+2
+θ+
+2
+′ = e−1
+0 θ+
+1
+39
+
+The action of the pure braid b3,4 ◦ b3,4 : CV (θ+) → CV (θ+) is defined by:
+g3,4 :
+
+
+
+
+
+x′
+1 = x1x2
+3 + x2x3 − x1 − θ+
+2 x3 + θ+
+1
+x′
+2 = −x1x3 − x2 + θ+
+2
+x′
+3 = x3
+As for the dynamics on CV I(θ), this dynamics and its inverse are polynomial dynamics.
+This is the only part of the dynamics on χV I which do not degenerate through the confluent
+morphisms from χV I to χV (see next paragraph).
+Definition 5.2 The dynamics generated by g3,4 is called the tame dynamics, and denoted by
+Tame(CV ) .
+5.2
+The confluent dynamic
+Using the birational map Φκ = Φ+
+κ , each dynamic hi,j on CV I(θκ) defined by the braids induces
+a family of birational dynamics gi,j(κ) = φ−1
+κ
+◦ hi,j ◦ φκ on χ+
+V (a). This one was previously
+obtained by M. Klimes by using an analytic confluent process in the spaces of connections.
+Definition 5.3 The confluent dynamic Conf(PV ) is the dynamic generated by g1,2(κ), g2,3(κ)
+and g3,1(κ) for κ is C∗.
+We still have the relation g1,2(κ) ◦ g2,3(κ) ◦ g3,1(κ) = id. Therefore it suffices to compute g1,2(κ)
+and g2,3(κ).
+Notice that, from the braid group relations, g1,2(κ) = g3,4(κ)−1.
+By a direct
+computation using proposition (3.36), it can be checked that
+Proposition 5.4 The dynamic g3,4(κ) = φ−1
+κ
+◦ h3,4 ◦ φκ on χV (a) do not depend on κ and
+generates the tame dynamic.
+Remark 5.5 Since the braid between s1 and s2 (or between s3 and s4) corresponds to a loop
+around 0 in the time space, g3,4 is conjugated through the Riemann-Hilbert correspondence to
+the non linear monodromy of the foliation FV over z = 0.
+Contrary to the previous case, the dynamics g2,3(κ) and g3,1(κ) are not generated by an
+automorphism of πV
+1 (Y, S): indeed on the groupoid level, ϕκ is not an isomorphism. Nevertheless
+they induce families of birational dynamics on CV (θ):
+Proposition 5.6 The family of dynamics g2,3(κ) is defined by:
+X1 = e0
+x2
+X2 = x2 − κ2
+x2
++ e−1
+0 κ2x1
+X3 = κ−2x2
+2x3 − (e−2
+0 κ2 − κ−2)x1x2 − 2e−1
+0 x2
+2 + (e−1
+0 θ2 + κ−2θ1)x2 + (e−1
+0 κ2 + e0κ−2)
+The second family g3,1(κ) is given by g1,3(κ) = g1,2 ◦ g2,3(κ).
+We first search for the matrix expressions of the confluent dynamics :
+40
+
+Proposition 5.7 The confluent dynamic (ϕ∗
+κ)−1 ◦ h2,3 ◦ ϕ∗
+κ :
+χκ
+V (a) → χκ
+V (a) is defined by
+([U1, M0, U2, M3]D → [V1, M0, V2, N3]D with
+V1 =
+
+
+
+1
+0
+κ−1e0s1θ3 + κ−1e−1
+0 γ3 − κe−1
+0 γ3 + κ−1e0s1s2γ3
+1
+
+
+ ,
+V2 =
+
+
+
+1
+κ−1e0s2θ3 + κ−1e0s2
+2γ3 − κ−1e0s2δ3 − κ−1e0β3 + κe−1
+0 β3 + κe−1
+0 s2δ3
+0
+1
+
+
+ ,
+N3 =
+1
+θ3 + s2γ3
+
+
+
+θ2
+3 + θ3γ3s2 + β3γ3 + γ3δ3s2
+−e0κ−1(θ3s2 + γ3s2
+2β3 − δ3s2)
+κe−1
+0 γ3
+1
+
+
+ .
+Proof. We consider an element ρ of χV (a) given by its normalized representation in the
+canonical Cartan-Borel decomposition:
+ρ = [U1, M0, U2, M3, M4], (U1, M0, U2, M3) ∈ U −×D×U + ×SL2(C), M4 = (U1M0U2M3)−1.
+We set:
+U1 =
+
+
+
+1
+0
+u1
+1
+
+
+ , M0 =
+
+
+
+e0
+0
+0
+e−1
+0
+
+
+ , U2 =
+
+
+
+1
+u2
+0
+1
+
+
+ , M3 =
+
+
+
+θ3
+β3
+γ3
+δ3
+
+
+ .
+The map Φκ is defined by:
+Φκ([U1, M0, U2, M3, M4]) = [M1, M2, M3, M4] with M1 = U1Dκ, M2 = D−1
+κ M0U2.
+Therefore
+M1 =
+
+
+
+κ
+0
+κu1
+κ−1
+
+
+ , M2 =
+
+
+
+κ−1e0
+κ−1e0u2
+0
+κe−1
+0
+
+
+ , M3 =
+
+
+
+θ3
+β3
+γ3
+δ3
+
+
+ .
+According to [53], the dynamic of h2,3 is given by
+M1 → (M2M3)−1M1(M2M3)
+M2 → M2
+M3 → M3
+M4 → (M2M3)−1M4(M2M3)
+Since each data is given up to a common conjugacy, we can also use:
+M1 → M′
+1 = M2−1M1M2
+M2 → M′
+2 = M3M2M3−1
+M3 → M′
+3 = M3
+M4 → M′
+4 = M2−1M4M2
+41
+
+Now, in order to compute φ−1
+κ (M′
+1, M′
+2, M′
+3, M′
+4), we make use of the matrix P given by Lemma
+(3.34), defined by a mixed basis for the pair (M′
+1, M′
+2). Let (u, v) a basis of the representation
+of the initial object such that the representation Φκ(ρ) is given by the matrices Mi: u is an
+eigenvector of M2 for the eigenvalue e2 = κ−1e0 and v is an eigenvector of M1 for the eigenvalue
+e1−1 = κ−1. The vectors
+u′ = M3 · u′, v′ = M2−1 · v
+are eigenvectors for M′
+2 (with eigenvalue κ−1e0) and for M′
+1 (with eigenvalue κ−1). The change
+of basis is given by the matrix
+P =
+
+
+
+θ3
+−κ−1e0u2
+γ3
+κ−1e0
+
+
+
+which is invertible outside
+κ−1e0(b3 + u2γ3) = 0.
+Using Lemma (3.8), this locus is given by x2 = 0. Therefore the dynamic g2,3(κ) is a rational
+dynamic with polar set x2 = 0. We have:
+P −1(M′
+1M′
+2)P =
+
+
+
+e0
+κ−1e2
+0u2b3 + κ−1e2
+0u2
+2γ3 − κ−1e2
+0u2δ3 − κ−1e2
+0β3 + κβ3 + κu2δ3
+κ−1e2
+0u1b3 + κ−1γ3 − κγ3 + κ−1e2
+0u1u2γ3
+⋆
+
+
+ .
+where the coefficient ⋆ is not specified.
+The LDU-decomposition of this matrix is given by
+(V1, M0, V2) with
+V1 =
+
+
+
+1
+0
+κ−1e0u1b3 + κ−1e−1
+0 γ3 − κe−1
+0 γ3 + κ−1e0u1u2γ3
+1
+
+
+ ,
+V2 =
+
+
+
+1
+κ−1e0u2b3 + κ−1e0u2
+2γ3 − κ−1e0u2δ3 − κ−1e0β3 + κe−1
+0 β3 + κe−1
+0 u2δ3
+0
+1
+
+
+ .
+we have:
+N3 = P −1M′
+3P =
+1
+b3 + u2γ3
+
+
+
+b2
+3 + b3γ3u2 + β3γ3 + γ3δ3u2
+−e0κ−1(b3u2 + γ3u2
+2β3 − δ3u2)
+κe−1
+0 γ3
+1
+
+
+
+which proves the proposition. ✷
+Proof of Proposition (5.6). The image of x1 = δ3 is given by X1 = N3[2, 2] =
+1
+b3+u2γ3 . According
+to Lemma (3.8), we have:
+N3[2, 2] =
+1
+b3 + u2γ3
+=
+1
+(a3 − x1) + e−1
+0 (x2 − e0a3 + e0x1) = e0
+x2
+.
+42
+
+The image of x2 = δ4 is given by N4[2, 2] where N4 = P −1M′
+4P. We have
+N4[2, 2] = −e−2
+0 κ2γ3β4 + b3δ4 + u2γ3δ4
+b3 + u2γ3
+.
+From M4 = (U1M0U2M3)−1 we obtain that β4 = −e0(β3 + u2δ3). Therefore,
+γ3β4 = −e0γ3β3 − e0u2γ3δ3
+= −e0(b3δ3 − 1) − e0u2γ3δ3
+= −e0((a3 − x1)x1 − 1) − x1(x2 + e0x1 − e0a3)
+= e0 − x1x2.
+From Lemma (3.8) and the above equality we obtain
+X2 = N4[2, 2] = x2 − κ2x−1
+2
++ e−1
+0 κ2x1.
+We can obtain the last component by computing X3 = tr(M′′
+1 M′′
+2 ) = tr(M′
+2
+−1M′
+1M′
+2M′
+3M′
+2M′
+3
+−1),
+or by using the equation of the cubic surface:
+X3 = −X2
+1 − X2
+2 + θ1X1 + θ2X2 − θ4
+X1X2 − e0
+.
+✷
+5.3
+The canonical dynamics on CV (θ)
+In [40], Martin Klimes has computed the dynamics obtained on CV (θ) from the wild dynamics
+(non linear stokes operators and non linear exponential tori of one irregular singularity of the
+Painlev´e foliation) through the Riemann-Hilbert morphism. It turns out that this dynamics
+is a rational one on CV (θ). We enhance here that this dynamics was already present on the
+cubic surface in a canonical way, using the log-canonical coordinates introduced in subsection
+3.4. This section is completely independent of the other ones excepted 3.4, and only deals with
+dynamics on a singular cubic surface. Nevertheless the terminology (canonical Stokes operators
+etc...) is deeply influenced by the confluent dynamics. We do not need here any restriction on
+the parameters.
+Let y be a log-canonical function on CV (θ): there exists z such that (y, z) –or (z, y)– is a
+log-canonical system of coordinates. We consider the logarithmic hamiltonian function Hy on
+CV (a) such that
+dHy = dy
+y .
+Let Xy be the hamiltonian vector field related to Hy for the symplectic form ωV :
+ωV (Xy, ·) = −dy
+y .
+Through the symplectic morphism (y, z), the image of this vector field is z ∂
+∂z (or −z ∂
+∂z if y is
+completed in log-canonical system of coordinates by the left side). Therefore its flow z(t) = z0e±t
+is globally defined on C, and it can be factorized through a multiplicative action of C∗ by setting
+λ = et. Let Ty be this multiplicative family of (rational symplectic) automorphisms on CV .
+43
+
+Definition 5.8 Ty = {tλ :
+(z′, y) �→ (λ−1z′, y)} = {tλ :
+(y, z) �→ (y, λz)} is the exponential
+torus related to the log canonical function y.
+Remark 5.9
+1. Each element of Ty keeps invariant y, and the set of rational invariant
+functions of Ty is C(y).
+2. Consider the log-canonical pentuple (z0, y1, z1, y2, z2). The elements of Ty1 keep invariant
+the reducible locus z0 = 0 and z1 = 0, and the elements of Ty2 keep invariant the reducible
+locus z1 = 0 and z2 = 0.
+The subgroups of Symp = Symp(C2, ωlog): Bi, B♮
+i, Ui, i = 1, 2 are defined in the Appendix.
+They induce subgroups of Symp(CV , ωV ) by using a system of log-canonical coordinates. Given
+a log canonical function z = zk in S, one can complete z into two log-canonical systems, either
+by the left side: (y, z) or by the right side: (z, y′).
+Lemma 5.10
+1. We fix a canonical triple (y, z, y′).
+The pull-back of B2 (resp.
+B♮
+2, U2)
+by (y, z) and the pull-back of B1 (resp. B♮
+1, U1) by (z, y′) define the same subgroup of
+Sym(CV , ωV ).
+2. We fix a canonical triple (z, y, z′). The pull-back of B2 (resp. B♮
+2, U2) by (z, y) and the
+pull-back of B1 (resp. B♮
+1, U1) by (y, z′) define the same subgroup of Sym(CV , ωV ).
+The subgroups obtained at the first point of Lemma (5.10) are denoted by Bz, B♮
+z, Uz, and the
+subgroups obtained at the second point are denoted by By, B♮
+y, Uy.
+Definition 5.11 Given a canonical system of coordinates (y, z) we will say that By and Bz are
+the opposite Borel subgroups associated to (y, z).
+Proof of Lemma (5.10). 1- The elements of the pullback of B2 by (y, z) are given by
+(y, z) → (r(z)y, λz), r ∈ C(z)∗, λ ∈ C∗,
+and the elements of the pullback of B1 by (z, y′) are given by
+(z, y′) → (λz, r(z)y′), r ∈ C(z)∗, λ ∈ C∗.
+Therefore using yy′ = z + e0, the elements of the first group also write
+(z, y′) → (λz, r′(z)y′), with r′(z) =
+λz + e0
+(z + e0)r(z),
+which proves the first point.
+2- The elements of the pullback of B2 by (z, y) are given by
+(z, y) → (r(y)z, λy), r ∈ C(y)∗, λ ∈ C∗,
+and the elements of the pullback of B1 by (y, z′) are given by
+(y, z′) → (λy, r(y)z′), r ∈ C(y)∗, λ ∈ C∗.
+Therefore using zz′ = Q(y), where Q = Q1 or Q2 is a polynomial, the elements of the first group
+also write
+(y, z′) → (λy, r′(y)z′), with r′(y) =
+Q(λy)
+Q(y)r(y),
+which proves the second point. ✷
+44
+
+Proposition 5.12 We have Z(Ty) = By, N(Yy) = By∪B−
+y , where B−
+y = {(y, z′) → (λy, r(y)z′−1), r ∈
+C(y)∗, λ ∈ C∗}.
+Proof. Let tλ : (y, z′) → (y, λz′). An element ρ of Symp(CV ), given by
+ρ : (y, z′) → (Y (y, z′), Z′(y, z′))
+commutes with tλ if and only if, for any λ in C∗,
+Y (y, λz′) = Y (y, z′),
+Z′(y, λz′) = λZ′(y, z′).
+We claim that any rational function r in K(x) over a field K ⊃ C which satisfies r(λx) = λr(x)
+for any λ in C∗ is a constant function. Indeed if it was not a constant function it would admit
+an infinite number of zeroes or poles in K. By applying this remark to z′ �→ Y (y, z′) and to
+z′ �→ Z′(y, λz′)z′−1, we conclude that Y do not depend on z′ and that Z′ is linear in z′. There
+exists two rational functions q(y) and r(y) such that
+Y (y, λz′) = q(y),
+Z′(y, λz′) = r(y)z′.
+We have
+dY
+Y
+∧ dZ′
+Z′ = q′(y)dy
+y ∧ (r′(y)dy
+y
++ dz
+z ) = q′(y)dy
+y ∧ dz
+z .
+Therefore q′(y) = 1, and q(y) = µ in C∗. The elements of Z(Ty) write
+(y, z′) → (µy, r(y)z′)
+which prove that Z(Ty) = By.
+Suppose now that an element ρ is in the normalizer of Ty.
+Then either it commutes whith
+each element of Ty, or it anti-commutes with each element of Ty. In this last case a similar
+computation proves that ρ : (y, z′) → (µy, r(y)z′−1). ✷
+Proposition 5.13 Let T = (y, z, y′) be a log-canonical triple in the cluster sequence S. There
+exists a unique element sT of Uz such that sTTys−1
+T
+= Ty′. This operator is given by:
+sT : (y, z) → (y(1 + e−1
+0 z), z) or by
+(z, y′) → (z, y′(1 + e−1
+0 z)−1) = (z, e0y−1)
+and we have: s−1
+T
+: (y, z) → (y(1 + e−1
+0 z)−1, z) = (e0y′−1, z).
+Proof. We have:
+Ty = {ty(λ) : (y, z) → (y, λz)},
+Ty′ = {ty′(λ) : (z, y′) → (λ−1z, y′)}
+= {ty′(λ) : (y, z) → (y1 + e−1
+0 λ−1z
+1 + e−1
+0 z
+, λ−1z)},
+Uz = {ur : (y, z) → (r(z)y, z), r ∈ C(z)∗, r(0) = 1}.
+Therefore the element s of Uz: (y, z) → (y(1 + e−1
+0 z), z) satisfies for all λ in C∗
+s ◦ ty(λ) ◦ s−1 = ty′(λ−1).
+In order to prove the unicity of s, suppose that there exist another s′ in Uz which conjugates Ty
+and Ty′. We have : s−1 ◦ s′(Ty) = Ty i.e. s−1 ◦ s′ belongs to the normalizer N(Ty) of Ty. We
+have N(Ty) ∩ Uz = {id} : this is a direct consequence of Proposition (5.12).✷
+45
+
+Definition 5.14 The automorphism sT is the canonical Stokes operator induced by the triple
+T = (y, z, y′).
+Proposition 5.15
+1. The automorphism sT writes in the coordinates h = log y, l = log(1 +
+e−1
+0 z): (h, l) → (h + l, l).
+2. sT is a pseudo-generator of Uz, that is to say the family {ty(λ) ◦ sT ◦ ty(λ)−1, λ ∈ C∗}
+generates Uz. Consequently, B♮
+z :=< Uz, Ty >=< sT, Ty >.
+3. Let sk be the canonical Stokes operator related to the triple (yk, zk, yk+1). We have: σ1 ◦
+s1 ◦ σ−1
+1
+= s−1
+2 ; g ◦ sk ◦ g−1 = sk+2.
+4. For i = 1, 2, si keeps invariant the lines zi = 0, and the restriction of si on each line is a
+translation. More generally, sk keeps invariant zk = 0, and the restriction of sk to the set
+of rational curves zk = 0 has no fixed point.
+Proof.
+1. This first point is trivial.
+2. We have:
+{ty(λ) ◦ sT ◦ ty(λ)−1, λ ∈ C∗} = {(y, z) → (y(1 + νz), z), ν ∈ C∗}
+The statement comes from the fact that any rational function r such that r(0) = 1 writes
+r(z) = �
+i(1 − µiz) �
+j(1 − νjz)−1 where the µ−1
+i
+are the zeroes of r and the ν−1
+j
+are the
+poles of r.
+3. We have:
+s1 : (y1, z1) → (y1(1 + e−1
+0 z1, z1)
+s2 : (z2, y3) → (z2, y3(1 + e−1
+0 z2)−1).
+Since σ1 : (y1, z1) → (y3, z2), we have σ1 ◦ s1 ◦ σ−1
+1
+= s−1
+2 .
+The automorphism g−1 send the triple (yk, zk, yk+1) on (yk+2, zk+2, yk+3). Therefore sk =
+g−1 ◦ sk+2 ◦ g.
+4. From the previous point, it suffices to prove the result for s1. We lift the expression of s1
+in the coordinate system (y1, z1, x3):
+s1 : (y1, z1, x3) → (e−1
+0 y1(z1 + e0), z1, X3).
+The equation of the character variety CV (θ) is given by
+x3z1 + y2
+1 + (z1 + e0)2y−2
+1
+− θ1y1 − θ2(z1 + e0)y−1
+1
++ θ4 = 0,
+which is equivalent to
+x3z1 + y−2
+1 z2
+1 + 2e0y−2
+1 z1 − θ2y−1
+1 z1 = −y2
+1 − e2
+0y−2
+1
++ θ1y1 + θ2e0y−1
+1
+− θ4.
+We have:
+X3z1 + e−2
+0 y2
+1(z1 + e0)2 + e2
+0y−2
+1
+− θ1e−1
+0 y1(z1 + e0) − θ2e0y−1
+1
++ θ4 = 0.
+46
+
+Therefore,
+X3z1 = −e−2
+0 y2
+1z2
+1 − 2e−1
+0 y2
+1z1 + θ1e−1
+0 y1z1 − y2
+1 − e2
+0y−2
+1
++ θ1y1 + θ2e0y−1
+1
+− θ4
+= −e−2
+0 y2
+1z2
+1 − 2e−1
+0 y2
+1z1 + θ1e−1
+0 y1z1 + x3z1 + y−2
+1 z2
+1 + 2e0y−2
+1 z1 − θ2y−1
+1 z1.
+We obtain:
+X3 = −e−2
+0 y2
+1z1 − 2e−1
+0 y2
+1 + θ1e−1
+0 y1 + x3 + y−2
+1 z1 + 2e0y−2
+1
+− θ2y−1
+1 .
+Therefore the restriction of s1 to z1 = 0 is given by
+x3 → x3 + θ1e−1
+0 y1 − θ2y−1
+1 .
+On each component of z1 = 0, y1 is constant (= e±
+3 or e0e±1
+4 ) and this map is a translation.
+✷
+Definition 5.16 The canonical dynamics induced by the canonical triple T = (y, z, y′) is the
+subgroup Dyn(T) =< Ty, sT, Ty′ > of Symp(CV ) generated by Ty, sT, and Ty′.
+Proposition 5.17 We have : Dyn(T) =< Ty, sT >= B♮
+z.
+Proof. The first equality is a consequence of the relation sTTys−1
+T
+= Ty′ obtained in Propo-
+sition (5.13), and the second one was obtained at the second point of Proposition (5.15). ✷
+Definition 5.18 The canonical dynamics Dyn(CV ) induced by the fundamental canonical se-
+quence {y0, z0, y1, z1, y2, z2, y3} is the rational symplectic dynamic generated by:
+g : (y2, z2) → (y0, z0), Ty1, s1 = s(y1,z1,y2), s2 = s(y2,z2,y3).
+The subgroup Tame(CV ) of Dyn(CV ) generated by g is the tame canonical dynamics.
+Remark 5.19
+1. From Proposition (3.25), the canonical sequence is almost unique, that is
+we can only replace zk with czk. This action is an element of Tyk and therefore do not
+change the dynamics. This dynamics only depends on the polynomial FV , which justify the
+notation.
+2. Dyn(CV ) also contains Ty0 = g∗Ty2 and s0 = s(y0, z0, y1) = g∗s2, and more generally all
+the Tyk and all the sk for k in Z.
+3. Dyn(CV ) =< B♮
+z1, B♮
+z2, g > .
+Proposition 5.20 The element �m of Dyn(CV ) such that g = s2 ◦ �m ◦ s1 is defined by
+(y2, z2) → (y2, e2
+0z2y−4
+2 ).
+Proof. We have
+s∗
+1z2 = s∗
+1Q2(y2)z−1
+1
+= Q2(e0y−1
+1 )z−1
+1
+= e2
+0y−4
+1 Q1(y1)z−1
+1
+= e2
+0y−4
+1 z0.
+Therefore,
+g−1∗s∗
+1z2 = e2
+0y−4
+3 z2,
+s∗
+2g−1∗s∗
+1z2 = e−2
+0 y4
+2z2.
+We also have
+s∗
+2g−1∗s∗
+1y2 = s∗
+2g−1∗(e0y−1
+1 ) = s∗
+2(e0y−1
+3 ) = y2,
+which proves that �m−1 = s1 ◦ g−1 ◦ s2 is defined by (y2, z2) → (y2, e−2
+0 z2y4
+2). ✷
+Definition 5.21 The element �m in Dyn(CV ) is the canonical formal monodromy.
+Note that �m = t2(e2
+0y−4
+2 ). In particular, �m is in the centralizer Z(Ty2) = By2 of Ty2, and
+preserves y2.
+47
+
+5.4
+Comparison between the confluent and the canonical dynamics
+We have previously found that the confluent dynamics and the canonical dynamics are both
+extensions of the tame dynamics.
+In order to compare Dyn(CV ) and Conf(PV ), we write
+the confluent dynamic in the canonical coordinates.
+The confluent dynamic is generated by
+g1,2 = g−1
+3,4 given by Proposition (5.4), the family g2,3(κ) given by Proposition (5.6) and the
+family g3,1(κ), which satisfies the relation g1,2 ◦ g2,3(κ) ◦ g3,1(κ) = id.
+Proposition 5.22 We have:
+(i) We have g1,2 = g. Hence g1,2 : (yn, zn) → (yn−2, zn−2).
+(ii) The automorphism g2,3(κ) and its inverse are given in canonical coordinate systems by
+g2,3(κ) : (z1, y2) → (κ2z1y−2
+2 , (1 + κ2e−1
+0 z1y−2
+2 )y2),
+g3,2(κ) : (y1, z1) → (y1 + e0κ−2z1y−1
+1 , e2
+0κ−2z1y−2
+1 ).
+(iii) The automorphism g3,1(κ) and its inverse are given in canonical coordinate systems by
+g3,1(κ) : (z2, y3) → (κ2z2y−2
+3 , (1 + κ2e−1
+0 z2y−2
+3 )y3),
+g1,3(κ) : (y2, z2) → (y2 + e0κ−2z2y−1
+2 , e2
+0κ−2z2y−2
+2 ).
+Proof.
+(i) We have g = σ1 ◦ σ2, with
+σ1(x1, x2, x3) = (x1 − FV,x1, x2, x3) = (−x1 − x2x3 + θ1, x2, x3)
+σ2(x1, x2, x3) = (x1, x2 − FV,x2, x3) = (x1, −x2 − x1x3 + θ2, x3).
+Therefore,
+(σ1 ◦ σ2)∗x1 = σ∗
+2(−x1 − x2x3 + θ1)
+= −x1 + x2x3 + x1x2
+3 − θ2x3 + θ1 = (g1,2)∗x1.
+(σ1 ◦ σ2)∗x2 = σ∗
+2(x2)
+= −x2 − x1x3 + θ2 = (g1,2)∗x2.
+Hence, g = g1,2.
+(ii) From Proposition (5.6) we have:
+g2,3(κ)∗z1 = e0x−1
+2 (x2 − κ2x−1
+2
++ e−1
+0 κ2x1) − e0
+= e0 + κ2x−2
+2 (x1x2 − e0) − e0
+= κ2z1y−2
+2 .
+g2,3(κ)∗y2 = g2,3(κ)∗((z1 + e0)y−1
+1 )
+= (κ2z1y−2
+2
++ e0)e−1
+0 y2
+= y2(1 + e−1
+0 κ2z1y−2
+2 ).
+We obtain g3,2(κ) by reversing the map (z1, y2) → (Z1, Y2):
+z1 = κ−2Y 2
+2 (1 + e−1
+0 Z1)−2, y2 = Y2(1 + e−1
+0 Z1)−1,
+and by using the change of canonical coordinates (z1, y2) → (y1 = (z1 + e0)y−1
+2 , z1).
+48
+
+(iii) We have g1,3(κ) = g ◦ g2,3(κ). We set Z2 := κ2z2y−2
+3 , Y3 := y3
+�
+1 + κ2e−1
+0 z2y−2
+3
+�
+. We have
+to prove that g1,3(κ)∗Z2 = z2 and g1,3(κ)∗Y3 = y3.
+g1,3(κ)∗Z2 = g2,3(κ)∗ ◦ g∗(κ2z2y−2
+3 ) = g2,3(κ)∗(κ2z0y−2
+1 ).
+g1,3(κ)∗Y3 = g2,3(κ)∗ ◦ g∗Y3 = g2,3(κ)∗ �
+y1(1 + κ2e−1
+0 z0y−2
+1 )
+�
+.
+We compute g2,3(κ)∗z0 and g2,3(κ)∗y1. We have z0 = z−1
+1 Q1(y1), therefore :
+g2,3(κ)∗z0 = (g2,3(κ)∗z0)−1 Q1 (g2,3(κ)∗y1) = κ−2z−1
+1 y2
+2Q1
+�
+e0y−1
+2
+�
+.
+We recall (cf. the remark 3.29) that Q1(e0t−1) = e−2
+0 t4Q2(t). We obtain :
+g2,3(κ)∗z0 = κ−2e2
+0y−2
+2 z−1
+1 e2
+0y4
+2Q1
+�
+e0y−1
+2
+�
+= κ−2e2
+0y−2
+2 z−1
+1 Q2(y2) = e2
+0κ−2y−2
+2 z2.
+Since y1 = (z1 + e0)y−1
+2 , by using (ii) we have
+g2,3(κ)∗y1 = (κ2z1y−2
+2
++ e0)(y2 + κ2e−1
+0 z1y−1
+2 )−1 = e0y−1
+2 .
+Finally,
+g1,3(κ)∗Z2 = g2,3(κ)∗(κ2z0y−2
+1 )
+= κ2(e2
+0κ−2y−2
+2 z2)(e0y−1
+2 )−2 = z2.
+g1,3(κ)∗Y3 = g2,3(κ)∗ �
+y1(1 + κ2e−1
+0 z0y−2
+1 )
+�
+= e0y−1
+2
+�
+1 + κ2e−1
+0 (e2
+0κ−2z2y−2
+2 )(e0y−1
+2 )−2�
+= e0y−1
+2 (1 + e−1
+0 z2) = e0y−1
+2
+�
+1 + e−1
+0 (y2y3 − e0)
+�
+= y3.
+Therefore, g3,1(κ) :
+(z2, y3) → (Z2, Y3). The computation of its inverse g3,1(κ) in the chart
+(y2, z2) is similar to the computation of the inverse of the g2,3(κ) in point (ii).✷
+Remark 5.23 The transformations g2,3(κ) and g3,1(κ) are conjugated by the map ξ : (y1, z2) →
+(y2, z3). Nevertheless, this conjugation ξ does not belong to the canonical dynamics.
+The following equalities are consequences of the points (ii) and (iii) in Proposition (5.22):
+Corollary 5.24 For all κ in C∗ we have:
+1. g2,3(κ) = g2,3(1) ◦ ty2(κ2) = ty1(κ−2) ◦ g2,3(1);
+2. g1,3(κ) = g1,3(1) ◦ ty2(κ2) = ty3(κ−2) ◦ g1,3(1).
+In particular, g2,3(κ) conjugates Ty1 and Ty2: g2,3(κ) ◦ ty2(κ2) ◦ g2,3(κ) = ty1(κ−2), and g3,1(κ)
+conjugates Ty2 and Ty3: g3,1(κ) ◦ ty3(κ2) ◦ g1,3(κ) = ty1(κ−2).
+From these preliminary results we immediately obtain:
+Proposition 5.25
+1. The tame dynamic generated by g = σ1 ◦ σ2 is included in Conf(PV ).
+2. For any n in Z, Tyn ⊂ Conf(PV ).
+49
+
+Proof. The first point is a direct consequence of g = g1,2.
+The second one is obtained for n = 1 or n = 2 from the first item of Corollary (5.24). The others
+tori are obtained from Ty1 or Ty2 by a conjugation with an element of the tame dynamic. ✷
+Nevertheless, the canonical Stokes operators do not belong to Conf(PV ). The reason is the
+following one: both g2,3(κ) and s1 keep invariant ∆ :
+z1 = 0. The restriction of s1 to each
+component of ∆ is a translation: see Proposition (5.15). One can compute the restriction of
+g2,3(κ) on each each line of ∆.
+g2,3(κ)∗x3 = κ−2x2
+2x3 + β(κ, x1, x2),
+where β(κ, x1, x2) only depends on κ, x1, and x2. On ∆, x1 = e±1
+3 , e0e±1
+4 , x2 = e0e±1
+3 , e±1
+4 ,
+therefore the restriction of the g2,3(κ) are affine transformations on each line. The same property
+holds for g3,1(κ). Nevertheless, we cannot find κ in C∗ such that κ−2x2
+2 = 1 simultaneously on
+each component of ∆.
+This remark suggests to introduce an extension of the confluent dynamics. The only way to
+obtain from the family g2,3(κ) an element which is a translation on ∆ is to introduce a functional
+time for the elements of Tt2, setting κ = x2. Therefore if we expect to obtain the canonical stokes
+operators from the confluent dynamic, we have to extend it by the element ty2(y−2
+2 ).
+Another motivation in order to introduce this element is purely algebraic and makes use of
+the structure induced by the symplectic Cremona group. We recall the subgroups of Bir(CV )
+generated from the Borel-Cartan structure of Symp(C2) by the canonical triple (y1, z1, y2) :
+By1 = {by1(λ, r) : (y1, z1) → (λy1, z1r(y1)), r ∈ C(y1)∗, λ ∈ C∗}
+By2 = {by2(λ, r) : (z1, y2) → (z1r(y2), λ−1y2), r ∈ C(y2)∗, λ ∈ C∗}
+Ty1 = {ty1(λ) : (y1, z1) → (y1, λz1), λ ∈ C∗}
+Ty2 = {ty2(λ) : (z1, y2) → (λ−1z1, y2), λ ∈ C∗}
+Uz1 = {uz1(r) : (z1, y2) → (z1, y2r(z1)), r ∈ C(z1)∗, r(0) = 1}.
+and we set :
+Ty2 (C(y2)) = {(z1, y2) → (z1r(y2), y2), r ∈ C(y2)∗}
+= {by2(1, r), r ∈ C(y2)∗} ⊂ By2;
+Ty1 (C(y1)) = {(z1, y2) → (y1, z1r(y1)), r ∈ C(y1)∗}
+= {by1(1, r), r ∈ C(y1)∗} ⊂ By1.
+Proposition 5.26
+(i) There exists a unique pair (by2, uz1) in Ty2 (C(y2)) × Uz1, such that :
+g2,3(1) = uz1 ◦ by2. Moreover by2 = ty2(y−2
+2 ) ∈ Ty2 (C(y2)) and uz1 = s−1
+1 .
+(ii) There exists a unique pair (by1, vz1) in Ty1 (C(y1)) × Uz1, such that :
+g3,2(1) = vz1 ◦ by1. Moreover by1 = ty1(e−2
+0 y2
+1) ∈ Ty1 (C(y1)) and vz1 = s1.
+Proof. (i) We set :
+by2 : (z1, y2) �→ (z1y−2
+2 , y2),
+uz1 : (z1, y2) �→ (z1, (1 + e−1
+0 z1)y2).
+50
+
+We have g2,3(1) = uz1 ◦ by2. The unicity of this decomposition is a consequence of Ty2 (C(y2)) ∩
+Uz1 = {id}. One recognizes here that by2 = ty2(y−2
+2 ) and uz1 = s−1
+1 .
+(ii) We set :
+by1 : (y1, z1) �→ (y1, e2
+0z1y−2
+1 )
+vz1 : (y1, z1) �→
+�
+(1 + e−1
+0 z1)y1, z1
+�
+.
+We have g3,1(1) = vz1 ◦ by1. The unicity of this decomposition is a consequence of Ty1 (C(y1)) ∩
+Uz1 = {id}. One recognizes here that by1 = ty1(e−2
+0 y2
+1) and vz1 = s1. ✷
+By using g2,3(κ) = g2,3(1) ◦ ty2(κ2), and Z(Ty2) = By2, we obtain:
+g2,3(κ) = uz1 ◦ by2 ◦ ty2(κ2) = uz1 ◦ ty2(κ2) ◦ by2.
+Since g3,1(κ) is conjugated to g2,3(κ) by (z1, y2) → (z2, y3) we have a similar decomposition of
+the family g3,1(κ) in Uz2 × B1
+y3 × Ty3:
+g3,1(κ) = uz2 ◦ by3 ◦ ty3(κ2) = uz2 ◦ ty3(κ2) ◦ by3, uz2 = s−1
+2 , by3 = ty3(y−2
+3 ).
+Notice that b−1
+y2 = ty2(y2
+2) : (z1, y2) → (z1y2
+2, y2) is a ramified blowing-up in the canonical chart
+(z1, y2). We extend the confluent dynamic by this element:
+Definition 5.27 The extended confluent dynamic is defined by
+Conf ♯(PV ) =< Conf(PV ), ty2(y2
+2) > .
+We recall that, according to Proposition (5.20), the element �m of Dyn(CV ) such that g = s2◦ �m◦s1
+is defined by �m = ty2(e−2
+0 y4
+2). We consider a square root of �m defined by:
+�m1/2 : (y2, z2) → (y2, e−1
+0 y2
+2z2).
+Definition 5.28 The extended confluent canonical dynamic is defined by
+Dyn♯(CV ) =< Dyn♯(CV ), �m1/2 > .
+Theorem 5.29 Conf ♯(PV ) = Dyn♯(CV ).
+Proof.. We first prove that Dyn(CV )♯ ⊂ Conf ♯(PV ). Dyn♯(CV ) is generated by g, Ty1, s1,
+s2 and �m1/2. From Proposition (5.26), since s−1
+1
+= g2,3(1) ◦ ty2(y2
+2), Conf ♯(PV ) contains s1. We
+have:
+�m = ty2(e−2
+0 y4
+2) =
+�
+ty2(e−1
+0 ) ◦ ty2(y2
+2)
+�◦2 .
+Hence, Conf ♯(PV ) also contains �m. According to the relation g = s2 ◦ �m ◦ s1, s2 also belongs
+to Conf ♯(PV ).
+Finally since �m1/2 = ty2(e0) ◦ ty2(y2
+2), we obtain the inclusion Dyn(CV )♯ ⊂
+Conf ♯(PV ).
+Now we prove that Conf ♯(PV ) ⊂ Dyn(CV )♯. The relation
+g2,3(κ) = s−1
+1
+◦ ty2(y−2
+2 ) ◦ ty2(κ2)
+proves that g2,3(κ) belongs to Dyn(CV ) for any κ in C∗. Since g1,2 = g belongs to Dyn(CV ),
+from the relation g1,2 ◦ g2,3(κ) ◦ g3,1(κ) = id, g3,1(κ) also belongs to Dyn(CV ) for any κ in C∗.
+Finally ty2(y2
+2) = �m1/2 ◦ tt2(e−1
+0 ) belongs to Dyn(CV )♯.✷
+51
+
+5.5
+Comparison with the wild dynamics
+The wild dynamics is a pseudogroup of non linear dynamics induced by the Painlev´e V foliation
+PV (κ) in a neighborhood of each singular point of saddle-node type. We will recall here the
+construction of its generators: the non linear stokes operators, non linear tori, and formal and
+analytic non linear monodromy. Through the Riemann-Hilbert map RHV it induces an (a priori)
+local dynamics on CV (θ). The second author formulates the Wuhan conjecture which claims that
+this dynamics extends into a symplectic rational dynamics. This conjecture has been proved
+for the Painlev´e V equation by M. Klimes in [40] (there is also a very clear presentation of this
+work in Klimes’s lecture [41]). His method uses a description of the confluence on the foliations
+of the Hamiltonian systems on the left side and on the linear isomonodromic systems and on
+the associated character variety on the right side. An essential tool is the discretization of the
+confluence as indicated in section 2.1 on the baby model of confluence x(x − ε) → x2y′ + y = 0,
+first introduced by C. Zhang for the confluence for the hypergeometric equations [65]. We will
+present here this result of M. Klimes, and compare this dynamics with the canonical dynamics
+Dyn(CV ) on the cubic surfaces.
+Definition of the wild dynamics. The Painlev´e V foliation is given under its hamiltonian
+form by (5). From section 4.3, the irregular singular points si all appear in the Boutroux chart.
+Around each irregular singular point of Painlev´e V of saddle-node type si the formal and sectoral
+normal forms are given by [40]:
+Theorem 5.30 Let α0 = 2α1 + α2 − 1.
+1- In a neighborhood of a saddle-node si, the hamiltonian system (5) can be brought to a
+formal normal form :
+x2 du
+dx = (1 − α0x + 4xu1u2)
+
+
+
+1
+0
+0
+−1
+
+
+ u,
+u =
+
+
+
+u1
+u2
+
+
+
+(13)
+by means of a formal transversally symplectic change of coordinates :
+
+
+
+q
+p
+
+
+ := �Ψ(u, x) =
+�
+k≥0
+ψ(k)(u)xk
+where the ψ(k) are analytic on a polydisc P = {|u1|, |u2| < δ} (δ > 0).
+2- The formal normal form (13) is integrable in closed form :
+�
+u1(x, c1) = c1e−1/xx−α0+4c1c2
+u2(x, c2) = c2e1/xxα0−4c1c2.
+(14)
+and this local hamiltonian vector field (for du1 ∧ du2 and h = x−2(1 − α0x)u1u2, also admits the
+analytic first integral u1u2.
+3- The formal series �Ψ ∈ O(P)[[x]] is uniformly 1-summable with a pair of 1-sums Ψ↑(u, x),
+Ψ↓(u, x), defined respectively above the sectors
+U ↑ := {| arg x − π/2| < π − η, |x| < δ′} and U ↓ := {| arg x − π/2| < π − η, |x| < δ′}
+for some 0 < η < π/2 arbitrary small and some δ′ > 0 (depending on η), and u ∈ P.
+52
+
+Remark 5.31 The formal Takano’s normal form [59] used here to define the non linear Stokes
+operators will not suffice for the other Painlev´e equations in order to obtain overlapping open sec-
+tors. We will have to make use of the Bittman’s normal forms, which are no longer polynomials:
+see [4], [5] and [6].
+Corollary 5.32 On each sector U • (• =↑ or ↓),
+1. the system (5) has a unique analytic bounded solution f • which is the 1-sum of the formal
+solution �f : u1(x, 0) = u2(x, 0) = 0. We call these solutions the sectoral center solutions.
+They are pole-free on the corresponding sector and they are characterized by this property.
+2. the system (5) has a 2-parameters family of solutions �fc1,c2:
+(q•, p•)(x, c1, c2) = Ψ•(u(x, c1, c2), x).
+We consider the left and right intersection sectors of the overlapping sectors U ↑ and U ↓
+V l := U ↑ ∩ U ↓ ∩ {ℜx < 0} and V r := U ↑ ∩ U ↓ ∩ {ℜx > 0}.
+The non linear Stokes multipliers are defined by
+S2 = (Ψ↑)−1 ◦ Ψ↓ on V r, S1 = (Ψ↑)−1 ◦ Ψ↓ on V l.
+The formal non linear monodromy is defined by the action induced by x �→ e2iπx on the space
+of formal solutions �fc1,c2:
+�
+N : (c1, c2) �→ (e2iπ(−α0+4c1c2)c1, e2iπ(α0−4c1c2)c2).
+The formal non linear exponential torus is defined by the analytic symplectic symmetries of the
+formal normal forms (14):
+tα : (c1, c2) �→
+�
+eα(c1c2)c1, e−α(c1c2)c2
+�
+,
+α ∈ O(C, 0).
+Using Ψ↑ and Ψ↓, the formal exponential torus induces two sectoral exponential tori T ↑ and T ↓
+and the formal first integral h = c1c2 induces 2 sectoral first integrals h↑ and h↓.
+Definition 5.33 The wild dynamics Wild(PV , f ↑) based in the central solution f ↑ is the pseudo-
+group generated by S1, S2, �
+N, and T = {tα}.
+Wild(PV , f ↑) also contains the actual monodromy N generated by a loop around x = ∞,
+according to the relation S2 ◦ �
+N ↑ ◦ S1 = �
+N, where �
+N ↑ = Ψ↑ ◦ �
+N ◦ (Ψ↑)−1.
+The wild dynamics through the Riemann-Hilbert map RHV . We fix a value x0 in a
+neighborhood of ∞, and we consider the Okamoto space of initial condition Vx0 over x0. Let
+m• = (q•, p•)(x, 0, 0)
+be the two points on Vx0 corresponding to the two central varieties in a neighborhood of a singular
+point s. We denote by (c•
+1, c•
+2)(q, p) the inverse maps of (q•, p•)(x0, c1, c2). The equations c•
+2 = 0
+define two germs of curves δ• in m•. The equations c•
+1 = 0 define two germs of curves d• in m•.
+On the rightside, we consider the 12 lines ∆·, Dr
+· , Dl· in the configuration of lines in χV : see
+figure 7. We can group them in four triples (∆i, Dri, Dli) such that ∆i∩Dri ̸= ∅ and ∆i∩Dli ̸= ∅.
+We set pri = ∆i ∩ Dri, pli = ∆i ∩ Dli.
+53
+
+Now we consider the confluent process between PV I and PV defined by
+tV I = 1 + εtV ,
+α3 = 1/ε,
+α2,V I = ˜α2 = −1
+ε + α2,V ,
+x = 1
+tV
++ ε, α1,V I = α1,V
+(15)
+which sends the three fixed singularities to tV = −1/ε, 0, ∞. This change of variables transforms
+εHV I into Hε
+V I and the PV I Hamiltonian system into :
+dq
+dt = ∂Hε
+V I
+∂p
+,
+dp
+dt = −∂Hε
+V I
+∂q
+,
+When ε → 0, the simple singular points −1/ε and ∞ merge into a double singular point, an
+irregular singularity at infinity, and the limit of Hε
+V I is HV . The four pairs of singularities over
+−1/ε and ∞ merges to 4 saddle-nodes (the confluent saddle-nodes) among the five saddle nodes
+si over ∞. In what follows, we suppose that s is one of these 4 singularities. We summarize the
+results of Martin Klimes [40] by the following theorem:
+Theorem 5.34 We consider the Riemann-Hilbert map RHV between an open set in the space
+of initial condition Vx0 induced by a neighborhood of a confluent singularity s and the character
+variety χV .
+The choice of s determines a triple of lines (∆, Dr, Dl) among the four triples
+(∆i, Dr
+i , Dl
+i) such that
+1. RHV (m↑) = ∆ ∩ Dr = pr, RHV (m↓) = ∆ ∩ Dl = pl.
+2. RHV (δ↑) = (∆, pr) (the germ of line at pr), RHV (δ↓) = (∆, pl). Furthermore, the germs δ↑
+and δ↓ extend to a same analytic curve δ in Vx0 isomorphic to C, such that RHV (δ) = ∆.
+3. RHV (d↑) = (Dr, pr), RHV (d↓) = (Dl, pl). Furthermore the germs of curves d↑ and d↓
+extends to 2 curves dr and dl isomorphic to C in Vx0, such that RHV (dr) = Dr and
+RHV (dl) = Dl.
+4. Let (y1, z1, y2) be the triple of canonical coordinates (3.25). We have:
+RH∗
+V y1 = y1(pr)eh↑, RH∗
+V y2 = y2(pr)e−h↓, RH∗
+V z1 = e0(e(h↑−h↓) − 1).
+5. Let g, s1, s2, be the generators of the canonical dynamics defined in 5.18, let �m be the
+canonical formal monodromy defined by 5.20 and Ty1, Ty2 the canonical exponential tori.
+We have:
+RH∗
+V g = N, RH∗
+V S1 = s1, RH∗
+V S2 = s2, RH∗
+V �m = �
+N, RH∗
+V T ↓ = Ty1, RH∗
+V T ↑ = Ty2.
+Therefore the dynamics induced by Wild(PV , f ↑) through the Riemann-Hilbert correspondence
+RHV extends to a rational symplectic dynamics Wild(PV ) on χV , and we have Wild(PV ) =
+Dyn(CV ). This proves the Wuhan conjecture of the second author for the PV foliation.
+The main ingredients of the proof of this result in [40] are:
+- an unfolded version of Theorem 5.30 for the hamiltonian system related to Hε
+V I;
+- a discretization of the confluent parameter by setting ε−1 = ε−1
+0 +n, n ∈ Z+, along which the
+monodromies of the unfolded system, a priori divergent, converge. The choice of this sequence
+depends on a parameter κ = e2iπ/ε.
+- a confluence on the rightside between χV I and χV which allows to compare these limits to
+the wild dynamics Wild(PV , f ↑).
+54
+
+6
+Conclusion and open questions
+Conclusion. Several dynamics related to the Painlev´e V foliation through the Riemann-Hilbert
+map RHV has been described here on the wild character variety χV (a) identified to the cubic
+symplectic surface CV (θ) for generic parameters θ. The tame dynamics is induced here by only
+one braid. We first extended the tame dynamics by using a confluent dynamics obtained from
+the Painlev´e VI dynamics by a birational confluent morphism between χV I and χV . This one,
+first discovered by M. Klimes, is obtained here by a confluent process between the corresponding
+groupoids (after an extension with families of exponential loops).
+We have constructed the canonical symplectic birational dynamics defined on the cubic
+symplectic surface CV (θ). This one is the pullback of symplectic dynamics on C2 by a canonical
+sequence of coordinates satisfying cluster-type relations. This birational dynamics coincides with
+the dynamics obtained by M. Klimes from the wild dynamics defined by a confluent irregular
+singular point of the foliation. After and extension by one element, confluent dynamics and
+canonical dynamics also coincide.
+The cubic symplectic surface CV (θ) contains a skeleton generated by 18 lines and their im-
+ages through the action of the tame dynamics. We have characterized this set by a condition of
+reducibility of the corresponding linear representations. Some intersections between these lines
+correspond to particular solutions of the Painlev´e V equation : Kaneko solutions, or central so-
+lutions. The restriction of the dynamics on this skeleton is an important tool in the comparisons
+between these dynamics.
+Open problems. The equations of the character varieties χJ for the other Painlev´e equations
+PJ, J = Vdeg, IV , III(D6), III(D7), III(D8), II(JM), II(FN), I (with the notations of [54])
+are known. We already know that each of them can be obtained as a quotient of a space of
+linear representations of a wild groupoid, and that there exists canonical cluster sequences of
+coordinates on each χJ, inducing a canonical birational symplectic dynamic DynJ which can be
+explicitely computed. We conjecture that for all J:
+- All the lines in χJ are a reducibility locus of some path in the groupoid;
+- the wild dynamics of all the Painlev´e equations (induced by non linear monodromies, non
+linear Stokes operators and exponential tori) around an irregular singular point obtained by a
+confluent process coincide with these canonical birational symplectic dynamics DynJ on χJ.
+We already know that these dynamics coincide on the skeleton generated by the lines for PI and
+PII. With M. Klimes we also conjecture that
+- There is a diagram of families of birational symplectic confluences between the χJ, which par-
+allels the diagram of confluence of Ohyama and Okumura [51].
+We expect that we can prove this fact by a confluent process between the groupoids (extended
+by families of exponential loops).
+Finally we mention here the initial motivation of this study. In [14] S. Cantat and F. Loray
+proved the irreducibility of PV I for generic parameters, by using the Malgrange closure of its
+dynamics [45]. The aim of the Wuhan conjectures of the second author was to extend their
+proof in the general case. In the PV I case, the dynamics obviously belongs to the Malgrange
+groupoid which is a closure of its holonomy pseudogroup. In the general case, we conjecture that
+the wild dynamics is contained in the Malgrange pseudogroup, but now it is far from obvious.
+Modulo this conjecture, we would obtain a proof of the irreducibility of the Painlev´e V equation
+for generic values of the parameters. Indeed since the extended canonical dynamic contains the
+confluent dynamic, it contains a copy of the dynamics of a PV I and we can use Theorem D of
+[14] in order to prove that its Malgrange closure is maximal.
+55
+
+Appendix. Structure of the symplectic Cremona group
+The Cremona group Cr = Cr2 is the group of birational automorphisms of P2(C). It does not
+depend on the representative X in the birational class of P2: C2, P1 × P1...
+The symplectic Cremona group is the subgroup Symp of Cr ≃ Bir(P1(C) × P1(C)) of the
+elements which preserve the differential form ω = du
+u ∧ dv
+v . The Cremona group is an old topic
+which appears at the end of the XIX-th century. One can find an introductive text in [43]. For
+a study of its subgroups, see [21]. To the contrary, the study of its symplectic subgroup Symp
+has been developed only since 2005, first by A. Usnich [62, 63] and then J. Blanc [7]. We present
+here without proofs the main results about subgroups of Symp. Some of them are new results.
+Our terminology is inspired by the algebraic groups, due to some similarities in this non linear
+infinite dimensional context. We denote
+T = {t(λ, µ) : (u, v) �→ (λu, µv), (λ, µ) ∈ C∗ × C∗}.
+We have T ⊂ Symp. We denote by Z(T) its centralizor, and N(T) its normalizor in Symp. We
+consider the subgroup W of Symp of the monomial maps:
+W = {(u, v) �→ (uavb, ucvd),
+
+
+
+a
+b
+c
+d
+
+
+ ∈ SL2(Z)}.
+Proposition 6.1 We have:
+1. T is an algebraic abelian maximal subgroup of Symp;
+2. Z(T) = T;
+3. N(T) = W ⋉ T.
+Proof. T is an algebraic abelian maximal subgroup of Cr [56], which preserve ω.
+The centralizor and normalizor of T in Cr has been computed by [20], since C is algebraically
+closed. We have: NSymp(T) = NCr(T) ∩ Symp = W. ✷
+Furthermore, according to a theorem of J. Blanc [7], Symp is generated by N(T) and the order
+five element p defined by:
+(u, v) �→
+�
+v, 1 + v
+u
+�
+.
+The Cartan subgroups.
+Definition 6.2 A Cartan subgroup of Symp is an algebraic abelian maximal subgroup of rank
+two in Symp. T is the standard Cartan subgroup of Symp, and W is the Weyl group associated
+to T.
+We have T = T1 × T2, where
+T1 = {t1(λ) = t(λ, 1), λ ∈ C∗}, T2 = {t2(µ) = t(1, µ), µ ∈ C∗}.
+All the Cartan subgroups are conjugated. 8
+8We have no reference for this fact; the proof will be delivered in a forthcoming publication. This proof uses
+a result of J. Blanc [7].
+56
+
+Borel subgroups. Let dJ1 be the subgroup of Cr ≃ Bir(P1 × P1) of the De Jonqui`eres maps,
+i.e. the rational maps which preserve the first projection:
+dJ1 = {dj1 : (u, v) �→ (a(u), b(u, v)), a ∈ PGL2(C), b(·, v) ∈ PGL2(C(u))}.
+Let dJS1 be the subgroup in Symp of the symplectic De Jonqui`eres maps. For any λ in C∗, any
+r in C(X)∗, the maps:
+dj1(λ, r) : (u, v) �→ (λu, r(u)v), σ : (u, v) �→ (u−1, v−1)
+are examples of elements in dJS1. We set
+B1 = {dj1(λ, r), λ ∈ C∗, r ∈ C(u)∗} =< T1, T2(C(u)) >, B−
+1 = {b1 ◦ σ, b1 ∈ B1}.
+Proposition 6.3 We have dJS1 = B1 ∪ B−
+1 = B1 ⋊ Z/2Z.
+We also define dJ2, dJS2, B2 with respect to the second projection. We have B1 ∩ B2 = T, and
+any pair (wB1w−1, w′B2w′−1), w, w′ ∈ W also satisfies this equality.
+Definition 6.4 (B1, B2) is the standard pair of Borel subgroups.
+Any pair (wB1w−1, w′B2w′−1), w, w′ ∈ W is a pair of Borel subgroups related to the standard
+Cartan subgroup T.
+Notice that the Borel subgroups are no longer algebraic subgroups since there are infinite di-
+mensional groups. It can be proved that B1 and B2 generate Symp.
+Definition 6.5 The subgroup U1 of B1 of the “unipotent” elements is defined by:
+U1 = {dj1(1, r), r(0) = 1}.
+Proposition 6.6 We have:
+1. U1 is an abelian subgroup;
+2. U1 is generated by the family dj1(1, 1 + νu), ν ∈ C∗, or by one element dj1(1, 1 + u) and
+its conjugations with an element of T1 (we say that this element is a pseudo generator of
+U1);
+3. [B1, B1] = {dj1(1, r), r(0) = 1, deg(r) = 0.} Therefore, [B1, B1] ⊂ U1 and B1 is meta-
+abelian.
+In this non linear context, we clearly have some analogies with the structure of the algebraic
+group SL2(C) but we also have some differences:
+- the inclusion [B1, B1] ⊂ U1 is a strict one, [B1, B1] is an invariant subgroup of U1 and we have
+U1/[B1, B1] =< dj1(1, 1 + u) >≃ Z.
+- let B♮
+1 =< U1, T1 >. B♮
+1 is also a meta-abelian group which contains U1, but it contains only a
+maximal torus of rank 1. We call B♮
+1 a Borel of rank one.
+57
+
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+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Ramis†  January 23, 2023  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The leaves of the Painlev´e foliations appear as the isomonodromic deformations  of a rank 2 linear connection on a moduli space of connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore they are the fibers of  the Riemann-Hilbert correspondence that sends each connection on its monodromy data, and  this correspondence induces a conjugation between the dynamics of the foliation and a dynamic  on a space of representations of some fundamental groupoid (a character variety).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
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+page_content='5  The Laurent property  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
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+page_content='  32  ∗Institut of Mathematics of Toulouse, 118 route de Narbonne, 31062 Toulouse Cedex, France  emmanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  40  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3  The canonical dynamics on CV (θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  43  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4  Comparison between the confluent and the canonical dynamics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  48  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='5  Comparison with the wild dynamics  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  52  6  Conclusion and open questions  55  Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Structure of the symplectic Cremona group  56  Bibliography  58  2  Introduction  The transverse dynamics of an holomorphic foliation is usually defined by its holonomy groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In general we use a specialization of this groupoid along a particular leaf L: the holonomy group  of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This one is a representation of the fundamental group of the leaf on a germ of transversal,  defined by the analytic continuation of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It only describes the transverse structure  of the foliation in a neigbourhood of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The Painlev´e foliations are holomorphic foliations defined by the Painlev´e equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' These  ones are order two non linear ordinary differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Their writings are available for  example in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We only recall here the Painlev´e VI and Painlev´e V families:  (PVI)  d2w  dt2 = 1  2  � 1  w +  1  w − 1 +  1  w − t  � �dw  dt  �2  −  � 1  w +  1  t − 1 +  1  w − t  � dw  dt  + w(w − 1)(w − t)  2t2(t − 1)2  �  (α4 − 1)2 − α2  3  t  w2 + α2  1  t − 1  (w − 1)2 + (1 − α2  2) t(t − 1)  (w − t)2  �  ,  (1)  (PV ) d2w  dt2 =  � 1  2w +  1  w − 1  � �dw  dt  �2  − 1  t  dw  dt + (w − 1)2  t2  �α2  1  2 w − α2  3  2w  �  + (1 − α1 − 2α2 − α3)w  t − 1  2  w(w + 1)  w − 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (2)  They define families of one dimensional foliation on {(t, w, w′) ∈ C3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' They share with all the  Painlev´e differential equations type three properties, which will allow us to give a more global  and explicit description of their dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let us detail these properties for the Painlev´e VI  foliation:  1- The Painlev´e property : all the solutions (w(t), w′(t)) have a meromorphic continuation  over the universal covering the time space T = P1(C) punctured at the fixed singularities 0, 1 and  ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This property generalizes the property of holomorphic extension of the solutions of linear  differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The search of non linear differential equations satisfying this property was  the initial motivation of Painlev´e and then Gambier in order to construct new transcentendal  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It gives rise to the six well known families of Painlev´e equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  According to this property, the movable singularities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' the singularities of the solutions  outside the fixed singularities defined by the equation itself, are only poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Okamoto has  introduced a semi-compactification process in order to get a foliation transverse to a fibration  over the time space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' After this process, one can define the dynamic of the Painlev´e foliation by  its non linear monodromy, which is defined by an action of the fundamental group of the time  space T, over the Okamoto fiber (the space of initial values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We call it the tame dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2- The isomonodromic property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In order to study the non linear monodromy of this  foliation, it is useful to introduce a conjugacy between this dynamical system and a simpler one:  the Riemann-Hilbert map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let us recall the different ingredients involved by this property for  the Painlev´e VI equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The moduli space of connections MV I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We consider the family of linear connections on the  trivial bundle over P1 defined the family SV I of rank 2 trace free differential systems  (A) : dY  dx =  � 4  �  i=1  Ai  x − pi  �  Y, pi ̸= pj for i ̸= j, Ai ∈ sl2(C),  4  �  i=1  Ai = 0, x ∈ P1,  3  up to the global gauge action: Y �→ Y · P, P ∈ SL2(C), and to a change of the independent  variable x �→ ϕ(x), ϕ ∈ Aut(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The extension of the system to x = ∞ ∈ P1 is defined by the  change of variable z = x−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The family SV I is a 13 dimensional space (including the variables  pi), and the above quotient is a 7 dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The local parameter space is defined by  α(A) = (det(Ai) = α2  i /4, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 4),  where ±αi/2’s are the eigenvalues of each residue matrix Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The time parameter is the cross  ratio t ∈ T = P1 \\{0, 1, ∞} of the configurations of the singular points S = {p1, p2, p3, p4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Over  a value α, the fiber MV I(α) is a 3 dimension space, and t(A) defines a fibration of MV I(α) over  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The character variety χV I as representations of a fundamental groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For a system A in  SV I we can define its monodromy representation by using the analytic continuation of a local  matrix solution Y0 along any element γ of the fundamental group π1(P1\\S, x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We consider here  another (equivalent) description of this monodromy, introduced by the authors in [53], essential  for what will follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Instead of considering as usually a fundamental group, we consider a  fundamental groupoid, which allows to make use of several base points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A groupoid is a small  category whose morphisms are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For details on groupoids see [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The fundamental  groupoid π1(X) of a variety X is the groupoid whose objects are the points of X and the  morphisms the paths between two points up to homotopy, with the obvious composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In  what follows, “paths” always means “path up to an homotopy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We consider the variety X  obtained by 4 real blowing up of the singular points pi in P1, and we choose a base point si on  each divisor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' a direction out of pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let S = {s1, s2, s3, s4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The groupoid πV I  1 (X, S) is the  restriction of the fundamental groupoid π1(X) to this finite set of objects S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' πV I  1 (X, S) has a  presentation given by the following picture:  s1  s2  s4  s3  γ1,1  γ2,2  γ3,3  γ4,4  γ1,2  γ2,3  γ4,1  γ3,4  Figure 1: A presentation of the groupoid πV I  1 (X, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The morphisms are generated by the local loops γi,i based in si homotopic to each divisor and  the paths γi,i+1 from si to si+1 satisfying the following relations:  Rext : γ1,2γ2,3γ3,4γ4,1 = ⋆1 (the trivial loop based in s1)  Rint : γ1,1γ1,2γ2,2γ2,3γ3,3γ3,4γ4,4γ4,1 = ⋆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  A linear representation ρ of πV I  1 (X, S) is a morphism of groupoid from πV I  1 (X, S) into the  category of vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In the Painlev´e context, we only consider linear representations ρ  4  such that ρ(s) = Vs is a 2-dimensional vector space and ρ(γs,t) belongs to the set of linear  morphisms which preserve the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  A choice of a basis in each Vs defines a map ρ(γs,t) ≃ Mγs,t in SL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We obtain a representation  of the groupoid πV I  1 (X, S) in the group SL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If we change the basis in each Vs, we obtain  another representation ρ′ of πV I  1 (X, S) in the group SL2(C) which is equivalent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' for any s,  there exist matrices Ps such that  M′  γs,t = P −1  s  Mγs,t · Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore a rank two trace free linear representation of πV I  1 (X, S) is also a class of equivalent  representations of πV I  1 (X, S) in SL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1 The character variety χV I is the categorical quotient 1 of the set of rank 2 trace  free linear representations up to the above equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The local data of an element ρ in χV I is the restriction of ρ to the groupoid πV I,loc  1  (X, S)  obtained by forgetting the generators γi,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It is characterized by the class of each ρ(γi,i), or  by a = (ai = tr(ρ(γi,i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=', 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We denote by χV I(a) the corresponding fiber in χV I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The character variety as a family of cubic surfaces CV I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Given a presentation of the groupoid  πV I  1 (X, S) by figure (1), we define the trace coordinates (a, x) by  ai(ρ) = tr(ρ(γi,i)), i = 1, · · · 4, xk(ρ) = tr(ρ(γi,iγi,jγj,j)), {i, j, k} = {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The trace coordinates do not change in a class of equivalent representations in SL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore  it can be convenient to make use of normalized representations: A representation ρ of the  groupoid πV I  1 (X, S) in the group SL2(C) is normalized if for any indexes i, j, i ̸= j we have:  ρ(γi,j) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' One can always choose the representation of the objects in order to get a normalized  representation by choosing arbitrarily the representation of a first one and by representing the  further consecutive ones such that ρ(γi,i+1) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A normalized representation is given by 4  matrices Mi = ρ(γi,i) unique up to a common conjugacy which satisfy, according to the relation  Rint, M1M2M3M4 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We recover here the usual monodromy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The trace coordinates of  a normalized representation ρ are defined by  ai = tr(Mi), i = 1, · · · 4, x1 = tr(M2M3), x2 = tr(M3M1), x3 = tr(M1M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='2 The trace map TrV I defined by (a, x) sends each χV I(a) on the cubic surface  CV I(θ) in C3 defined by the equation FV I(x, θ) = 0 with  FV I(x, θ) = x1x2x3 + x2  1 + x2  2 + x2  3 − θ1x1 − θ2x2 − θ3x3 + θ4,  and θi = aia4 + ajak, {i, j, k} = {1, 2, 3}, θ4 = a1a2a3a4 + �  i a2  i − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  From [31], this trace map is an homeomorphism only on an open set χ0  V I(a) in the character  variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For a generic value of the local data a, we have χ0  V I(a) = χV I(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This condition implies  that Mi ̸= ±I for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The Riemann-Hilbert correspondence RHV I : MV I → χV I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let A in MV I(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We choose  a representative of each base point si by a fundamental system of solutions Ys of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The  1This means that we consider the affine variety defined by the ring of invariants functions over the set of linear  representations: see [12] or [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  5  representation of a path γ from s to t is obtained by comparing the analytic continuation �Ys  γ  of Ys along γ with Yt:  ρA(γ) = Mγ ⇔ Yt = �Ys  γ · Mγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  A change of representation of the objects, or a gauge action on A gives an equivalent represen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The Riemann-Hilbert map RHV I is defined by RHV I(A) = [ρA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The map induced by  RHV I on the local parameters is defined by  RHloc  V I(αi) = 2 cos(παi) = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We do not discuss here about the surjectivity of RHV I (the so called Riemann-Hilbert prob-  lem) or its properness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For a survey on this wide subject, see [29] for the Painlev´e VI case, and  [54] for the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Isomonodromic families and Painlev´e VI equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' An isomonodromic family on MV I is a  fiber of the map RHV I, parametrized by a variable t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' a family of connections defined by  dY  dx = A(x, t) · Y such that the monodromy representation is locally constant in χV I along this  family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The relationship with the Painlev´e VI equation was discovered by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Fuchs in 1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3 There exists coordinates w, w′, t over a Zariski open set in MV (α) such that the  isomonodromic families are the solutions the Painlev´e VI equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  A sketch of proof [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Given a fundamental system of solutions Y (x, t), by isomonodromy,  d  dtY (x, t) and Y (x, t) have the same behaviour along any continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore the quotient  B(x, t) = d  dtY (x, t) · Y (x, t)−1  is univalued outside the fixed singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since the singular points are regular singular  points (see section 2 for the definition) B extends to the singular set in a meromorphic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore Y (x, t) satisfies two rational linear systems dY  dx = A(x, t)·Y and dY  dt = B(x, t)·Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The  compatibility condition requires that the two operators d/dx − A and d/dt − B must commute :  dA  dt − dB  dz + [B, A] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (3)  The pair (A, B) is called an isomonodromic Lax pair, and the above equation the Schelsinger  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If A is irreducible, B is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Now, one can find coordinates such that the Schelsinger  equation (3) is equivalent to the Painlev´e VI equation: see [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  3- The Hamiltonian property : Any Painlev´e differential equation is equivalent to a (non  autonomous) hamiltonian system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This fact was first remarked by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Malmquist [46], and ex-  tended by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Okamoto in [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For example the family of Painlev´e VI equations is equivalent to  ˙p = −∂HV I  ∂q  , ˙q = ∂HV I  ∂p  ,  (4)  with  HV I(α) = q(q − 1)(q − t)  t(t − 1)  �  p2 − (α3  q +  α1  q − 1 + α2 − 1  q − t )p +  β  4q(q − 1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  with β = (α1 + α2 + α3 + α4 − 2)(α1 + α2 + α3 − α4), and where the numbers ±αi/2 are the  eigenvalues of the residues matrices Ai2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2The numbering is chosen here such that the two first indices correspond to the singularities 1 and t which are  confluent in the usual confluent process of the Painlev´e equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  6  We denote by PV I(α) the corresponding family of Painlev´e vector fields, and by FV I(α) the  family of dimension one holomorphic foliations on C3  (p,q,t) ⊂ C2 × P1 (where P1 is the complex  one dimensional projective space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The fibers t = 0, 1, ∞ are singular and the foliation has 4  singular points on each of this fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Remark 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4 Painlev´e sixth equation was not discovered by Painlev´e and Gambier, using the  Painlev´e property, but by Richard Fuchs, using an isomonodromic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Fuchs consid-  ered the monodromy preserving deformation of a second order Fuchsian differential equation with  four regular singular points and an apparent singularity3 : Dy = y′′ + a1(x, t)y′ + a2(x, t) = 0,  where a1 and a2 are rational functions on x depending holomorphically of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We write the  Riemann scheme of Dy = 0 :  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  x = 0  x = 1  x = t  x = q  x = ∞  0  0  0  0  κ0  κ1  κ2  κ3  2  κ0 + κ4  \uf8fc  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8fd  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8fe  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In this scheme the exponents κi, i = 0, 1, 2, 3, 4 are complex parameters subject to the Fuchs  relation :  κ1 + κ2 + κ3 + κ4 + 2κ0 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The κi, i = 1, 2, 3, 4 are related to the eigenvalues ±αi/2 of the Fuchsian systems (A) by  κ1 = α1, κ2 = α2, κ3 = α3 and κ4 = α4 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  For a proof, see [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It is easy to derive the Hamiltonian formulation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' [48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The variable q  is the position of the apparent singularity, the variable p is defined by p = Resx=qa2(x, t)dx and  the hamiltonian by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The family of Painlev´e V equations (2) is equivalent to  ˙p = −∂HV  ∂q , ˙q = ∂HV  ∂p ,  (5)  with HV (α) = t−1 (p(p + t)q(q − 1) − α1p(q − 1) + α2qt − α3pq) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have used here the notations of [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The values  alphai/2 are not here eigenvalues of the residues matrices of the connection but are directly  related to them by an affine invertible map whose expression in available in the appendix C of  [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We denote by PV (α) the corresponding families of Painlev´e vector fields, and by FV (α) the  family of dimension one holomorphic foliations on C3  (p,q,t) ⊂ C2 × P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' There is here only two  singular fibers over t = 0 and t = ∞, with 2 singular points over 0, and 5 singular points over  ∞, 3 of them are of saddle-node type i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' with a vanishing eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This construction provides a symplectic structure ΩV I(α) on MV I(α) given by dq ∧dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This  Poisson structure on MV I also comes from a general construction of Atiyah and Bott [1]: the  symplectic structure on the infinite dimensional space of all the connections induces a Poisson  structure by a symplectic reduction under the gauge action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' There also exists several direct  3Respectively x = 0, 1, t, ∞ and x = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  7  finite dimensional constructions, by using either ciliated fat graphs [24], [2], or quasi-hamiltonian  geometry and quivers varieties [10], or symplectic reduction of multi-Poisson structures [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  On the right-hand side, the space of linear representations χV I has also a Poisson structure  which induces a symplectic form ωχV I(a) on each fiber χV I(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This fact was first proved by  Goldman in [26], and makes use of the Poincar´e-Lefschetz duality on the tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It can  be extended to irregular cases by using the concept of decorated character variety : [15], or  quasi-hamiltonian geometry [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  There also exists a Poisson structure on the family of cubic surfaces CV I(θ) given by:  ωV I(θ) = dx1 ∧ dx2  FV I,x3  = dx2 ∧ dx3  FV I,x1  = dx3 ∧ dx1  FV I,x2  ,  where FV I,xi = ∂FV I/∂xi = xjxk + 2xi − θi = ∂FV I/∂xi for i=1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This structure obtained  by the Poincar´e residue of the volume form coincide with the Goldman structure: see [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Finally K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='Iwasaki has proved that the Riemann-Hilbert map RHV I is a Poisson morphism  from (MV I, ΩV I) to (CV I, 2iπωV I): [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' One can also find a proof of this fact in [40] obtained  from the Jimbo’s asymptotic formula [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This result has been extended for the Riemann-Hilbert  map in a local irregular case by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Boalch: see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  One can summarize these three properties for the Painlev´e foliation by the following state-  ment: the dynamics of the Painlev´e VI foliation FV I(κ) is conjugated through the Riemann-  Hilbert map to the dynamics on CV I(θ) induces by the automorphisms of πV I  1 (X, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Since the inner automorphisms γi,j �→ αi,iγi,jα−1  j,j acts trivially on the character variety, this  action factorizes to the quotient  Out(πV I  1 (X, S)) = Aut(πV I  1 (X, S))/Inn(πV I  1 (X, S))  which is defined by pure braids over S in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This braid group is generated by three elementary  braids b1,2, b2,3 and b3,1 such that b1,2 · b2,3 · b3,1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The action of each generator on χV I has  been first computed by Dubrovin and Mazzocco [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' See also Iwasaki [31], or Cantat and Loray  [14] or the authors in [53] for a simple computation using groupoids instead of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have  Proposition 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='5 The action of bi,j is given by the automorphism:  hi,j :  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  xi → −xi + xjxk + xix2  k − θjxk + θi  xj → −xj − xixk + θj  xk → xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This dynamics has the remarkable “tame property”: any element and its inverse is given by  polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Contents of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Our aim is to present the main tools that will allow us to  extend this description for all the Painlev´e equations, by considering here the first case of PV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In this case, which can be obtained from PV I, by a confluent process, new difficulties arise:  – The monodromy around the two singular fibers 0 and ∞ do not describe the whole trans-  verse structure of the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This monodromy will only give a “tame” dynamics generated  by just one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This is a consequence of the “irregular” type of the singularities of the  foliation over ∞ which are now saddle-nodes, with non linear Stokes phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  – On the other side, the class of connections on which PV naturally arises is now a class of  irregular connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The representation of such a connection must take into account beyond  the usual monodromy, new operators related to such singular points: the linear Stokes operators  and exponential tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  8  We solve these problems by constructing a wild fundamental groupoid πV  1 (X, S) with its  linear representations (modulo equivalence):  the wild character variety χV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The action of  Out(πV  1 (X, S)) allows us to recover the tame dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In order to complete this dynamics, one can construct a family of (non invertible) confluent  morphisms from πV  1 (X, S) to πV I  1 (X, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The key point is that these morphisms induce families  of birational maps between the corresponding character varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This will allow us to define  and compute a confluent dynamics Conf(PV ) on χV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This one was previously found by Martin  Klimes in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Our technique based here on wild fundamental groupoids allows us to expect a  generalization for all the other Painlev´e equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Notice hat the dynamics that we obtain is  no longer polynomial but only a rational one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We will also discuss about a canonical dynamic which exists on the cubic surfaces CV , with-  out any reference to a Painlev´e equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The central idea is that here the cubic surface is  symplectically birationally equivalent to  �  C2, du  u ∧ dv  v  �  by a sequence of log-canonical coordi-  nates which satisfies some exchange relations, which appear in cluster algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The pull-back  of the symplectic Cremona group Symp  �  C2, du  u ∧ dv  v  �  delivers a very rich structure denoted by  Dyn(CV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We will compare Conf(PV ) and Dyn(CV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The wild dynamics of the Painlev´e V foliation is defined by the non linear monodromy, non  linear Stokes operators and non linear tori in a neighborhood of a saddle-node singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Following M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Klimes [40], we will recall the definition of this dynamics and its image through the  Riemann-Hilbert map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We also recover here the canonical symplectic dynamics, thus confirming  in this case the rationality conjecture of the second author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The last part is dedicated to open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We would like to thank Martin Klimes for the discussions we had  together on this subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' His work [40] is a primary source of inspiration for this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  9  1  The moduli space MV  According to [54], the moduli space of connections on which lives the Painlev´e V foliation is  defined by:  SV =  �dY  dx = A(x) · Y, A(x) =  A0  x − s0  +  A1  x − s1  + A∞, A0, A1, A∞ ∈ sl2(C)  �  such that A∞ is a non trivial semi-simple element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The singularities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' the points s such that  A is not holomorphic around s are s0, s1 and ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We suppose that the eigenvalues ±t of A∞ are  distinct (t ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' One can extend the basis to P1(C) by setting z = x−1:  dY  dz = −z−2A(z−1) · Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  As explained in the introduction, we consider this family up to a gauge action Y = PZ, which  acts on the system by A → AP = P −1AP − P −1 dP  dx , and up to a change of independent variable  ϕ(x), ϕ in the M¨obius group Aut(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1  The local classification  Suppose that x = 0 is a singular point of a germ of connection defined by  xp+1 dY  dx = A(x) · Y, A ∈ sl2(C{x}), A0 = A(0) ̸= 0, p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The local formal classification is the classification of such a system up to a gauge transformation  Y = PZ, with P in SL2 (C((x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The integer p is called the Poincar´e rank of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It is  not a local gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If p = 0 the singular point is called a fuchsian singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Given  an element of SV , the points s0 and s1 are fuchsian singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The local formal classification  around such a point is given by the following general result (cf [64]):  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1 We consider the fuchsian system xY ′ = A(x) · Y, A0 = A(0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Suppose that the eigenvalues of A0 do not differ from a positive integer (the non resonant  case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' There exists a local meromorphic gauge transformation P in SL2 (C((x))) which  conjugates the fuchsian singular system to xZ′ = A0 · Z  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In this case, the system (1) admits a local fundamental solution Y = P(x)xA0, P in  SL2 (C((x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Furthermore if A is a convergent data, then P is also a convergent matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The general case reduces to the non resonant case by using shearing transformations, which can  eventually modify the Poincar´e rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We also obtain a local fundamental system Y = P(x)xL  for some constant matrix L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In any cases, the sectoral local solutions admits a moderate growth :  fuchsian singularities are regular singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The classification of non fuchsian singularities  is given by the Hukuhara Turritin Theorem [3] :  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='2 We consider a sl2-system zp+1Y ′ = A(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='Y, A0 = A(0) ̸= 0, p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' There exists  a formal meromorphic gauge transformation Y = PZ and a ramification z = yk such that the  differential system has the canonical form  yZ′ = (L + D1y−1 + · · · + Dmy−m) · Z,  where the Di’s are diagonal matrices, and L commutes with the Di’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  10  If the irregular part D1y−1 + · · · + Dmy−m vanishes, we still get a regular singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Other-  wise, the singular point is irregular, and the degree r = m/k is called the Katz invariant of the  irregular singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3 If the eigenvalues of A0 are distincts, we do not need to use a ramification z = yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Otherwise, for example if A0 is a nilpotent element of sl2(C), we first have to use shearing  transformations in order to modify the leading term A0, which may change the Poincar´e rank  and requires a ramification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This may occur for some Painlev´e families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Given an element A of SV , the singular point x = ∞ (z = 0) is an irregular singular point  with Katz rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since the eigenvalues ±t of A∞ are distinct, we do not need here a ramification  of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The explicit computation of the residue matrix L∞ gives:  L∞ =  \uf8eb  \uf8ec  \uf8ed  a0 + a1  0  0  −a0 − a1  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore the system (A) has a formal fundamental matrix solution around z = 0 given by  �Y (z) = P(z)zL∞ exp Q(z), Q(z) = diag(αz−1, −αz−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='2  The global gauge moduli space  We want to compute the categorical quotient SV //Sl2(C), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e the affine variety defined by the  ring of invariant functions under the action of the constant gauge action of Sl2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' One can  first use a constant gauge transformation in order to diagonalize A∞:  A∞ =  \uf8eb  \uf8ec  \uf8ed  t/2  0  0  −t/2  \uf8f6  \uf8f7  \uf8f8 , Ai =  \uf8eb  \uf8ec  \uf8ed  ai  bi  ci  −ai  \uf8f6  \uf8f7  \uf8f8 , i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Let S′  V be the subset of SV defined be the above writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For a fixed singular set s0, s1, ∞, it is  a 7-1=6 dimensional variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The remaining gauge action which normalizes the diagonal matrix  A∞ is the group N =< T, P > generated by  T = {Tm =  \uf8eb  \uf8ec  \uf8ed  m  0  0  m−1  \uf8f6  \uf8f7  \uf8f8 , m ∈ C∗} and P =  \uf8eb  \uf8ec  \uf8ed  0  1  −1  0  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This group acts on sl2(C) by  Tm ·  \uf8eb  \uf8ec  \uf8ed  a  b  c  −a  \uf8f6  \uf8f7  \uf8f8 =  \uf8eb  \uf8ec  \uf8ed  a  bm2  cm−2  −a  \uf8f6  \uf8f7  \uf8f8 , P ·  \uf8eb  \uf8ec  \uf8ed  a  b  c  −a  \uf8f6  \uf8f7  \uf8f8 · P −1 =  \uf8eb  \uf8ec  \uf8ed  −a  −c  −b  a  \uf8f6  \uf8f7  \uf8f8 = −AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  After this first reduction, SV //SL2(C) = S′  V //N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The following functions are invariant by the  action of N:  α0 = a2  0 + b0c0  α1 = a2  1 + b1c1  α∞ = (a0 + a1)2  τ = a0t  β0 = b0c1 + b1c0  β1 = t(b0c1 − b1c0)  (6)  11  Notice that the three coordinates α = (αi), i = 0, 1, ∞ are local invariants around each singular  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' As mentioned in the introduction the αi parameters used in the expressions of the Painlev´e  V equation and in its hamiltonien HV are not exactly equal to the present parameters but only  related to the eigenvalues of the Ai by an affine invertible map : see [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4 SV //SL2(C) = Spec(α0, α1, α∞, τ, β0, β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  It suffices to prove that the map (α0, α1, α∞, τ, β0, β1) :  S′V //SL2(C) → C6 is  invertible over the Zariski open set a0 ̸= 0, b0c0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Given a value (α0, α1, α∞, τ, β0, β1), we  first choose arbitrarily t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Now a0 is uniquely determined by ta0 = τ, and b0c0 is uniquely  determined by α0−a2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The values a2  0 and b0c0 determine a unique class [A0] modulo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It suffices  to prove that the choice of A0 in this class determines in a unique way the triple (A0, A1, A∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  A∞ is uniquely defined by a0t = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The two last equations define a linear system in (b1, c1)  which admits a unique solution for b0c0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The relation  2a0a1 = α2  ∞ − α2  0 − α2  1 + b0c0 + b1c1  determines a unique value of a1 for a0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Another choice of A0 in [A0] will give an equivalent  triple (A0, A1, A∞) modulo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  This quotient is endowed with a Poisson structure whose Casimir functions are α0, α1, α∞  and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore, each fiber has a canonical symplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' See the introduction for  references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3  The moduli space of connections MV  We now consider the action of a global change of independent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have previously  fixed one singular point at ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The positions s0 and s1 are now free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' One can use a  translation x → x+λ in order to get s0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This action do not modify A0 A1 and A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Now we  want to normalize the singular point s1 to the value 1, by using the last change of independent  variable that we can use: x �→ µx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since s1 ̸= s0 = 0, we can introduce the parameter s−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The  linear system  dY = (A0 +  A1  1 − s1x−1 + xA∞)dx  x ) · Y  is changed in  dY = (A0 +  A1  1 − s1µ−1x−1 + xµA∞)dx  x ) · Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This action do not modify A0 and A1 but modify A∞ in µA∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore it keeps invariant the  local variables αi and modify the others variables by:  (τ, β0, β1, s−1  1 ) → (µβ0, β1, µβ2, µs−1  1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The quotient of this action is the weighted projective space P3  (1,0,1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The usual choice of µ such  that s1 = 1, corresponds to a choice of chart in P3  (1,0,1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='5 The moduli space of connections MV (α) induced by SV for fixed local invari-  ants is isomorphic to P3  (1,0,1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This space is identical to that obtained by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Chiba in [17] when he seeks a  natural compactification on which lives the vector field PV by using its Newton polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since one weight vanishes, the quotient space is isomorphic to P2(C) × C, and it is not a  compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This fact is shared with the quotient spaces corresponding to the equations  PV I and PIII,D6, while we obtain a compact weighted projective space for the other families  (I), (II), (IV ), (III, D7), (III, D8): [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' To deal with this problem, we will use the  trick introduced by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Chiba, who introduces two copies of the previous space, glued by a  convenient B¨acklund transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2  The wild fundamental groupoid πV  1 (X, S)  The fundamental group of π1(P1 \\{s0, s1, s∞}, b0) acts on a local fundamental system Y0 around  b0 by analytic continuation of Y0 along a path in P1 \\ {s0, s1, s∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We first enlarge this usual  monodromy representation by considering new operators around the irregular point s∞: the  formal monodromy, the Stokes operators and the exponential torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We first describe theses  operators in the present context, by using a point of view very similar to that introduced by  Stokes himself in [58], starting from a formal solution and using the summability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1  Stokes operators, formal monodromy and exponential tori  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A formal power series �f = �  i≥0 aixi is 1/k-Gevrey if there  exists M > 0 and A > 0 such that for all i |ai| ≤ MAi(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' )1/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let V be a sector at the origin and f an holomorphic function on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' f is 1/k-Gevrey  asymptotic to �f if for every strict subsector W in V , there exists MW > 0 and AW > 0  such that for all n ≥ 1  | (f(x) −  n−1  �  i=0  aixi |≤ MW An  W (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' )1/k|x|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We denote by C[[x]]1/k the algebra of series which are 1/k-Gevrey, A1/k(V ), the algebra  of holomorphic functions on V which are 1/k-Gevrey asymptotic to some formal series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We  mention here the following facts (for the proofs see [44]):  If f is 1/k-Gevrey asymptotic to �f, then �f is a 1/k-Gevrey series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore the Taylor  map T : A1/k(V ) → C[[x]]1/k is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  For V “narrow” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' with an opening ≤ π/k, the Taylor map T is surjective: this is a  Gevrey extension of the Borel-Ritt theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  For V “large” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' with an opening > π/k, the Taylor map T is injective: this is a Gevrey  extension of the Watson Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Any formal solution of any linear or non linear analytic differential equation is 1/k-Gevrey  for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This a theorem of Maillet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In the linear case, the second author gives the  optimal Gevrey index by using the Newton polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In particular, for a linear system  xk+1dY/dx = A(x) · Y , this optimal order is the Katz rank of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let d ∈ S1 a direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  A formal power series �f = �  k≥0 aixi is k-  summable in the direction d if there exists a sector Vd bisected by d, whose opening is  greater than π/k and an holomorphic function f on V which is 1/k-Gevrey asymptotic to  �f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A formal power series �f is k-summable if �f is k-summable in any direction d excepted a  finite number of directions (the singular directions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Let C{x}1/k,d ⊂ C[[x]]1/k be the algebra of k-summable series in the direction d, and C{x}1/k ⊂  C[[x]]1/k the algebra of k-summable series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since Vd is a large sector, the summation operator  Σd : C{x}1/k,d → A(Vd) is an injective morphism of C-differential algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3 Let x2dY/dx = A(x)Y , A(x) in sl2(C{x}), be a germ at x = 0 of meromorphic  differential system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We suppose that the eigenvalues of A(0) are distincts (which avoids a rami-  fication of the independent variable), and that the singularity is irregular, with a Katz rank equal  to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The formal series appearing in a formal fundamental system of solutions �Y are 1-summable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For any non singular direction d, the operator Σd extends to a unique differential morphism  from the algebra C{x}1[xλ, x−λ, exp t/x, exp(−t/x)] into O(Vd), such that this morphism  induces the identity map on C[xλ, x−λ, exp t/x, exp(−t/x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This allows us to define Σd(�Y )  for a formal matrix solution �Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Given a formal fundamental system of solutions  �Y (x) = P(x)xL exp Q(x), Q(x) = diag(tx, −tx),  the singular directions d on which Σd(�Y ) is not defined are characterized by arg(x) = d if  and only if exp(2tx) has a maximal decay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' by tx ∈ R−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Let Σ be the set of singular directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The Stokes operators {Sd, d ∈ Σ} are defined in the  following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We choose a regular direction d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For any singular direction d, we choose two  regular directions d− and d+ such that the arc (d−, d+) contains the only one singular direction  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We choose a determination �Y0 of �Y around a d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This choice induces a determination �Y − of �Y  around the direction d− (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' a determination �Y + of �Y around d+) by analytic continuation of  the logarithm along the positive arc (d0, d−) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (d0, d+)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The summations Y − = Σd−(�Y −)  and Y + = Σd+(�Y +) are defined on large sectors V −  d and V +  d which intersects non trivially on an  open sector around d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore there exists some constant matrix Sd such that Y + = Y −Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  One can easily check that Sd does not depend on the choices of d+ and d− around d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A change  of choice for �Y or for its determination �Y0 around a d0 will modify the family {Sd, d ∈ Σ} by a  common conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4 Starting from a formal solution �Y0, the Stokes operator Sd are defined by the  successive operations: analytic continuation of the formal solution �Y0 along an arc from d0 to  d−, summation at d−, analytic continuation of Y − until d+ on V + ∩ V −, anti-summation at d+  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Taylor expansion of the actual solution Y +) and finally analytic continuation of the formal  solution �Y + from d+ to d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The Stokes operators at each singular direction are defined by the above  composition of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Given a formal solution �Y0, they are characterized by matrices  Ud, up to a common conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The formal monodromy is the operator generated by a loop �γ around 0 acting on a de-  termination of �Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  For a given determination �Y0, it is defined by a matrix denoted by  �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  14  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The geometric monodromy is the operator generated by a loop γ around 0 acting on a  determination of an actual solution Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For the actual solution Y0 obtained by summation  of �Y0, it is defined by a matrix denoted by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The (formal) exponential torus is an action of the algebraic group C∗ on �Y0 induced by  a rescaling of the exponential map: exp q → τ exp q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If �Y0 = PxL exp Q is chosen such  that Q is diagonal, the action of τ is given by a diagonal matrix Tτ = diag(τ, τ −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In the  general case Tτ is a Cartan component of SL2(C)  The motivation for the introduction of the exponential torus action is given by the following  heuristic interpretation in a one dimensional context: we consider the equation x2y′ + y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We introduce an unfolding of the irregular singularity at the origin : we replace D = x2d/dx + 1  by Dε := x(x − ε)d/dx + 1 (ε ∈ C∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Then the irregular singularity is replaced by two regular  singularities at 0 and ε, with respective exponents 1/ε and −1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' An important point to notice  is the coupling between the relative positions of the singularity and the exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The general  solution of Dεy = 0 is y = Cx1/ε(x − ǫ)−1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It is invariant by the monodromy action of a  simple loop around the two points 0 and ε but the monodromy action of a simple loop around  0 passing between the two points will transform f into e−2iπ/εf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' When ε → 0, this monodromy  action does not have a limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' However we can replace the continuous limit by a discrete limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We fix ε0 ∈ C∗ and we define a sequence (εn)n∈N by  1  εn :=  1  ε0 + n, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Then the sequence  (e−2iπ/εn)n∈N is constant and we can interpret f �→ fτ0, with τ0 := e−2iπ/ε0 as the action of a  simple loop passing between two infinitely near points4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' As the choice of ε0 is arbitrary, then τ0  is arbitrary in C∗ and we can consider the exponential torus as a generalized monodromy group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This group is no longer discrete, it is an algebraic group of dimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The “monodromy of  a loop between the two infinitely near singularities” can be interpreted as a “random point” in  this group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In the unfolding bifurcation Dε there is a breaking of symmetry, the choice of ε fix  a point e−2iπ/ε ∈ C∗ and the exponential torus is replaced by the ordinary monodromy group  generated by f �→ e−2iπ/εf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The random point is replaced by a true point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let d1, d2 = d1+π be the 2 singular directions of the differential system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have M = �  M · Ud2 · Ud1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If �Y0 = PxL exp Q is chosen such that Q is diagonal, Ud1 is a lower triangular unipotent  matrix and Ud2 is a upper triangular unipotent matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In the general case, Ud1 and Ud2  are in the two Borel components of the Cartan component of the exponential torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In the present case (a non ramified case), the formal monodromy commutes with the expo-  nential torus (and belongs to it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This fact is no longer true in the ramified case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  A Borel-Cartan configuration is a triple of subgroups (B−, C, B+) in SL2(C) such that C is a  maximal torus, isomorphic to the algebraic group C∗, and B− and B+ are two maximal solvable  subgroups which contains C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Any Borel-Cartan configuration is conjugated to the canonical  one given by (T −, D, T +) where D is the subgroup of diagonal matrices, T − (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' T +) the  subgroup of lower (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' upper) triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The unipotent subgroups of B− and B+  are respectively denoted by U − and U +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The triple (Ud1, �  M, Ud2) and the triples (Ud1, Tτ, Ud2) take their values in a unipotent Borel-  Cartan configuration (U −, C, U +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The local wild dynamics of the irregular connection is the subgroup (well defined up to  a conjugation) of SL2(C) generated by the Stokes operators, the formal monodromy and the  4The idea of considering an irregular singularity as a pack of infinitely near regular singularities is due to Ren´e  Garnier in 1919 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It will be used as a guideline in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  15  exponential torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' According to a result of the second author [55], the Zariski closure of the  local dynamics in SL2(C) is the differential Galois group of the local linear differential system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  For the study of isomonodromic deformations, we need a parametric version of Theorem  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' They are parametric variants of Gevrey power series and of Gevrey expansions (with  analytic parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' One supposes that the estimates in definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1) are uniform on the  parameter space U ⊂ Cp, or more generally on every compact K of U : one replaces MW and  AW by MW,K and AW,K (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' uniform on the parameter space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For an open set U ⊂ Cp,  we will denote O(U)[[x]]1/k the algebra of Gevrey series of order 1/k uniformly on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' They  are also a similar parametric variant of k-summability using the uniformly parametrized 1/k-  Gevrey expansions : the definition is the same mutatis mutandis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We will speak of uniform  k-summability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Then the sums of the uniformly k-summable series in a direction d are analytic  in the variable and in the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='5 For more information on these subjects cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='7 Let U ⊂ Cp be an open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Let D ⊂ C be an open disc centered at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Let [A] :  x2dY/dx = A(x, t)Y , where A ∈ sl2 (O(D × U)), be a parametrized meromorphic  differential system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We suppose that, for all t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' tp) ∈ U, the eigenvalues of A(0, t) are  distinct and that the singularity is irregular, with a Katz rank equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (i) Locally on U, the system [A] admits a formal fundamental solution analytic in the param-  eters :  ˆF = P �HxLeQ, where P ∈ SL2 (O(V )), �H ∈ SL2 (O(V )[[x]]1), �H(0, t) = I (for  every t ∈ V ), L ∈ sl2(C), Q = (q, −q), with q ∈ x−1O(V )[x−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (ii) Let t0 ∈ U and V ⊂ U a neighborhood of t0 such that the system admits on V a formal  fundamental solution analytic in the parameters as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let t1 ∈ U and d a nonsingular  direction for the system x2dY/dx = A(x, t1)Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Then there exists a neighborhood W ⊂ V  of t1 such that P �H is uniformly 1-summable in the direction d on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (i) Let t0 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If an open polydisc V centered at t0 is sufficiently small, then there  exists P ∈ GL (O(V )) conjugating the restriction of A0 = A(0, t) to V to a diagonal matrix B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore we are reduced to the case where A0 is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Then we can use a parametrized  variant of the splitting lemma of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It reduces the system to two parametrized formal one  dimensional equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The result follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (ii) A first proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We use a parametrized generalization of the Improved Splitting Lemma of  [3] (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1, Lemma 11, page 124).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The proof is the same mutatis mutandis6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  A second proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We imitate a proof of the unparamatrized version based on Gevrey asymp-  totics (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' [44]), replacing the Gevrey Main Asymptotic Existence Theorem by a parametrized  version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='2  The local wild fundamental groupoid  Following an idea which appears in a correspondence between P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Deligne B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Malgrange and the  second author [19], the Stokes operators, formal and geometric monodromy around an irregular  point appear as a representation of a “local wild fundamental groupoid”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We want to encode  all these operators by loops, or paths if we allow several base points, in a “halo” (see also [44])  around the singular point whose internal boundary describes the formal world (represented by  5when these sums exist for all values of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In general the singular directions move with the  parameters and it is necessary to reduce the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  6It uses the interpretation of 1-summability in terms of the Borel-Laplace method and delicate estimates in  the Borel plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  16  determinations of formal solutions) and its exterior the analytic world (represented by sectoral  holomorphic solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We consider a small disc ∆0 around x = 0, and we perform a real blowing up E : X0 → ∆0  of x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let D = E−1(0) ⊂ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We draw a second divisor �D inside the disc bounded by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We mark two opposite directions by drawing two points p1 and p2 in the annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let A be the  punctured annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We consider the groupoid π1(X0, D ∪ �D) whose objects are the points of  D ∪ �D and whose morphisms are the paths or loops up to homotopy between two objects inside  the punctured annulus A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let S0 = {s1, s2} two opposite directions distinct from p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='8 The local wild groupoid π♭  1(X0, S0) is the restriction of π1(X0, D ∪ �D) over S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We define a presentation of π♭  1(X0, S0) in the following way:  s1  s2  p2  p1  r+  1  r−  1  α1  �γ−  1  �γ+  1  r1  r2  �γr  1,2  γr  1,2  �γl  1,2  γl  1,2  Figure 2: A presentation of π♭  1(X0, S0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We choose rays r−  i , r+  i on the left and right side of pi for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We choose two opposite  base points s1 and s2 on D on the orthogonal direction to (p1, p2), endpoints of 2 rays r1 and  r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let �γ−  i (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='�γ+  i ) be the arc from si to the origin of r−  i (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' r+  i ) in �D, and αi the arc on D  from the end of r−  i to the end of r+  i on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The two Stokes loops are the loops based in si (see  figure above):  σi = r−1  i  �γ−  i · r−  i · αi · (r+  i )−1 · (�γ+  i )−1ri, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Let γr  1,2 and γl  1,2 the lower and upper arcs from s1 to s2 on D, and �γr  1,2 and �γl  1,2 their analog on  �D by using r1, an arc on �D, and r−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The local wild groupoid is generated by σ1, σ2, γr  1,2, γl  1,2,  �γr  1,2 and �γl  1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let γ1,1 = γr  1,2 · γl  1,2, �γ1,1 = �γr  1,2 · �γl  1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' From the above picture we have the relation  γ1,1 = σ1 · �γr  1,2 · σ2 · �γl  2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In order to include the exponential torus in a linear representation of this groupoid, we introduce  a (non discrete) extension π1(X0, S0) of π♭  1(X0, S0): we glue a representation of (C∗, ×) by adding  two collections of loops �t1,1(κ), κ ∈ C∗ based in �s1, and �t2,2(κ), κ ∈ C∗ based in �s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We denote:  t1,1(κ) = r−1  1  �t1,1(κ) · r1( based in s1), t2,2(κ) = r−1  2  �t2,2(κ) · r2( based in s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  17  We require the relations Rloc:  (i) ∀κ ∈ C∗, ∀κ′ ∈ C∗, ti,i(κκ′) = ti,i(κ) · ti,i(κ′), i = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (ii) ∀κ ∈ C∗, [[σi, ti,i(κ)], σi] = ⋆i, i = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (iii) ∀κ ∈ C∗, �γi,i · ti,i(κ) · �γ−1  i,i ti,i(κ)−1 = ⋆i, i = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (iv) t1,1(κ) · �γl  1,2 · t2,2(κ) · �γr  2,1 = ⋆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='9 The (complete) wild local groupoid is the groupoid π1(X0, S0) generated by  π♭  1(X0, S0) and the families T1,1 = {t1,1(κ), κ ∈ C∗} and T2,2 = {t2,2(κ), κ ∈ C∗}, with the  above relations Rloc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  s1  s2  p2  p1  t1,1(κ)  t2,2(κ)  Figure 3: A presentation of π1(X0, S0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3  The global wild fundamental groupoid  The local wild fundamental groupoid suffices to deal with the Euler equation, or the Kummer  equations by adding one base point [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Nevertheless for the Painlev´e equations, we have to  consider several regular and irregular singularities, with connecting data between them (and in  some cases an order two ramification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In order to represent the global monodromy data together with the wild local dynamic of a  connection in MV , we consider the following groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We begin with B = P1\\{p0, p1, p∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We  perform real blowing up at each of these points, and we obtain a variety X with divisors D0, D1  and D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We choose two opposite base points s1 and s2 on D∞, a base point s3 on D0, and s4 on  D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let S = {s1, s2, s3, s4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We consider the groupoid whose morphisms are the paths between  the base points up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Inside (D∞, s1, s2), we glue the local wild groupoid π♭  1(X0, S0)  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' the complete local wild groupoid π1(X0, S0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We obtain a new groupoid denoted by  πV,♭  1 (X, S) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' πV  1 (X, S) for the complete version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A presentation of this groupoid is given  by the figure:  18  s3  s4  γ2,3  γ4,1  γ3,4  p1  p2  s1  s2  γ3,3  γ4,4  Figure 4: A presentation of πV  1 (X, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The relations of this presentation are generated by :  – the local relations Rloc previously defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  – Rext: γl  1,2 · γ2,3 · γ3,4 · γ4,1 = ⋆1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  – Rint: γr  1,2 · γ2,3 · γ3,3 · γ3,4 · γ4,4 · γ4,1 = ⋆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We denote by πV,κ  1  (X, S) the fiber of the morphism K :  πV  1 (X, S) → C∗ induced by  K(t1,1(κ)) = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4  A confluent morphism of groupoid  The comparison of πV,κ  1  (X, S) with πV I  1 (X, S) is a central point in order to define the confluent  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For this purpose, we introduce another presentation of this groupoid, setting:  t1,2(κ) = t1,1(κ−1) · �γr  1,2 = �γl  1,2 · t2,2(κ−1),  γ2,3(κ) = t2,2(κ−1) · σ−1  2  γ2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We obtain the following figure  19  ≃  t1,2(κ)  t1,1(κ)  t2,2(κ)  γ2,3(κ)  σ1  σ2  t1,2(κ)  t1,1(κ)  σ1  t2,2(κ)  σ2  γ2,3(κ)  Figure 5: Another presentation for πV,κ  1  (X, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This last figure looks like the figure (1) for the groupoid πV I  1 (X, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' More precisely, we denote  by s′  i the objects of πV I  1 (X, S), and we consider a presentation {γ′  i,j, R′  int, R′  ext} of πV I  1 (X, S)  given by figure (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We consider the groupoid morphism ϕκ from πV I  1 (X, S) to πV,κ  1  (X, S) which  sends s′  i on si and which is defined by:  ϕκ(γ′  3,3) = γ3,3, ϕκ(γ′  4,4) = γ4,4, ϕκ(γ′  3,4) = γ3,4, ϕκ(γ′  4,1) = γ4,1,  ϕκ(γ′  1,1) = σ1 · t1,1(κ), ϕκ(γ′  2,2) = σ2 · t2,2(κ),  ϕκ(γ′  1,2) = t1,2(κ), ϕκ(γ′  2,3) = γ2,3(κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This morphism ϕκ is an injective morphim but not a surjective one: the pre-image of a  Stokes loop or a loop of an exponential tori are not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='10 The confluent morphism from ϕ :  πV I  1 (X, S) → πV  1 (X, S) is defined by the  family ϕ = (ϕκ)κ∈C∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  There exists a similar result in the context of the hypergeometric equation and the confluent  hypergeometric equation in Kummer form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3  The character variety χV  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1  Definition of χV  The class of rank 2 linear representations of a fundamental groupoid in SL2(C) has been defined  in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Here we are only interested in a subclass of linear representations ρ which  satisfy the following property  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1 A representation ρ of πV  1 (X, S) satisfies the property (⋆) if there exists a Borel-  Cartan configuration (B−, C, B+) such that:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ρ(t1,1(κ), κ ∈ C∗) = C,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ρ(σ1) ∈ U −, ρ(�γl  1,2σ2�γl  2,1) ∈ U +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  If this property holds for ρ it still holds for ρ′ equivalent to ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  20  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='2 The character variety χV is the categorical quotient of the set of linear repre-  sentations πV  1 (X, S) which satisfy the property (⋆) through the equivalence of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  A local representation is a representation over the sub-groupoid restricted to the local loops,  generated by �γ1,1, γ3,3, γ4,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  It is characterized by a = (a0, a3, a4) with a0 = tr(ρ(�γ1,1)) =  tr(ρ(�γ2,2)), a3 = tr(ρ(γ3,3)), a4 = tr(ρ(γ4,4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Any representation ρ in χV determines a unique  local representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The fiber of this map is denoted by χV (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The inclusion of πV,♭  1 (X, S) in πV  1 (X, S) induces by restriction a map r whose image is denoted  by χ♭  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3 For a such that a0 ̸= ±2, r is a (2:1) map over the open set of χ♭  V defined by  ρ(σ1) or ρ(σ2) is a non trivial element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If we suppose that σ1 is non trivial, one can choose �  Y0 such that  ρ(σ1) =  \uf8eb  \uf8ec  \uf8ed  1  0  u1  1  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In this case, ρ(t1,1(κ), κ ∈ C∗) = C is the diagonal subgroup D of SL2(C), and we have  ρ(t1,1(κ)) =  \uf8eb  \uf8ec  \uf8ed  f(κ)  0  0  f(κ−1)  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  From the relation t1,1(κ)t1,1(κ′) = t1,1(κκ′) we deduce that f belongs to Aut(C∗, ×), and there-  fore f(κ) = κ or f(κ) = κ−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4 The character variety χV (a) = χ+  V (a) ∪ χ−  V (a) with  χ+  V (a) = {(U1, M0, U2, M3, M4, Dκ)}//SL2(C),  χ−  V (a) = {(U1, M0, U2, M3, M4, D−1  κ )}//SL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The two copies χ+  V (a) and χ−  V (a) coincide over a such that e0 = e−1  0  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' such that a0 = ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='2  The character variety χV and cubic surfaces  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='5 A representation of πV,♭  1 (X, S) in SL2(C) is normalized if,  ρ(γl  1,2) = ρ(γ2,3) = ρ(γ3,4) = I,  where the γi,j are generators of the presentation introduced in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  There always exists a normalized representation in each class of equivalent representations in  SL2(C), obtained by choosing a representation of an initial object –say s2– and by representing  the successive others objects by analytic continuation along γ2,3, γ3,4 and γl  2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' From the relation  Rext, for a normalized representation we also have ρ(γ4,1) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='6 A normalized representation of πV,♭  1 (X, S) is characterized by the data  M0 = ρ(�γ1,1), U1 = ρ(σ1), U2 = ρ(�γl  1,2 · σ2 · �γl  2,1), M3 = ρ(γ3,3), M4 = ρ(γ4,4)  such that U1M0U2M3M4 = I, up to a common conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  21  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have to prove that, for a normalized representation ρ, the images of all the  generators of the groupoid by ρ are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since ρ(γl  1,2) = I, we have  ρ(�γl  1,2) = U2, ρ(�γr  1,2) = M0U2, ρ(γr  1,2) = U1M0U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The relation Rint gives U1M0U2M3M4 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A change of representation of the initial object will  change this data by a common conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  One can suppose, by using a conjugacy, that the Cartan subgroup C = {ρ(t1,1(κ)), κ ∈ C∗} ⊂  SL2(C) is the diagonal one D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' From the relation (iii) of the local groupoid presentation, �  M0  commutes with each element of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore from the relations (i) (ii) and (iii), we have:  U1 =  \uf8eb  \uf8ec  \uf8ed  1  0  u1  1  \uf8f6  \uf8f7  \uf8f8 , M0 =  \uf8eb  \uf8ec  \uf8ed  e0  0  0  e−1  0  \uf8f6  \uf8f7  \uf8f8 , U2 =  \uf8eb  \uf8ec  \uf8ed  1  u2  0  1  \uf8f6  \uf8f7  \uf8f8 ,  M3 =  \uf8eb  \uf8ec  \uf8ed  α3  β3  γ3  δ3  \uf8f6  \uf8f7  \uf8f8 , M4 =  \uf8eb  \uf8ec  \uf8ed  α4  β4  γ4  δ4  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This data is now defined up to a conjugacy by an element of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Finally we have  χ♭  V = {(U1, M0, U2, M3, M4) ∈ (U −, D, U +) × SL2(C)2, U1M0U2M3M4 = I}//D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This algebraic quotient is a 5 dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The function (a+, x+) = (e0, a3, a4, x+  1 , x+  2 , x+  3 ):  e0 = M0[1, 1], a3 = tr(M3), a4 = tr(M4),  x+  1 = M3[2, 2] = δ3, x+  2 = M4[2, 2] = δ4, x+  3 = tr(U1M0U2) = e0 + e−1  0  + e0u1u2,  is invariant under the action of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='7 The coordinates (a+, x+) define a map Tr+  V , invertible for a generic a, from  χ♭  V (a) to the family affine cubic surface CV (θ+) defined by  FV (θ+, x) = x1x2x3 + x2  1 + x2  2 − θ+  1 x1 − θ+  2 x2 − θ+  3 x3 + θ+  4 = 0,  where θ+  1 = a3 + e0a4, θ+  2 = a4 + e0a3, θ+  3 = e0, θ+  4 = e2  0 + e0a3a4 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='8 We have:  e0u1u2 = x3 − e0 − e−1  0  e0γ3s2 = x2 − e0a3 + e0x1  e0β3s1 = −x1x3 + e0x1 − x2 + a4  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have x3 = e0 + e−1  0  + e0u1u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' From U1M0U2M3 = M−1  4  we obtain  δ4 = α3e0 + γ3e0u2, α4 = β3e0s1 + δ3e0u1u2 + δ3e−1  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Using αi = ai − δi, we obtain the two other equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Proof of Proposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since α3δ3 − β3γ3 = 1, we have  (e2  0u1u2)(α3δ3) − (e0β3u1)(e0γ3u2) − e2  0u1u2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  22  If we replace e0s1s2, e0γ3s2 and e0β3s1 by the above expressions, we obtain the equation  FV (x, θ+) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This proves that the map (a+, x+) takes its values in the cubic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Given  a point in the cubic surface, we recover α3 = a3 − x1, α4 = a4 − x2, β3γ3 = 1 − x1(a3 − x1),  β4γ4 = 1 − x2(a4 − x2), γ3s2 = e−1  0 x2 − a3 + x1, and therefore, for generic values, a unique point  of χ♭  V (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='9 It might seen more natural to consider the trace coordinates:  y1(ρ) = tr(ρ(�γ1,1γ1,3γ3,3)), y2(ρ) = tr(ρ(�γ2,2γ2,4γ4,4)), y3(ρ) = tr(ρ(γ3,3γ3,4γ4,4))  which, for a normalized representation, give:  y1 = tr(M0M3), y2 = tr(M0M4), y3 = tr(M3M4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  But clearly, the coordinates y1 and y2 degenerate if M0 = ±I, since we have y1 = a3 and y2 = a4  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The coordinates  x+  1 = y1 − e0a3  e−1  0  − e0  = δ3, x+  2 = y2 − e0a4  e−1  0  − e0  = δ4, x+  3 = y3  give an extension to this exceptional case M0 = ±I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This choice is also the usual one in the  litterature: see [54] or [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since the coordinates (a+, x+) are directly related to the above trace  coordinates through an affine map, we still denote the corresponding map by Tr+  V , and we call it  a trace map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The normalization of the representation by (U1, M0, U2, M3, M4) used in order to construct Tr+  V  requires that U1 is a lower triangular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If we require that U1 must be an upper triangular  matrix, we obtain the data (U −  1 , M−  0 , U −  2 , M−  3 , M−  4 ) which is equivalent to the first data by the  conjugacy with the matrix  P =  \uf8eb  \uf8ec  \uf8ed  0  1  −1  0  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We define the trace map Tr−(ρ) = (a−, x−) = (e−1  0 , a3, a4, x−  1 , x−  2 , x−  3 ) with  e−1  0  = M−  0 [1, 1], x−  1 = M−  3 [1, 1] = α3, x−  2 = M−  4 [1, 1] = α4, x−  3 = tr(U −  1 M−  0 U −  2 ) = x+  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This map takes its values in the cubic surface CV (θ−):  FV (x, θ−) = x1x2x3 + x2  1 + x2  2 − θ−  1 x1 − θ−  2 x2 − θ−  3 x3 + θ−  4 = 0,  where θ−  1 = a3 + e−1  0 a4, θ−  2 = a4 + e−1  0 a3, θ−  3 = e−1  0 , θ−  4 = e−2  0  + e−1  0 a3a4 + 1 which can also  be identified to the categorical quotient χ♭  V (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The points p+ ∈ CV (θ+) and p− = Tr−  V ◦  (Tr+  V )−1(p+) correspond to the same representation in χ♭  V (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4) we know that χV (a) = χ+  V (a) ∪ χ−  V (a) where  χ+  V (a) = {(U1, M0, U2, M3, M4, Dκ)//SL2(C)},  χ−  V (a) = {(U1, M0, U2, M3, M4, D−1  κ )//SL2(C)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Using a conjugacy with P this second data is equivalent to (U −  1 , M−  0 , U −  2 , M−  3 , M−  4 , ρ(t1,1(κ)) =  Dκ), which define an element in CV (θ−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore χV (a) is identified through the trace coor-  dinates to the union of the two affine surfaces CV (θ−) ∪ CV (θ+)  We denote by Tr±,κ the restriction of Tr± over πV,κ  1  (X, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  23  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3  Lines and reducibility locus  For what follows in this article, excepted (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3), we will suppose that the parameters satisfy the  generic conditions:  e0 ̸= ±1, e3 ̸= ±1, e4 ̸= ±1, e0eε3  3 eε4  4 ̸= 1 for ε3 = ±1, ε4 = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The description of the lines in cubic surfaces can be found in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In particular the compacti-  fication of a generic element CV I(θ) in P3(C) contains 27 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In the figure below the central  triangle is the union of the three lines at infinity:  Le1,e2 Le−1  1 ,e−1  2  Le3,e4 Le−1  3 ,e−1  4  Le1,e−1  2 Le−1  1 ,e2 Le3,e−1  4 Le−1  3 ,e4  Le1,e3  Le2,e3  Le−1  1 ,e−1  3  Le−1  2 ,e−1  3  Le1,e3  Le−1  1 ,e−1  3  Le1,e−1  3  Le−1  1 ,e3  Le2,e4  Le−1  2 ,e−1  4  Le2,e−1  4  Le−1  2 ,e4  Le2,e−1  3  Le−1  2 ,e3  Le1,e4  Le−1  1 ,e−1  4  Le1,e−1  4  Le−1  1 ,e4  Figure 6: The 24 lines in CV I(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The equations of the lines are obtained from the following decomposition which appears in [40]:  FV I(x, θ) = (xk − cei,ej)(FV I,xk − xk + cei,ej) + lei,ejle−1  i  e−1  j ,  where FV I,xk is the partial derivative of FV I(x, θ) with respect to the variable xk, cα,β = αβ−1 +  α−1β and  lei,ej = eixi + ejxj − akeiej − a4, le−1  i  ,e−1  j  = e−1  i xi + e−1  j xj − ake−1  i e−1  j  − a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore the plane xk = cei,ej intersects the cubic surface along a degenerated conic, union of  the two lines  Lei,ej : xk = cei,ej = eie−1  j  + e−1  i ej and lei,ej = 0, Le−1  i  ,e−1  j  : xk = ce−1  i  ,e−1  j  and le−1  i  ,e−1  j  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The compactification of CV (θ) in P3(C) has a singular point of type A1 at infinity, and admits  21 lines, of which 18 are in the affine part:  24  ∆e4  ∆e−1  4  ∆e3  ∆e−1  3  Dl  e−1  4  Dr  e3  Dl  e4  Dr  e−1  3  Dl  e3  Dr  e−1  4  Dl  e−1  3  Dr  e4  De3,e4  De−1  3 ,e−1  4  De3,e−1  4 De−1  3 ,e4  Df0,f−1  0  Df−1  0  ,f0  Figure 7: The 18 lines in CV (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='10 The equations of these lines are given by  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' z0 = −x2  1x3 − x1x2 + θ2x1 − e0 = 0 for Dl  e3 ∪ Dl  e−1  3  ∪ Dl  e−1  4  ∪ Dl  e4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' z1 = x1x2 − e0 = 0 for ∆e3 ∪ ∆e−1  3  ∪ ∆e4 ∪ ∆e−1  4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' z2 = −x2  2x3 − x1x2 + θ1x2 − e0 = 0 for Dr  e−1  3  ∪ Dr  e3 ∪ Dr  e4 ∪ Dr  e−1  4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' x3 = ce3,e4 or ce3,e−1  4  or ce0,e−1  0  for the pairs of lines which cut the basis of the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In this section, we want to characterize the representations whose image is a point on a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='7  The local loops in πV I  1 (X, S) are the loops γi,i based in si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The local loops in πV  1 (X, S) are γ3,3  in s3, γ4,4 in s4, and a generic element of the torus t1,1(κ) in s1 and t2,2(κ) in s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In any cases,  the local loops at si are denoted below by γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='11 Let γ be a morphism of the groupoid πV I  1 (X, S) or πV  1 (X, S), defined by some  path from si to sj, j ̸= i, γi the local loop in si and γj the local loop in si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A linear representation ρ  of the groupoid in SL2(C) is reducible along γ if ρ(γ−1γiγ) and ρ(γj) have a common eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  If ρ is the linear representation associated with a regular connection ∇, and γ is represented  by analytic continuation, it means that there exists some local solution at si which is an eigen-  vector of the local monodromy around si and whose analytic continuation along γ is also an  eigenvector of the local monodromy around sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This semi-local solution in a neighbourhood of  γ has an abelian monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If ρ is reducible along the path γ, any equivalent representation is reducible  along γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The representation ρ is reducible along γ if and only ρ is reducible along γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If ρ(γi) = ±I, ρ is reducible along any path joining si to another base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  7We thanks Martin Klimes for discussions on this topics (see also [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let ρ be a representation in SL2(C) given by a choice of basis of solutions Xi on each Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We set ρ(γi) = Mi, ρ(γ) = Mγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ρ is reducible on γ if and only if the matrices M−1  γ MiMγ  and Mj have a common eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In particular, if ρ is a normalized representation associated to the presentation given by  figure (1) of the groupoid πV I  1 (X, S), with ρ(γi) = Mi and ρ(γi,j) = I for i ̸= j, ρ is  reducible on the generator path γi,j if and only if Mi and Mj have a common eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='13 A pair of matrices in SL2(C) is reducible if and only if they have a common  eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The reducibility locus associated to a path γ is the set R(γ) of linear  representations ρ in χV I which are reducible on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Given a presentation of πV I  1 (X, S), the generating reducibility locus is the union of the sets  R(γi,j) for all the generators γi,j, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The total reducibility locus is the set R of linear representations ρ in χV I such that there  exists a path between two distinct objects on which ρ is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='15 Let {γi,j), Ri} be the presentation of πV I  1 (X, S) given by the figure (1), and  TrV I : χV I(a) → CV I(θ) the trace map associated to this presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The 24 lines in CV I(θ)  are the reducibility locus of the 6 paths: γi,i+1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='., 4 and γ1,3 = γ1,2 · γ2,3, γ2,4 = γ2,3 · γ3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='16 Let M1 and M2 be two matrices in SL2(C) distinct from ±I, e1, e−1  1 , and e2,  e−1  2  their eigenvalues (we may have e1 = e−1  1  or e2 = e−1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let  cα,β = αβ−1 + α−1β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The pair (M1, M2) is reducible if and only if  tr(M1M2) = ce1,e2, or tr(M1M2) = ce1,e−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We choose a ”mixed basis” taking an eigenvector of M2, and an eigenvector of M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In such a basis, we have:  M1 =  \uf8eb  \uf8ec  \uf8ed  e1  0  f1  e−1  1  \uf8f6  \uf8f7  \uf8f8 ,  M2 =  \uf8eb  \uf8ec  \uf8ed  e2  f2  0  e−1  2  \uf8f6  \uf8f7  \uf8f8  or any similar writing obtained by changing e1 with e−1  1  or e2 with e−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let Di = diag(ei, e−1  i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have  tr(M1M2) = tr(D1D2) + f1f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Such a pair has a common eigenvector if and only if f1 = 0 or f2 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  if and only if  tr(M1M2) = tr(D1D2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have tr(D1D2) = ce1,e2 or tr(D1D2) = ce1,e−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Proof of Theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We consider a normalized representation ρ and the six (non ori-  ented) generating paths γi,j, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  For a normalized representation ρ, these paths satisfy:  ρ(γi,j) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore ρ is reducible over γi,j if and only if the pair of matrices (Mi, Mj) is  reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For {i, j} in {1,2,3}, according to Lemma (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='16), ρ is reducible on the path γi,j if and  only if xk = cei,ej or xk = cei,e−1  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore the reducibility locus R(γi,j) is given by the four  lines Le±1  i  ,e±1  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  26  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='17 The 12 Kaneko points are the intersections  pei,ej = Lei,ej ∩ Le−1  i  ,e−1  j , pei,e−1  j  = Lei,e−1  j  ∩ Le−1  i  ,ej, {i, j} ⊂ {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='18 The Kaneko points pei,ej and pei,e−1  j  correspond to a normalized monodromy  representation such that Mi and Mj commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In a ”mixed basis”,  Mi =  \uf8eb  \uf8ec  \uf8ed  ei  0  fi  e−1  i  \uf8f6  \uf8f7  \uf8f8 ,  Mj =  \uf8eb  \uf8ec  \uf8ed  ej  fj  0  e−1  j  \uf8f6  \uf8f7  \uf8f8  (7)  or the same writing changing ej in e−1  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The reducibility locus is given by fifj = 0, which  defines two components Lei,ej and Le−1  i  ,e−1  j  (or Lei,e−1  j  and Le−1  i  ,ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore pei,ej (or pei,e−1  j )  is defined by fi = 0 and fj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='✷  The Painlev´e VI foliation admits 12 singular points, 4 over each singular fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The local  description of the foliation in a neighborhood of these non linear singularities of “regular type”  can be found in the chapter 4 of [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The Painlev´e V foliation also admits 3 non linear regular  singular points over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The germ of Painlev´e equation admits a unique meromorphic solution  around such a singular point, defining an analytic leave for the germ of foliation: we call them  the central solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' These solutions have been studied by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Kaneko [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Furthermore, they  appears as the intersection of two codim 1 germs of invariant analytic surfaces [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The Riemann-Hilbert correspondence RHV I sends the 12 central solutions  to the 12 Kaneko points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The Riemann-Hilbert correspondence RHV I sends the local invariant varieties near a sin-  gular point to the 2 germs of lines intersecting at the corresponding Kaneko point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We consider the 4 singular points over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The non linear monodromy generated  by a simple loop around 0 will keep invariant each meromorphic solutions and the analytic  invariant surfaces which intersect on it, around this singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Through the Riemann-Hilbert  correspondence this non linear monodromy is conjugated to one of the polynomial dynamics hi,j  given by (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The central solutions are sent on fixed points of hi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since each hi,j fixes 4  central points, and has no more than 4 fixed points, this proves the first point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Now we search for invariant curves under the action of hi,j through the central point pei,ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Clearly the two lines Lei,ej and Le−1  i  ,e−1  j  are invariant since the union of the two lines is (xk =  cei,ej) ∩ (FV I(x, θ) = 0) which is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Suppose that there exists another local analytic  invariant curve transverse to dxk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It cut the plane (xk = c) on a finite set of points, for any  value c near from cei,ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This would create a periodic orbit on each (xk = c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The restriction of  hi,j on each (xk = c) is a family of affine maps, and for a generic affine map, there doesn’t exist  such a periodic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore the local invariant surfaces are sent on the germ of two lines  around each central point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='20 Using Proposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='18) and Theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='19), we recover here a result of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Kaneko [36]: the linear monodromy data preserved along a Kaneko solution is generated by  3 matrices with a pair of commuting matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In particular, it generates a solvable group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Nevertheless, the theorem of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Kaneko is much more precise: it describes explicitely this linear  solvable monodromy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  27  We have a result similar to Theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='15) for the Painlev´e V foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='21 Let {γi,j, R} be the presentation of πV  1 (X, S) given by figure (4), and Tr+  V :  χ+  V (a) → CV (θ+) the trace map associated to this presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Any line in CV (θ+) is the image  of a reducibility locus in χV (a) for some path in πV  1 (X, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The proof is similar to the one of Theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='15, but one needs to investigate different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We just mention here an example:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='22 The reducibility locus R(γr  1,2) is given by a pair of lines defined by (x3 = a0)∩CV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For a normalized representation ρ we have  ρ(γr  1,2) = U1M0U2, ρ(�γ1,1) = ρ(�γ2,2) = M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore, according to Remark (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='12), we have:  ρ ∈ R(γr  1,2) ⇔ ((U1M0U2)−1M0(U1M0U2), M0) is a reducible pair  ⇔ (M−1  0 U −1  1 M0U1, U2M0U −1  2 M−1  0 ) is a reducible pair  ⇔ u1u2 = 0  ⇔ tr(U1M0U2) = tr(M0),  ⇔ x3 = a0 = f 2  0 + f −2  0  where f 2  0 = e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The intersection (x3 = a0) ∩ CV is given by one pair of lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Indeed we have:  FV (x, θ) = (x3 − e0 − e−1  0 )(x1x2 − e0) + df0,f−1  0 df−1  0  ,f0,  with  df0,f−1  0  = f0x1 + f −1  0 x2 − f0a3, df−1  0  ,f0 = f −1  0 x1 + f0x2 − f0a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore the reducibility locus R(γr  1,2) is given by the pair of lines:  Df0,f−1  0  : x3 = e0 + e−1  0  and df0,f−1  0  = 0,  Df−1  0  ,f0 :  x3 = e0 + e−1  0  and df−1  0  ,f0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  The others cases are treated with similar computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Here we also have a correspondence  between the 3 Kaneko solutions described by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Kaneko and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Ohyama in and [37], and the 3  Kaneko points pe3,e4, pe3,e−1  4  and pe0,e−1  0  through RHV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The complete reducibility locus, defined by all the paths γ of the fondamental groupoid,  contains these lines and their images by the tame dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We don’t know if the lines and the  tame action generate the complete reducibility locus (maybe we have to add some parabolas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4  Log-canonical coordinates on CV (θ) and cluster sequences  Let CV (θ) be the family of symplectic cubic affine surfaces in C3 defined by FV (x, θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The  cubic surfaces CV (θ) are birational to C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Indeed, since the equation FV (θ) is affine in x3, the  map π3 : CV (θ) → C2 induced by (x1, x2, x3) → (x1, x2) is rationally invertible and therefore  birational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The polar locus of π−1  3  is given by FV,x3 = x1x2 − e0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Recall that CV (θ) has a  symplectic structure defined by  ωV (θ) = dxi ∧ dxj  ∂FV,θ/∂xk  ,  for any i, j, k = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We want here to introduce symplectic birational maps:  Notations (see Appendix):  28  ωlog = du  u ∧ dv  v : the log-canonical symplectic form on C2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Bir(C2) : the group of the birational automorphisms of the complex plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Symp(C2) = Symp+(C2) : the subgroup of Bir(C2) of the automorphisms which preserve  ωlog;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Symp−(C2) : the subset of Bir(C2) of the automorphisms ϕ such that ϕ∗ωlog = −ωlog;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Symp±(C2) : the subgroup of Bir(C2) generated by Symp+(C2) ∪ Symp−(C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A log-canonical system of coordinates on CV (θ) is a birational symplectic  morphism from (CV (θ), ωV (θ)) to (C2, ωlog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A log-canonical function on CV (θ) is a component of some log-canonical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A log-canonical sequence of coordinates is a sequence of coordinates such that two consec-  utive elements define a log-canonical system of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A log-anti-canonical system of coordinates on CV (θ) is a birational symplectic morphism  from (CV (θ), ωV (θ)) to (C2, −ωlog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='24  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If (x, y) is a log-anti-canonical system of coordinates, (y, x) and (x−1, y)  are log-canonical systems of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If (x, y) is a log-canonical system of coordinates, for any λ, µ in C∗, (λx, µy) is a log-  canonical system of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The cubic surface CV (θ) is symplectic birationally equivalent to (C2, ωlog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Indeed we have two  first examples of log-canonical systems of coordinates on CV (θ):  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='25 Let y1 = x1, y2 = x2 and z1 = FV,x3 = x1x2 − e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The pairs (y1, z1), and  (z1, y2) are log-canonical systems of coordinates on CV (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A map z1 such that both (y1, z1), and  (z1, y2) are log-canonical systems of coordinates is unique up to a multiplicative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have:  dy1  y1  ∧ dz1  z1  = dx1  x1  ∧ x1dx2 + x2dx1  z1  = dx1 ∧ dx2  x1x2 − e0  = ωV (θ),  and we have a similar computation for (z2, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' All the maps such that (y1, z) is a log-canonical  system of coordinates write z = c(y1)z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore the only maps such that (y1, z), and (z, y2)  are log-canonical systems of coordinates are z = cz1 for some c in C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Therefore (y1, z1, y2) is a log-canonical triple, which satisfies: z1 = y1y2−e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In order to construct  new log-canonical or log-anti-canonical systems of coordinates we consider the two maps:  σ1(x1, x2, x3) = (x1 − FV,x1, x2, x3) = (−x1 − x2x3 + θ1, x2, x3)  σ2(x1, x2, x3) = (x1, x2 − FV,x2, x3) = (x1, −x2 − x1x3 + θ2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='26 σ1 and σ2 define polynomial involutive automorphims of CV (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Furthermore they  are anti-symplectic automorphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' σ∗  i ωV (θ) = −ωV (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  29  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The involutive property is a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have FV ◦σi = FV , and therefore  the σi’s are polynomial automorphims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Finally, by using ωV (θ) =  dx2∧dx3  x2x3+2x1−θ1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ωV (θ) =  dx3∧dx1  x3x1+2x2−θ2), we obtain σ∗  1ωV (θ) = −ωV (θ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' σ∗  2ωV (θ) = −ωV (θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  We set:  g = σ1 ◦ σ2,  g−1 = σ2 ◦ σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  From Lemma (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='26), g is a polynomial symplectic automorphisms of (CV (θ), ωV (θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore,  starting from the log-canonical triple (y1, z1, y2), we obtain two other log-canonical triples by  applying σ∗  i , i = 1, 2 and reversing the order of the triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We set:  (y2, z2, y3) := (σ∗  1y2, σ∗  1z1, σ∗  1y1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (y0, z0, y1) := (σ∗  2y2, σ∗  2z1, σ∗  2y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore we obtain a log-canonical sequence of length seven:  H : (y0, z0, y1, z1, y2, z2, y3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We call it the fundamental log-canonical heptuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Note that we have:  (y2, z2, y3) = g−1∗(y0, z0, y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We extend the fundamental log-canonical sequence to an infinite log-canonical sequence by  setting for any k in Z:  (y2k, z2k, y2k+1, z2k+1, y2k+2, z2k+3, y2k+3) := (g−k∗)(y0, z0, y1, z1, y2, z2, y3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This sequence satisfies the following relations:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='27 [exchange relations] let AV = Z[e±  0 , e±  3 , e±  4 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let P, Q1 and Q2 in AV [t]  defined by  P(t) = t + e0,  Q1(t) = (t − e0e−1  4 )(t − e0e4)(t − e−1  3 )(t − e3),  Q2(t) = (t − e0e−1  3 )(t − e0e3)(t − e−1  4 )(t − e4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  For all k in Z we have  ykyk+1 = P(zk)  z2kz2k+1 = Q1(y2k+1)  z2k+1z2k+2 = Q2(y2k+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='28 We have :  σ∗  1x1 = x−1  2  �  e0 + z−1  1 Q2(x2)  �  ,  and σ∗  2x2 = x−1  1  �  e0 + z−1  1 Q1(x1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (8)  with :  Q2(t) = t4 − θ2t3 + θ4t2 − e0θ1t + e2  0 = (x − e0e−1  3 )(x − e0e3)(x − e−1  4 )(x − e4),  (9)  Q1(t) = t4 − θ1t3 + θ4t2 − e0θ2t + e2  0 = (x − e0e−1  4 )(x − e0e4)(x − e−1  3 )(x − e3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (10)  30  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have :  x2σ∗  1x1 = −x1x2 − x2  2x3 + θ1x2  = −e0 − z1 + x2  2z−1  1 (x2  1 + x2  2 − θ1x1 − θ2x2 + θ4) + θ1x2  = −e0 − z1 + z−1  1  �  (e0 + z1)2 + x4  2 − θ1x2(e0 + z1) − θ2x3  2 + θ4x2  2  �  + θ1x2  = e0 + z−1  1  �  e2  0 + x4  2 − θ1x2 − θ2x3  2 + θ4x2  2  �  = e0 + z−1  1 Q2(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Hence σ∗  1x1 = e0x−1  2  + x−1  2 z−1  1 Q2(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The proof of the other equality is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If we develop the right term of (9), we get :  t4 − (a4 + e0a3)t3 + (e0a3a4 + e2  0 + 1)t2 − e0(a3 + e0a4)t + e2  0  = t4 − θ2t3 + θ4t2 − e0θ1t + e2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  If we develop the right term of (10), we get :  t4 − (a3 + e0a4)t3 + (e0a3a4 + e2  0 + 1)t2 − e0(a4 + e0a3)t + e2  0  = t4 − θ1t3 + θ4t2 − e0θ2t + e2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Proof of Proposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' From lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='28, we get : z2 = x2σ∗  1x1 − e0 = z−1  1 Q2(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Hence z1z2 = Q2(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Similarly : z0 = σ∗  2x2 − e0 = z−1  1 Q1(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Hence z0z1 = Q1(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The  others relations are obtained by applying g−k∗ to the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have Q1 = e−2  0 t4Q2(e0t−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' According to Remark (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='10), z2k = 0 are the equations of the lines Dr  , Dl  , or their images  by the dynamic < g >, and z2k+1 = 0 are the equations of the lines ∆· or their images by  the dynamic < g >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='5  The Laurent property  The exchange relations obtained at Proposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='27) suggests that the sequence of log-canonical  systems of coordinates is related to a structure of generalized cluster algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Without going  into the theory of cluster algebras (for details see [38]), we just recall here the fact that under  some hypothesis they satisfy the “Laurent property” (see also [15]):  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A rational map r ∈ C(x, y) satifies the Laurent property if its polar set  is included in xy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A birational map r satifies the Laurent property if both r and r−1 have the Laurent property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let X be an affine surface, and let (yn, zn) be a sequence of algebraic morphisms from X  to C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This sequence satisfies the Laurent property, if given an element (yn, zn), any other  regular function ym (or zm) = r(xn, yn) satisfies the Laurent property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We must remark that the Laurent property is not stable by composition or inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It  turns out that in a cluster sequence some simplications arise from the exchange relations and  give this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' A direct proof of this fact needs heavy computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We can prove here the  Laurent property for the canonical sequence by an argument which avoid these computations  with the exchange relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  31  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='31 The log-canonical sequence satisfies the Laurent property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We first prove that all the log-canonical variables are polynomials in (x1, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This  is true for (y1, z1, y2) = (x1, x1x2 − e0, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Recall that σ1, σ2 (and therefore g) are polynomials  in (x1, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' All the other canonical variables are obtained by the action of a word in σ∗  1, σ∗  2  from a variable in this triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore they are still polynomials in (x1, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We set:  pn : CV (θ) → C2, u = yn(x), v = zn(x),  qn : CV (θ) → C2, u = zn(x), v = yn+1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We claim that pn and qn satisfy the Laurent property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since pn is polynomial, we only have to  prove that the polar set of p−1  n  is included in uv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This is true for p1 = (y1, z1):  x1 = y1  x2 = (z1 + e0)y−1  1  x3 = z−1  1 (−y2  1 − (z1 + e0)y−2  1  + θ1y1 + θ2(z1 + e0)y−1  1  − θ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This is also true for q1 = (z1, y2) by using a similar computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let  ι : (u, v) → (v, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have :  q0 = ι ◦ p1 ◦ σ2, p2 = ι ◦ q1 ◦ σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore, q0 and p2 also satisfy the Laurent property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since the property is true for p1 and  p2, and since the tame dynamic g is a polynomial automorphism, it remains true for any pn =  g−1∗pn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since the property is true for q0 and q1 it remains true for any qn = g−1∗qn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Finally,  since pn satisfies the Laurent property and pm is polynomial, the map  pm ◦ p−1  n  : (ym, zm) = (r(yn, zn), s(ym, zm))  satisfies the Laurent property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='✷  From the above proof, one can remark that the Laurent property comes from the fact that the  antisymplectic tame dynamics (σ1 and σ2) and the symplectic dynamics (g) are automorphisms  of the cubic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='6  Families of confluent and diffluent morphisms  These families were first discovered by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Klimes in [40] through an analytic confluent process,  both on Painlev´e equations and on character varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We recover them by using a groupoid  point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='10) we have introduced a family of morphisms ϕκ : πV I  1 (X, S) →  πV,κ  1  (X, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore we obtain a family  ϕ∗  κ : χκ  V (a) → χV I(aκ)  ρ → ρ ◦ ϕκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  On the restriction over χ±  V we have 2 families ϕ±∗  κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='32 The morphisms ϕ±∗  κ  are invertible on the open set in χV I(aκ) defined by the  elements (M1, M2, M3, M4) such that (M1, M2) is a non reducible pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  32  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The matrix expression of ϕ+∗  κ  is  [U1, M0, U2, M3, M4]D → [U1Dκ, Dκ−1M0U2, M3, M4)]SL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Given (M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' M4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' and a parameter κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' we search for matrices M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' U1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' U2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' M′  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' M′  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (U1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' U2) in U − × D × U + and a matrix Q in SL2(C) such that  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  U1Dκ = Q−1M1Q  D−1  κ M0U2 = Q−1M2Q  M′  3 = Q−1M3Q  M′  4 = Q−1M4Q  (11)  For a given local data (a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' a3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' a4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' and a value κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' we have only two choices for the eigenvalues  (e0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' e−1  0 ) of M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' solutions of κ−1e0 + κe−1  0  = a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' one centered around a2κ and the other around  a2κ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' induced by the change κ → κ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let e±  1 1, e±  2 be the eigenvalues of M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The two  solutions for M0 are: M0 = diag(e1e2, e−1  1 e−1  2 ), or M0 = diag(e−1  1 e−1  2 , e1e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We choose the first  one (the second will give a pre-image for ϕ−∗  κ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We recall the LDU decomposition in SL2(C):  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='33 Let M be an element of SL2(C) such that M[1, 1] ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' There exists a unique  triple (L, D, U) in U − × D × U + such that M = L × D × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have  \uf8eb  \uf8ec  \uf8ed  a  b  c  d  \uf8f6  \uf8f7  \uf8f8 =  \uf8eb  \uf8ec  \uf8ed  1  0  l  1  \uf8f6  \uf8f7  \uf8f8 ×  \uf8eb  \uf8ec  \uf8ed  e  0  0  e−1  \uf8f6  \uf8f7  \uf8f8 ×  \uf8eb  \uf8ec  \uf8ed  1  u  0  1  \uf8f6  \uf8f7  \uf8f8 ⇔  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  a = e  b = eu  c = el  d = e−1 + elu  which system has the unique solution (l, e, u) = (c/a, a, b/a) if a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  We call the above decomposition the LDU-decomposition of M and we denote:  LDU(M) = (L, D, U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='34 Let M1 and M2 be two non vanishing matrices in SL2(C), M1 ̸= ±I, M2 ̸= ±I,  with eigenvalues (e1, e−1  1 ) and (e2, e−1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Suppose that the eigenvectors related to e−1  2  and to e1  are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' There exists a matrix Q in SL2(C) such that the diagonal component of the  decomposition LDU of Q−1M1M2Q is diag(e1e2, e−1  1 e−1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The hypothesis of the Lemma gives the existence of a ”mixed basis” (u, v) given by  an eigenvector u of M2 related to the eigenvalue e−1  2 , and an eigenvector v of M1 related to the  eigenvalue e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let Q be the matrix of this change of basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have:  Q−1M1Q =  \uf8eb  \uf8ec  \uf8ed  e1  0  c1  e−1  1  \uf8f6  \uf8f7  \uf8f8 , Q−1M2Q =  \uf8eb  \uf8ec  \uf8ed  e2  b2  0  e−1  2  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  33  Therefore  Q−1M1M2Q =  \uf8eb  \uf8ec  \uf8ed  e1e2  e1b2  e2c1  e−1  1 e−1  2  + b2c1  \uf8f6  \uf8f7  \uf8f8 =  \uf8eb  \uf8ec  \uf8ed  1  0  c1  1  \uf8f6  \uf8f7  \uf8f8 ·  \uf8eb  \uf8ec  \uf8ed  e1e2  0  0  e−1  1 e−1  2  \uf8f6  \uf8f7  \uf8f8 ·  \uf8eb  \uf8ec  \uf8ed  1  b2  0  1  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  ✷  End of the proof of Proposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Suppose that the pair of matrices (M1,κ, M2,κ) satisfies  the hypothesis of Lemma (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We choose the matrix Q given by this Lemma and we obtain  a solution of (11):  (U1, M0, U2) = LDU(Q−1M1,κM2,κQ), M′  3 = Q−1M3Q, M′  4 = Q−1M4Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  A change of mixed basis will modify Q by a multiplication on the righta side with a diagonal  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore the class [(U1, M0, U2, M′  3, M′  4)]D is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The representations ρ for which  the hypothesis of Lemma (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='34) is not satisfied are the representations for which the eigen-  directions related to e1,κ and e−1  2,κ are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This defines a component –here the line Le1,e2–  of the reducibility locus along the path γ1,2 in πV I  1 (X, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore Φκ is invertible outside  Le1,e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Note that for the other choices of κ or e0 we shall find one of the other components of  R(γ1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Using the trace maps, ϕ±∗  κ  induces two families of morphisms of cubic surfaces:  Φ±  κ = (Tr±κ  V )−1 ◦ ϕ±∗  κ  TrV I : CV (θ±) → CV I(κ(θ±))  where κ(θ+) = κ(e0, a3, a4) = (κ + κ−1, e0κ−1 + e−1  0 κ, a3, a4) and κ(a−) = κ(e−1  0 , a3, a4) =  (κ + κ−1, e−1  0 κ−1 + e0κ, a3, a4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='35 (Confluent and diffluent morphisms) The morphisms Φ±  κ are symplectic  birational morphisms from CV (θ±) to CV I(κ(θ±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have to prove that Φ+  κ has rational expression in trace coordinates and is invertible  in the class of birational transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='36 The map Φ+  κ : CV (θ+) to CV I(κ(θ+) is defined by  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  x1,κ = e−1  0 κx1 + κ−1x2  x2,κ = −e−1  0 κx1x3 + κ−1x1 − e−1  0 κx2 + a3κ + a4e−1  0 κ  x3,κ = x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We set M1,κ = U1Dκ, M2,κ = Dκ−1M0U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  x1,κ = tr(M2,κM3) = α3e0κ−1 + e0γ3s2κ−1 + δ3e−1  0 κ  x2,κ = tr(M3M1,κ) = α3κ + β3s1κ + δ3κ−1  x3,κ = tr(M1,κM2,κ) = e0s1s2 + e0 + e−1  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Using Lemma (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='8), we obtain the expressions of Φκ in trace coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='37 This map is invertible outside the lines Le1,κ,e2,κ ∪ Le−1  1,κ,e−1  2,κ in χV I(aκ) and Φ−1  κ  is given by  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  x1 = (−κx1,κ − e0κ−1x2,κ + a3e0 + a4)(x3,κ − ce1,κ,e2,κ)−1  x2 = (κx1,κx3,κ − e0κ−1x1,κ + κx2,κ − a3κ2 − a4κ2e−1  0 )(x3,κ − ce1,κ,e2,κ)−1  x3 = x3,κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  34  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In order to find Φ−1  κ  we solve the system  �  e−1  0 κx1 + κ−1x2 = x1,κ  (−e−1  0 κx3,κ) + κ−1)x1 − e−1  0 κx2 = x2,κ − a3κ − a4e−1  0 κ  For a fixed value of x3 = x3,κ this system is linear and invertible outside the set  e−1  0 (x3,κ − e−1  0 κ2 − e0κ−2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This set is the pair of lines x3,κ = e1,κe−1  2,κ + e1,κe−1  2,κ = ce1,κ,e2,κ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' the pair of lines Le1,κ,e2,κ ∪  Le−1  1,κ,e−1  2,κ in χV I(aκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Solving this system, we obtain the expression of Φ−1  κ  given in the statement  of the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='✷  Finally one can prove by using the above expressions that Φ+  κ is a symplectic morphism (see  also [40]) with respects to the symplectcic forms ωκ  V (θ+) and ωV I(θκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have a similar result  for Φ−  κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  4  The Painlev´e V vector field on MV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1  The Riemann-Hilbert map RHV  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Any local connection ∇, defined by a germ of irregular sl2-system of  Katz rank 1 induces a representation ρ∇ of the local wild groupoid π1(X0, S0) in SL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Any connection A in CV induces a representation ρA of πV  1 (X, S) in SL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The representation of the local wild fundamental groupoid induced by a local irregular  system is defined in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We first consider an extension of the set of base points to  any point of D or �D which is an extremity of the rays ri, r±  i , i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The points p1 and p2 are  defined by the singular directions of the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Any point of �D is arbitrarily represented  by a determination in the direction d of a formal fundamental system of solutions �Yd of the  connection and any point of D is represented by an actual holomorphic fundamental system  of solutions Yd on a small sector around d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Any arc γ on D with origin a and end point b is  represented, as usually by the comparison between the analytic continuation �  Ya  γ of Ya along γ  with of Yb, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' by the matrix Mγ such that  Yb = �  Ya  γ · Mγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Any arc on �D is represented in the same way by using the formal representations of its extrem-  ities, and the analytic continuation of the logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let rd be a regular ray whose origin and  end-point are represented by �  Yd and Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The morphism rd is represented by the comparison  between the summation of �  Yd (the summation replace here he analytic continuation) with Yd :  Yd = Sd(�  Yd) · Mrd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We represent the loops ti,i(κ) in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' One can suppose that the matrix �Ysi is a  diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Then the action of �ti,i(κ) on �Ysi is given by the product with the diagonal  matrix diag(κ−1, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='Therefore ρA (ti,i(κ)) = Dκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have defined the representation ρ∇ on all  the generators of π1(X0, S0) and therefore on π1(X0, S0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Given a global system A in SV , we complete the previous local wild representation to a  representation of πV  1 (X, S) by defining ρ∇(γi,j), (i, j) = (2, 3), (3, 4), (4, 1), (3, 3), (4, 4) in  35  the usual monodromic way, by analytic continuation along these paths or loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  A change  of representations of the objects, or a gauge action on the linear system, induce equivalent  representations, which end the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  We have defined a Riemann-Hilbert map  RHV : MV (α) → χV (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The general Riemann-Hilbert problem discusses about the surjectivity of RH: in the present  context, for any irreducible representation ρ in χV , there exists a unique connection in MV on  the trivial bundle whose linear representation is ρ: for details see [54], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='2  Isomonodromic families on MV  Any isomonodromic family on MV (α) is a fiber of the map RHV , parametrized by a variable t,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' a family of connections defined by  dY  dx = A(x, t) · Y  such that the monodromy and the Stokes operators are locally constant in χV (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='2 There exists an open Zariski set in MV (α) and coordinates p, q on the fiber  MV (α) such that the isomonodromic families are solutions the Painlev´e V hamiltonian differ-  ential system (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  A sketch of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We follow here an argument of [54] wich extends the argument of  Schlesinger for fuchsian connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Given a fundamental system of solutions Y (x, t), by isomon-  odromy, d  dtY (x, t) and Y (x, t) have the same behaviour by analytic continuation or under a Stokes  operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore the quotient  B(x, t) = d  dtY (x, t) · Y (x, t)−1  is univalued outside the fixed singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3 B(x, t) extends to the singular set in a meromorphic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Let U ⊂ C be an open subset, and let D be an open disc centered at ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Let  dY/dx = A(x−1, t)Y , where A ∈ sl2(O(D × U), be a parametrized meromorphic differential  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We suppose that, for all t ∈ U, the eigenvalues of A(0, t) are distinct and that the  singularity is irregular, with a Katz rank equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' From Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='7), the system dY/dx =  AY admits, in a small neighborhood of each t0 ∈ U a formal fundamental solution analytic in  t : �Y = �ΦxLeQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The polynomials Q and the matrix L are analytic in t and the entries of �Φ  belongs to O(V )[[x−1]], where V is a small open disc centered at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Moreover these coefficients  are 1-summable uniformly in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For a direction d non-singular at t0, if |t − t0| is small, then d  remains non-singular and the 1-sum is analytic in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Multiplying on the right by an invertible  diagonal matrix analytic in t, we get a fundamental solution with similar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  If the system is isomonodromic, then the matrix L is constant and it is possible to choose �Y  in such a way that the Stokes multipliers does not depend on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  By a simple calculation, we see that �B(x, t) =  d  dt �Y (x, t) · �Y (x, t)−1 does not contain expo-  nentials, and is invariant by the formal monodromy and by the (constant) Stokes multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore its entries belongs to O(D)((x−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' These entries are 1-summable uniformly in t and  36  are invariant by the Stokes multipliers, therefore they are meromorphic in x (with a pole at ∞)  and analytic in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Therefore Y (x, t) satisfies two rational linear systems dY  dx = A(x, t) · Y and dY  dt = B(x, t) · Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The compatibility condition requires that the two operators d/dx − A and d/dt − B have to  commute :  dA  dt − dB  dz + [B, A] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (12)  The pair (A, B) is called an isomonodromic Lax pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' If A is irreducible, B is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In [54], the  computation of B and the equivalence of the compatibility condition (12) with the hamiltonian  system of PV have been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3  Singularities of the Painlev´e V vector field in the Okamoto’s compactifi-  cation of MV (α)  The Okamoto’s compactification [49], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' All the Painlev´e foliations satisfy the Painlev´e  property: any solution admits an extension along any path is the basis given by the time variable  punctured by the fixed singular points, as a meromorphic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In a first step, we search for a compactification on which appear the “polar” singular set,  which corresponds to the poles of the meromorphic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Here, following H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Chiba [16] and  [17], we have to distinguish the equations PIV , PII and PI from PV I, PV and PIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For the first  triple we obtain this compactification by a convenient weighted projective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For the second  one, the natural weighted projective space presents a vanishing weight and we can’t catch all  the polar set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In this case we need to glue two copies of this weighted projective space by a  convenient B¨acklund transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In a second step one can remove the polar singular set by a convenient sequence of blowing  up’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In the version of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Chiba, it suffices to perform only one weighted blowing up whose  weights are given by the eigenvalues of the polar singular point, which are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The first step creates new singular points outside the polar singular set, over z = ∞, which  are saddle-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We describe here this set for the PV foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In this case, the weights are  ω = (1, 0, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Since the second weight vanishes, this space P3  (1,0,1,1) is not compact and is  isomorphic to P2 × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This space is here a manifold (an orbifold for other Painlev´e equations:  [Chiba1-2-4]), covered by 3 charts:  U1 = {(X1 : X2 : X3 : X4), X1 ̸= 0}, (X1 : X2 : X3 : X4) = (1 : u1 : u2 : x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  U3 = {(X1 : X2 : X3 : X4), X3 ̸= 0}, (X1 : X2 : X3 : X4) = (v1 : v2 : 1 : y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  U4 = {(X1 : X2 : X3 : X4), X4 ̸= 0}, (X1 : X2 : X3 : X4) = (w1 : w2 : z : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The change of charts are given by  w1 = v1y−1 = x−1, w2 = v2 = u1, z = y−1 = u2x−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The chart U4 is the ”initial chart”, before compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In this chart  the vector field corresponding to the hamiltonian zHV is defined by  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  z ˙w1 = −2w2  1w2 + w2  1 + w1z − 2w1w2z + (α1 + α3)w1 − α2z  z ˙w2 = 2w1w2  2 − 2w1w2 − w2z + w2  2z − (α1 + α3)w2 + α1  ˙z = z  37  For each vector field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' the plane z = 0 is invariant and the singular points of the restriction of  the vector field to this plane are given by  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  (−2w1w2 + w1 + (α1 + α3))w1 = 0  2w1w2  2 − 2w1w2 − (α1 + α3)w2 + α1 = 0  z = 0  For α1 ̸= ±α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' we obtain 2 singular points r1 and r2:  (w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' z) =  �  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  α1  α1 + α3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' z) =  �  α1 − α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  α1  α1 − α3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Now we search for singular points over z = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In the first chart the Painlev´e vector field is (see  [17]):  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ˙u1 = −2u1 + 2u2  1 − u1u2 + u2  1u2 + α1x − (α1 + α3)u1x  ˙u2 = −u2 + 2u1u2 − u2  2 + u2x + 2u1u2  2 − (α1 + α3)u2x + α2u2  2x  ˙x = x(−1 + 2u1 − u2 + 2u1u2 − (α1 + α3)x + α2u2x)  The plane (x = 0) is invariant and PV |(x=0) is the autonomous hamiltonian vector field  �  ˙u1 = −2u1 + 2u2  1 − u1u2 + u2  1u2 = u1(u1 − 1)(u2 + 2)  ˙u2 = −u2 + 2u1u2 − u2  2 + 2u1u2  2 = u2(2u1 − 1)(u2 + 1)  We have 5 singularities p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' s3 in this first chart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' which do not depend on the parameters  αi:  (u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' x) = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' −2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The singularities p1 and p2 are polar singular points: H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Chiba has proved in [17] that they  correspond to solutions given by Laurent series at a movable pole z = z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' One can remove these  singularities by a weighted blowing up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The eigenvalues of linear part of PV are (−2, −1, −1)  and (2, 1, 1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' They are analytically linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In the third chart –the Boutroux chart–,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' the Painlev´e vector field is given by  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ˙v1 = v1(1 + v1 − 2v2 − 2v1v2 + (α1 + α3 − 1)y) − α2y  ˙v2 = v2(−1 + v2 − 2v1 + 2v1v2 − (α1 + α3)y) + α1y  ˙y = −y2  The plane y = 0 is invariant and the restriction of the vector field at infinity is  �  ˙v1 = v1(1 + v1 − 2v2 − 2v1v2) = v1(v1 + 1)(1 − 2v2)  ˙v2 = v2(−1 + v2 − 2v1 + 2v1v2) = v2(v2 − 1)(2v1 + 1)  There are five singular points s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' s4 and s5 in y = 0 given by  (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' y) = (−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (−1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  38  The polar singular set of PV contains 4 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In order to catch the two missing polar singular  points, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Chiba introduces a second copy of �  P3(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1) glued to the first one by the B¨acklund  transformation π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' given in each chart by  π :( ˜w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜α0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜α3) = (z(w2 − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' −w1z−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' α0)  (˜v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜α0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜α3) = (v2 − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' −v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' α0)  (˜u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜α0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜α3) = (−u−1  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (u1 − 1)−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (u1 − 1)−1u−1  2 x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' α0)  By computing the singularities of the vector field in the chart ( ˜w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' we find two singular  points r3 and r4 given by:  ( ˜w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜z) = (˜α1 − ˜α3 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  ˜α1  ˜α1 − ˜α3 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ( ˜w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜w2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ˜z) = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' −  ˜α1  ˜α0 + ˜α2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have r4 = r1, and therefore we obtain 3 singular points of regular type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In the chart  (˜u1, ˜u2, ˜x), we find five singular points:  p3 : (˜u1, ˜u2, ˜x) = (0, 0, 0),  p4 : (˜u1, ˜u2, ˜x) = (1, 0, 0),  s1 : (˜u1, ˜u2, ˜x) = (1, −1, 0),  s2 : (˜u1, ˜u2, ˜x) = (1  2, −2, 0),  s4 : (˜u1, ˜u2, ˜x) = (0, −1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  According to [17], the singular points p3 and p4 are the two missing polar singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The  singularities s1, s2 and s4 was yet detected in the first copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The singular points in the Boutroux  chart (˜v1, ˜v2, y) = (v2 −1, −v1, y) are s1, s2, s3, s4 and s5 without new singular point (the change  of chart is polynomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have obtained:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4 The Painlev´e vector field admits 3 singular points r1, r2, r3 of regular type  over z = 0, 4 polar singular points pi, i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4 over z = ∞, which can be removed by a weighted  blowing-up, and 5 singular points si over z = ∞, of saddle-node type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  By a local study around each saddle-node singularity, one can recover the 5 solutions of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Parusnikova in Laurent series around z = ∞ for the Painlev´e V equation: [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  5  Dynamics on χV  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1  The tame dynamics  As mentionned in the introduction, the dynamic of the Painlev´e VI equation on χV I is induced  by the automorphisms of the groupoid, given by braids over 3 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Here since the singular  set reduces to 3 points, the braid group over two points has only one generator, which can  be represented by the pure braid b over s3 and s4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Furthermore one can consider the “half-  monodromy” defined the (non pure) braid b3,4 which permutes the positions of s3 and s4 and  induces an involution on the local parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' These actions induced by the trace coordinates  on CV (θ+) has been computed by the authors in [53] and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Klimes in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1 The action of the braid b3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4 : CV (θ+  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' θ+  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' e0) → CV (e−1  0 θ+  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' e−1  0 θ1+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' e−1  0 ) is  defined by:  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  x′  1 = e−1  0 (−x2 − x1x3 + θ+  2 )  x′  2 = e−1  0 x1  x′  3 = x3  and  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  e′  0 = e−1  0  θ+  1  ′ = e−1  0 θ+  2  θ+  2  ′ = e−1  0 θ+  1  39  The action of the pure braid b3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4 ◦ b3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4 : CV (θ+) → CV (θ+) is defined by:  g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4 :  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  x′  1 = x1x2  3 + x2x3 − x1 − θ+  2 x3 + θ+  1  x′  2 = −x1x3 − x2 + θ+  2  x′  3 = x3  As for the dynamics on CV I(θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' this dynamics and its inverse are polynomial dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This is the only part of the dynamics on χV I which do not degenerate through the confluent  morphisms from χV I to χV (see next paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='2 The dynamics generated by g3,4 is called the tame dynamics, and denoted by  Tame(CV ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='2  The confluent dynamic  Using the birational map Φκ = Φ+  κ , each dynamic hi,j on CV I(θκ) defined by the braids induces  a family of birational dynamics gi,j(κ) = φ−1  κ  hi,j ◦ φκ on χ+  V (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This one was previously  obtained by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Klimes by using an analytic confluent process in the spaces of connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3 The confluent dynamic Conf(PV ) is the dynamic generated by g1,2(κ), g2,3(κ)  and g3,1(κ) for κ is C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We still have the relation g1,2(κ) ◦ g2,3(κ) ◦ g3,1(κ) = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore it suffices to compute g1,2(κ)  and g2,3(κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Notice that, from the braid group relations, g1,2(κ) = g3,4(κ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  By a direct  computation using proposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='36), it can be checked that  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4 The dynamic g3,4(κ) = φ−1  κ  h3,4 ◦ φκ on χV (a) do not depend on κ and  generates the tame dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='5 Since the braid between s1 and s2 (or between s3 and s4) corresponds to a loop  around 0 in the time space, g3,4 is conjugated through the Riemann-Hilbert correspondence to  the non linear monodromy of the foliation FV over z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Contrary to the previous case, the dynamics g2,3(κ) and g3,1(κ) are not generated by an  automorphism of πV  1 (Y, S): indeed on the groupoid level, ϕκ is not an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Nevertheless  they induce families of birational dynamics on CV (θ):  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='6 The family of dynamics g2,3(κ) is defined by:  X1 = e0  x2  X2 = x2 − κ2  x2  + e−1  0 κ2x1  X3 = κ−2x2  2x3 − (e−2  0 κ2 − κ−2)x1x2 − 2e−1  0 x2  2 + (e−1  0 θ2 + κ−2θ1)x2 + (e−1  0 κ2 + e0κ−2)  The second family g3,1(κ) is given by g1,3(κ) = g1,2 ◦ g2,3(κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We first search for the matrix expressions of the confluent dynamics :  40  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='7 The confluent dynamic (ϕ∗  κ)−1 ◦ h2,3 ◦ ϕ∗  κ :  χκ  V (a) → χκ  V (a) is defined by  ([U1, M0, U2, M3]D → [V1, M0, V2, N3]D with  V1 =  \uf8eb  \uf8ec  \uf8ed  1  0  κ−1e0s1θ3 + κ−1e−1  0 γ3 − κe−1  0 γ3 + κ−1e0s1s2γ3  1  \uf8f6  \uf8f7  \uf8f8 ,  V2 =  \uf8eb  \uf8ec  \uf8ed  1  κ−1e0s2θ3 + κ−1e0s2  2γ3 − κ−1e0s2δ3 − κ−1e0β3 + κe−1  0 β3 + κe−1  0 s2δ3  0  1  \uf8f6  \uf8f7  \uf8f8 ,  N3 =  1  θ3 + s2γ3  \uf8eb  \uf8ec  \uf8ed  θ2  3 + θ3γ3s2 + β3γ3 + γ3δ3s2  −e0κ−1(θ3s2 + γ3s2  2β3 − δ3s2)  κe−1  0 γ3  1  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We consider an element ρ of χV (a) given by its normalized representation in the  canonical Cartan-Borel decomposition:  ρ = [U1, M0, U2, M3, M4], (U1, M0, U2, M3) ∈ U −×D×U + ×SL2(C), M4 = (U1M0U2M3)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We set:  U1 =  \uf8eb  \uf8ec  \uf8ed  1  0  u1  1  \uf8f6  \uf8f7  \uf8f8 , M0 =  \uf8eb  \uf8ec  \uf8ed  e0  0  0  e−1  0  \uf8f6  \uf8f7  \uf8f8 , U2 =  \uf8eb  \uf8ec  \uf8ed  1  u2  0  1  \uf8f6  \uf8f7  \uf8f8 , M3 =  \uf8eb  \uf8ec  \uf8ed  θ3  β3  γ3  δ3  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The map Φκ is defined by:  Φκ([U1, M0, U2, M3, M4]) = [M1, M2, M3, M4] with M1 = U1Dκ, M2 = D−1  κ M0U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore  M1 =  \uf8eb  \uf8ec  \uf8ed  κ  0  κu1  κ−1  \uf8f6  \uf8f7  \uf8f8 , M2 =  \uf8eb  \uf8ec  \uf8ed  κ−1e0  κ−1e0u2  0  κe−1  0  \uf8f6  \uf8f7  \uf8f8 , M3 =  \uf8eb  \uf8ec  \uf8ed  θ3  β3  γ3  δ3  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  According to [53], the dynamic of h2,3 is given by  M1 → (M2M3)−1M1(M2M3)  M2 → M2  M3 → M3  M4 → (M2M3)−1M4(M2M3)  Since each data is given up to a common conjugacy, we can also use:  M1 → M′  1 = M2−1M1M2  M2 → M′  2 = M3M2M3−1  M3 → M′  3 = M3  M4 → M′  4 = M2−1M4M2  41  Now, in order to compute φ−1  κ (M′  1, M′  2, M′  3, M′  4), we make use of the matrix P given by Lemma  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='34), defined by a mixed basis for the pair (M′  1, M′  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let (u, v) a basis of the representation  of the initial object such that the representation Φκ(ρ) is given by the matrices Mi: u is an  eigenvector of M2 for the eigenvalue e2 = κ−1e0 and v is an eigenvector of M1 for the eigenvalue  e1−1 = κ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The vectors  u′ = M3 · u′, v′ = M2−1 · v  are eigenvectors for M′  2 (with eigenvalue κ−1e0) and for M′  1 (with eigenvalue κ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The change  of basis is given by the matrix  P =  \uf8eb  \uf8ec  \uf8ed  θ3  −κ−1e0u2  γ3  κ−1e0  \uf8f6  \uf8f7  \uf8f8  which is invertible outside  κ−1e0(b3 + u2γ3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Using Lemma (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='8), this locus is given by x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore the dynamic g2,3(κ) is a rational  dynamic with polar set x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have:  P −1(M′  1M′  2)P =  \uf8eb  \uf8ec  \uf8ed  e0  κ−1e2  0u2b3 + κ−1e2  0u2  2γ3 − κ−1e2  0u2δ3 − κ−1e2  0β3 + κβ3 + κu2δ3  κ−1e2  0u1b3 + κ−1γ3 − κγ3 + κ−1e2  0u1u2γ3  ⋆  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  where the coefficient ⋆ is not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The LDU-decomposition of this matrix is given by  (V1, M0, V2) with  V1 =  \uf8eb  \uf8ec  \uf8ed  1  0  κ−1e0u1b3 + κ−1e−1  0 γ3 − κe−1  0 γ3 + κ−1e0u1u2γ3  1  \uf8f6  \uf8f7  \uf8f8 ,  V2 =  \uf8eb  \uf8ec  \uf8ed  1  κ−1e0u2b3 + κ−1e0u2  2γ3 − κ−1e0u2δ3 − κ−1e0β3 + κe−1  0 β3 + κe−1  0 u2δ3  0  1  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  we have:  N3 = P −1M′  3P =  1  b3 + u2γ3  \uf8eb  \uf8ec  \uf8ed  b2  3 + b3γ3u2 + β3γ3 + γ3δ3u2  −e0κ−1(b3u2 + γ3u2  2β3 − δ3u2)  κe−1  0 γ3  1  \uf8f6  \uf8f7  \uf8f8  which proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Proof of Proposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The image of x1 = δ3 is given by X1 = N3[2, 2] =  1  b3+u2γ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' According  to Lemma (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='8), we have:  N3[2, 2] =  1  b3 + u2γ3  =  1  (a3 − x1) + e−1  0 (x2 − e0a3 + e0x1) = e0  x2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  42  The image of x2 = δ4 is given by N4[2, 2] where N4 = P −1M′  4P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have  N4[2, 2] = −e−2  0 κ2γ3β4 + b3δ4 + u2γ3δ4  b3 + u2γ3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  From M4 = (U1M0U2M3)−1 we obtain that β4 = −e0(β3 + u2δ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore,  γ3β4 = −e0γ3β3 − e0u2γ3δ3  = −e0(b3δ3 − 1) − e0u2γ3δ3  = −e0((a3 − x1)x1 − 1) − x1(x2 + e0x1 − e0a3)  = e0 − x1x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  From Lemma (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='8) and the above equality we obtain  X2 = N4[2, 2] = x2 − κ2x−1  2  + e−1  0 κ2x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We can obtain the last component by computing X3 = tr(M′′  1 M′′  2 ) = tr(M′  2  −1M′  1M′  2M′  3M′  2M′  3  −1),  or by using the equation of the cubic surface:  X3 = −X2  1 − X2  2 + θ1X1 + θ2X2 − θ4  X1X2 − e0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  ✷  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3  The canonical dynamics on CV (θ)  In [40], Martin Klimes has computed the dynamics obtained on CV (θ) from the wild dynamics  (non linear stokes operators and non linear exponential tori of one irregular singularity of the  Painlev´e foliation) through the Riemann-Hilbert morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It turns out that this dynamics  is a rational one on CV (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We enhance here that this dynamics was already present on the  cubic surface in a canonical way, using the log-canonical coordinates introduced in subsection  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This section is completely independent of the other ones excepted 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4, and only deals with  dynamics on a singular cubic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Nevertheless the terminology (canonical Stokes operators  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=') is deeply influenced by the confluent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We do not need here any restriction on  the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Let y be a log-canonical function on CV (θ): there exists z such that (y, z) –or (z, y)– is a  log-canonical system of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We consider the logarithmic hamiltonian function Hy on  CV (a) such that  dHy = dy  y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Let Xy be the hamiltonian vector field related to Hy for the symplectic form ωV :  ωV (Xy, ·) = −dy  y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Through the symplectic morphism (y, z), the image of this vector field is z ∂  ∂z (or −z ∂  ∂z if y is  completed in log-canonical system of coordinates by the left side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore its flow z(t) = z0e±t  is globally defined on C, and it can be factorized through a multiplicative action of C∗ by setting  λ = et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let Ty be this multiplicative family of (rational symplectic) automorphisms on CV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  43  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='8 Ty = {tλ :  (z′, y) �→ (λ−1z′, y)} = {tλ :  (y, z) �→ (y, λz)} is the exponential  torus related to the log canonical function y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Each element of Ty keeps invariant y, and the set of rational invariant  functions of Ty is C(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Consider the log-canonical pentuple (z0, y1, z1, y2, z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The elements of Ty1 keep invariant  the reducible locus z0 = 0 and z1 = 0, and the elements of Ty2 keep invariant the reducible  locus z1 = 0 and z2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The subgroups of Symp = Symp(C2, ωlog): Bi, B♮  i, Ui, i = 1, 2 are defined in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  They induce subgroups of Symp(CV , ωV ) by using a system of log-canonical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Given  a log canonical function z = zk in S, one can complete z into two log-canonical systems, either  by the left side: (y, z) or by the right side: (z, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We fix a canonical triple (y, z, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The pull-back of B2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  B♮  2, U2)  by (y, z) and the pull-back of B1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' B♮  1, U1) by (z, y′) define the same subgroup of  Sym(CV , ωV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We fix a canonical triple (z, y, z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The pull-back of B2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' B♮  2, U2) by (z, y) and the  pull-back of B1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' B♮  1, U1) by (y, z′) define the same subgroup of Sym(CV , ωV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The subgroups obtained at the first point of Lemma (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='10) are denoted by Bz, B♮  z, Uz, and the  subgroups obtained at the second point are denoted by By, B♮  y, Uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='11 Given a canonical system of coordinates (y, z) we will say that By and Bz are  the opposite Borel subgroups associated to (y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof of Lemma (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 1- The elements of the pullback of B2 by (y, z) are given by  (y, z) → (r(z)y, λz), r ∈ C(z)∗, λ ∈ C∗,  and the elements of the pullback of B1 by (z, y′) are given by  (z, y′) → (λz, r(z)y′), r ∈ C(z)∗, λ ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore using yy′ = z + e0, the elements of the first group also write  (z, y′) → (λz, r′(z)y′), with r′(z) =  λz + e0  (z + e0)r(z),  which proves the first point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2- The elements of the pullback of B2 by (z, y) are given by  (z, y) → (r(y)z, λy), r ∈ C(y)∗, λ ∈ C∗,  and the elements of the pullback of B1 by (y, z′) are given by  (y, z′) → (λy, r(y)z′), r ∈ C(y)∗, λ ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore using zz′ = Q(y), where Q = Q1 or Q2 is a polynomial, the elements of the first group  also write  (y, z′) → (λy, r′(y)z′), with r′(y) =  Q(λy)  Q(y)r(y),  which proves the second point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  44  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='12 We have Z(Ty) = By, N(Yy) = By∪B−  y , where B−  y = {(y, z′) → (λy, r(y)z′−1), r ∈  C(y)∗, λ ∈ C∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let tλ : (y, z′) → (y, λz′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' An element ρ of Symp(CV ), given by  ρ : (y, z′) → (Y (y, z′), Z′(y, z′))  commutes with tλ if and only if, for any λ in C∗,  Y (y, λz′) = Y (y, z′),  Z′(y, λz′) = λZ′(y, z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We claim that any rational function r in K(x) over a field K ⊃ C which satisfies r(λx) = λr(x)  for any λ in C∗ is a constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Indeed if it was not a constant function it would admit  an infinite number of zeroes or poles in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' By applying this remark to z′ �→ Y (y, z′) and to  z′ �→ Z′(y, λz′)z′−1, we conclude that Y do not depend on z′ and that Z′ is linear in z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' There  exists two rational functions q(y) and r(y) such that  Y (y, λz′) = q(y),  Z′(y, λz′) = r(y)z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have  dY  Y  ∧ dZ′  Z′ = q′(y)dy  y ∧ (r′(y)dy  y  + dz  z ) = q′(y)dy  y ∧ dz  z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore q′(y) = 1, and q(y) = µ in C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The elements of Z(Ty) write  (y, z′) → (µy, r(y)z′)  which prove that Z(Ty) = By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Suppose now that an element ρ is in the normalizer of Ty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Then either it commutes whith  each element of Ty, or it anti-commutes with each element of Ty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In this last case a similar  computation proves that ρ : (y, z′) → (µy, r(y)z′−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='13 Let T = (y, z, y′) be a log-canonical triple in the cluster sequence S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' There  exists a unique element sT of Uz such that sTTys−1  T  = Ty′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This operator is given by:  sT : (y, z) → (y(1 + e−1  0 z), z) or by  (z, y′) → (z, y′(1 + e−1  0 z)−1) = (z, e0y−1)  and we have: s−1  T  : (y, z) → (y(1 + e−1  0 z)−1, z) = (e0y′−1, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have:  Ty = {ty(λ) : (y, z) → (y, λz)},  Ty′ = {ty′(λ) : (z, y′) → (λ−1z, y′)}  = {ty′(λ) : (y, z) → (y1 + e−1  0 λ−1z  1 + e−1  0 z  , λ−1z)},  Uz = {ur : (y, z) → (r(z)y, z), r ∈ C(z)∗, r(0) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore the element s of Uz: (y, z) → (y(1 + e−1  0 z), z) satisfies for all λ in C∗  s ◦ ty(λ) ◦ s−1 = ty′(λ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In order to prove the unicity of s, suppose that there exist another s′ in Uz which conjugates Ty  and Ty′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have : s−1 ◦ s′(Ty) = Ty i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' s−1 ◦ s′ belongs to the normalizer N(Ty) of Ty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We  have N(Ty) ∩ Uz = {id} : this is a direct consequence of Proposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='✷  45  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='14 The automorphism sT is the canonical Stokes operator induced by the triple  T = (y, z, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The automorphism sT writes in the coordinates h = log y, l = log(1 +  e−1  0 z): (h, l) → (h + l, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' sT is a pseudo-generator of Uz, that is to say the family {ty(λ) ◦ sT ◦ ty(λ)−1, λ ∈ C∗}  generates Uz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Consequently, B♮  z :=< Uz, Ty >=< sT, Ty >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let sk be the canonical Stokes operator related to the triple (yk, zk, yk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have: σ1 ◦  s1 ◦ σ−1  1  = s−1  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' g ◦ sk ◦ g−1 = sk+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For i = 1, 2, si keeps invariant the lines zi = 0, and the restriction of si on each line is a  translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' More generally, sk keeps invariant zk = 0, and the restriction of sk to the set  of rational curves zk = 0 has no fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This first point is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have:  {ty(λ) ◦ sT ◦ ty(λ)−1, λ ∈ C∗} = {(y, z) → (y(1 + νz), z), ν ∈ C∗}  The statement comes from the fact that any rational function r such that r(0) = 1 writes  r(z) = �  i(1 − µiz) �  j(1 − νjz)−1 where the µ−1  i  are the zeroes of r and the ν−1  j  are the  poles of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have:  s1 : (y1, z1) → (y1(1 + e−1  0 z1, z1)  s2 : (z2, y3) → (z2, y3(1 + e−1  0 z2)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Since σ1 : (y1, z1) → (y3, z2), we have σ1 ◦ s1 ◦ σ−1  1  = s−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The automorphism g−1 send the triple (yk, zk, yk+1) on (yk+2, zk+2, yk+3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore sk =  g−1 ◦ sk+2 ◦ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' From the previous point, it suffices to prove the result for s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We lift the expression of s1  in the coordinate system (y1, z1, x3):  s1 : (y1, z1, x3) → (e−1  0 y1(z1 + e0), z1, X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The equation of the character variety CV (θ) is given by  x3z1 + y2  1 + (z1 + e0)2y−2  1  − θ1y1 − θ2(z1 + e0)y−1  1  + θ4 = 0,  which is equivalent to  x3z1 + y−2  1 z2  1 + 2e0y−2  1 z1 − θ2y−1  1 z1 = −y2  1 − e2  0y−2  1  + θ1y1 + θ2e0y−1  1  − θ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have:  X3z1 + e−2  0 y2  1(z1 + e0)2 + e2  0y−2  1  − θ1e−1  0 y1(z1 + e0) − θ2e0y−1  1  + θ4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  46  Therefore,  X3z1 = −e−2  0 y2  1z2  1 − 2e−1  0 y2  1z1 + θ1e−1  0 y1z1 − y2  1 − e2  0y−2  1  + θ1y1 + θ2e0y−1  1  − θ4  = −e−2  0 y2  1z2  1 − 2e−1  0 y2  1z1 + θ1e−1  0 y1z1 + x3z1 + y−2  1 z2  1 + 2e0y−2  1 z1 − θ2y−1  1 z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We obtain:  X3 = −e−2  0 y2  1z1 − 2e−1  0 y2  1 + θ1e−1  0 y1 + x3 + y−2  1 z1 + 2e0y−2  1  − θ2y−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore the restriction of s1 to z1 = 0 is given by  x3 → x3 + θ1e−1  0 y1 − θ2y−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  On each component of z1 = 0, y1 is constant (= e±  3 or e0e±1  4 ) and this map is a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  ✷  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='16 The canonical dynamics induced by the canonical triple T = (y, z, y′) is the  subgroup Dyn(T) =< Ty, sT, Ty′ > of Symp(CV ) generated by Ty, sT, and Ty′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='17 We have : Dyn(T) =< Ty, sT >= B♮  z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The first equality is a consequence of the relation sTTys−1  T  = Ty′ obtained in Propo-  sition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='13), and the second one was obtained at the second point of Proposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='18 The canonical dynamics Dyn(CV ) induced by the fundamental canonical se-  quence {y0, z0, y1, z1, y2, z2, y3} is the rational symplectic dynamic generated by:  g : (y2, z2) → (y0, z0), Ty1, s1 = s(y1,z1,y2), s2 = s(y2,z2,y3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The subgroup Tame(CV ) of Dyn(CV ) generated by g is the tame canonical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' From Proposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='25), the canonical sequence is almost unique, that is  we can only replace zk with czk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This action is an element of Tyk and therefore do not  change the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This dynamics only depends on the polynomial FV , which justify the  notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Dyn(CV ) also contains Ty0 = g∗Ty2 and s0 = s(y0, z0, y1) = g∗s2, and more generally all  the Tyk and all the sk for k in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Dyn(CV ) =< B♮  z1, B♮  z2, g > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='20 The element �m of Dyn(CV ) such that g = s2 ◦ �m ◦ s1 is defined by  (y2, z2) → (y2, e2  0z2y−4  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have  s∗  1z2 = s∗  1Q2(y2)z−1  1  = Q2(e0y−1  1 )z−1  1  = e2  0y−4  1 Q1(y1)z−1  1  = e2  0y−4  1 z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore,  g−1∗s∗  1z2 = e2  0y−4  3 z2,  s∗  2g−1∗s∗  1z2 = e−2  0 y4  2z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We also have  s∗  2g−1∗s∗  1y2 = s∗  2g−1∗(e0y−1  1 ) = s∗  2(e0y−1  3 ) = y2,  which proves that �m−1 = s1 ◦ g−1 ◦ s2 is defined by (y2, z2) → (y2, e−2  0 z2y4  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='21 The element �m in Dyn(CV ) is the canonical formal monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Note that �m = t2(e2  0y−4  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In particular, �m is in the centralizer Z(Ty2) = By2 of Ty2, and  preserves y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  47  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4  Comparison between the confluent and the canonical dynamics  We have previously found that the confluent dynamics and the canonical dynamics are both  extensions of the tame dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In order to compare Dyn(CV ) and Conf(PV ), we write  the confluent dynamic in the canonical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The confluent dynamic is generated by  g1,2 = g−1  3,4 given by Proposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4), the family g2,3(κ) given by Proposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='6) and the  family g3,1(κ), which satisfies the relation g1,2 ◦ g2,3(κ) ◦ g3,1(κ) = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='22 We have:  (i) We have g1,2 = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Hence g1,2 : (yn, zn) → (yn−2, zn−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (ii) The automorphism g2,3(κ) and its inverse are given in canonical coordinate systems by  g2,3(κ) : (z1, y2) → (κ2z1y−2  2 , (1 + κ2e−1  0 z1y−2  2 )y2),  g3,2(κ) : (y1, z1) → (y1 + e0κ−2z1y−1  1 , e2  0κ−2z1y−2  1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (iii) The automorphism g3,1(κ) and its inverse are given in canonical coordinate systems by  g3,1(κ) : (z2, y3) → (κ2z2y−2  3 , (1 + κ2e−1  0 z2y−2  3 )y3),  g1,3(κ) : (y2, z2) → (y2 + e0κ−2z2y−1  2 , e2  0κ−2z2y−2  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (i) We have g = σ1 ◦ σ2, with  σ1(x1, x2, x3) = (x1 − FV,x1, x2, x3) = (−x1 − x2x3 + θ1, x2, x3)  σ2(x1, x2, x3) = (x1, x2 − FV,x2, x3) = (x1, −x2 − x1x3 + θ2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore,  (σ1 ◦ σ2)∗x1 = σ∗  2(−x1 − x2x3 + θ1)  = −x1 + x2x3 + x1x2  3 − θ2x3 + θ1 = (g1,2)∗x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (σ1 ◦ σ2)∗x2 = σ∗  2(x2)  = −x2 − x1x3 + θ2 = (g1,2)∗x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Hence, g = g1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (ii) From Proposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='6) we have:  g2,3(κ)∗z1 = e0x−1  2 (x2 − κ2x−1  2  + e−1  0 κ2x1) − e0  = e0 + κ2x−2  2 (x1x2 − e0) − e0  = κ2z1y−2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  g2,3(κ)∗y2 = g2,3(κ)∗((z1 + e0)y−1  1 )  = (κ2z1y−2  2  + e0)e−1  0 y2  = y2(1 + e−1  0 κ2z1y−2  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We obtain g3,2(κ) by reversing the map (z1, y2) → (Z1, Y2):  z1 = κ−2Y 2  2 (1 + e−1  0 Z1)−2, y2 = Y2(1 + e−1  0 Z1)−1,  and by using the change of canonical coordinates (z1, y2) → (y1 = (z1 + e0)y−1  2 , z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  48  (iii) We have g1,3(κ) = g ◦ g2,3(κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We set Z2 := κ2z2y−2  3 , Y3 := y3  �  1 + κ2e−1  0 z2y−2  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have  to prove that g1,3(κ)∗Z2 = z2 and g1,3(κ)∗Y3 = y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  g1,3(κ)∗Z2 = g2,3(κ)∗ ◦ g∗(κ2z2y−2  3 ) = g2,3(κ)∗(κ2z0y−2  1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  g1,3(κ)∗Y3 = g2,3(κ)∗ ◦ g∗Y3 = g2,3(κ)∗ �  y1(1 + κ2e−1  0 z0y−2  1 )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We compute g2,3(κ)∗z0 and g2,3(κ)∗y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have z0 = z−1  1 Q1(y1), therefore :  g2,3(κ)∗z0 = (g2,3(κ)∗z0)−1 Q1 (g2,3(κ)∗y1) = κ−2z−1  1 y2  2Q1  �  e0y−1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We recall (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' the remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='29) that Q1(e0t−1) = e−2  0 t4Q2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We obtain :  g2,3(κ)∗z0 = κ−2e2  0y−2  2 z−1  1 e2  0y4  2Q1  �  e0y−1  2  �  = κ−2e2  0y−2  2 z−1  1 Q2(y2) = e2  0κ−2y−2  2 z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Since y1 = (z1 + e0)y−1  2 , by using (ii) we have  g2,3(κ)∗y1 = (κ2z1y−2  2  + e0)(y2 + κ2e−1  0 z1y−1  2 )−1 = e0y−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Finally,  g1,3(κ)∗Z2 = g2,3(κ)∗(κ2z0y−2  1 )  = κ2(e2  0κ−2y−2  2 z2)(e0y−1  2 )−2 = z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  g1,3(κ)∗Y3 = g2,3(κ)∗ �  y1(1 + κ2e−1  0 z0y−2  1 )  �  = e0y−1  2  �  1 + κ2e−1  0 (e2  0κ−2z2y−2  2 )(e0y−1  2 )−2�  = e0y−1  2 (1 + e−1  0 z2) = e0y−1  2  �  1 + e−1  0 (y2y3 − e0)  �  = y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore, g3,1(κ) :  (z2, y3) → (Z2, Y3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The computation of its inverse g3,1(κ) in the chart  (y2, z2) is similar to the computation of the inverse of the g2,3(κ) in point (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='✷  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='23 The transformations g2,3(κ) and g3,1(κ) are conjugated by the map ξ : (y1, z2) →  (y2, z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Nevertheless, this conjugation ξ does not belong to the canonical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The following equalities are consequences of the points (ii) and (iii) in Proposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='22):  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='24 For all κ in C∗ we have:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' g2,3(κ) = g2,3(1) ◦ ty2(κ2) = ty1(κ−2) ◦ g2,3(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' g1,3(κ) = g1,3(1) ◦ ty2(κ2) = ty3(κ−2) ◦ g1,3(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In particular, g2,3(κ) conjugates Ty1 and Ty2: g2,3(κ) ◦ ty2(κ2) ◦ g2,3(κ) = ty1(κ−2), and g3,1(κ)  conjugates Ty2 and Ty3: g3,1(κ) ◦ ty3(κ2) ◦ g1,3(κ) = ty1(κ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  From these preliminary results we immediately obtain:  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The tame dynamic generated by g = σ1 ◦ σ2 is included in Conf(PV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For any n in Z, Tyn ⊂ Conf(PV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  49  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The first point is a direct consequence of g = g1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The second one is obtained for n = 1 or n = 2 from the first item of Corollary (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The others  tori are obtained from Ty1 or Ty2 by a conjugation with an element of the tame dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Nevertheless, the canonical Stokes operators do not belong to Conf(PV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The reason is the  following one: both g2,3(κ) and s1 keep invariant ∆ :  z1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The restriction of s1 to each  component of ∆ is a translation: see Proposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' One can compute the restriction of  g2,3(κ) on each each line of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  g2,3(κ)∗x3 = κ−2x2  2x3 + β(κ, x1, x2),  where β(κ, x1, x2) only depends on κ, x1, and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' On ∆, x1 = e±1  3 , e0e±1  4 , x2 = e0e±1  3 , e±1  4 ,  therefore the restriction of the g2,3(κ) are affine transformations on each line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The same property  holds for g3,1(κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Nevertheless, we cannot find κ in C∗ such that κ−2x2  2 = 1 simultaneously on  each component of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  This remark suggests to introduce an extension of the confluent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The only way to  obtain from the family g2,3(κ) an element which is a translation on ∆ is to introduce a functional  time for the elements of Tt2, setting κ = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Therefore if we expect to obtain the canonical stokes  operators from the confluent dynamic, we have to extend it by the element ty2(y−2  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Another motivation in order to introduce this element is purely algebraic and makes use of  the structure induced by the symplectic Cremona group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We recall the subgroups of Bir(CV )  generated from the Borel-Cartan structure of Symp(C2) by the canonical triple (y1, z1, y2) :  By1 = {by1(λ, r) : (y1, z1) → (λy1, z1r(y1)), r ∈ C(y1)∗, λ ∈ C∗}  By2 = {by2(λ, r) : (z1, y2) → (z1r(y2), λ−1y2), r ∈ C(y2)∗, λ ∈ C∗}  Ty1 = {ty1(λ) : (y1, z1) → (y1, λz1), λ ∈ C∗}  Ty2 = {ty2(λ) : (z1, y2) → (λ−1z1, y2), λ ∈ C∗}  Uz1 = {uz1(r) : (z1, y2) → (z1, y2r(z1)), r ∈ C(z1)∗, r(0) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  and we set :  Ty2 (C(y2)) = {(z1, y2) → (z1r(y2), y2), r ∈ C(y2)∗}  = {by2(1, r), r ∈ C(y2)∗} ⊂ By2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Ty1 (C(y1)) = {(z1, y2) → (y1, z1r(y1)), r ∈ C(y1)∗}  = {by1(1, r), r ∈ C(y1)∗} ⊂ By1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='26  (i) There exists a unique pair (by2, uz1) in Ty2 (C(y2)) × Uz1, such that :  g2,3(1) = uz1 ◦ by2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Moreover by2 = ty2(y−2  2 ) ∈ Ty2 (C(y2)) and uz1 = s−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (ii) There exists a unique pair (by1, vz1) in Ty1 (C(y1)) × Uz1, such that :  g3,2(1) = vz1 ◦ by1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Moreover by1 = ty1(e−2  0 y2  1) ∈ Ty1 (C(y1)) and vz1 = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' (i) We set :  by2 : (z1, y2) �→ (z1y−2  2 , y2),  uz1 : (z1, y2) �→ (z1, (1 + e−1  0 z1)y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  50  We have g2,3(1) = uz1 ◦ by2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The unicity of this decomposition is a consequence of Ty2 (C(y2)) ∩  Uz1 = {id}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' One recognizes here that by2 = ty2(y−2  2 ) and uz1 = s−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (ii) We set :  by1 : (y1, z1) �→ (y1, e2  0z1y−2  1 )  vz1 : (y1, z1) �→  �  (1 + e−1  0 z1)y1, z1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have g3,1(1) = vz1 ◦ by1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The unicity of this decomposition is a consequence of Ty1 (C(y1)) ∩  Uz1 = {id}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' One recognizes here that by1 = ty1(e−2  0 y2  1) and vz1 = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  By using g2,3(κ) = g2,3(1) ◦ ty2(κ2), and Z(Ty2) = By2, we obtain:  g2,3(κ) = uz1 ◦ by2 ◦ ty2(κ2) = uz1 ◦ ty2(κ2) ◦ by2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Since g3,1(κ) is conjugated to g2,3(κ) by (z1, y2) → (z2, y3) we have a similar decomposition of  the family g3,1(κ) in Uz2 × B1  y3 × Ty3:  g3,1(κ) = uz2 ◦ by3 ◦ ty3(κ2) = uz2 ◦ ty3(κ2) ◦ by3, uz2 = s−1  2 , by3 = ty3(y−2  3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Notice that b−1  y2 = ty2(y2  2) : (z1, y2) → (z1y2  2, y2) is a ramified blowing-up in the canonical chart  (z1, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We extend the confluent dynamic by this element:  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='27 The extended confluent dynamic is defined by  Conf ♯(PV ) =< Conf(PV ), ty2(y2  2) > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We recall that, according to Proposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='20), the element �m of Dyn(CV ) such that g = s2◦ �m◦s1  is defined by �m = ty2(e−2  0 y4  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We consider a square root of �m defined by:  �m1/2 : (y2, z2) → (y2, e−1  0 y2  2z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='28 The extended confluent canonical dynamic is defined by  Dyn♯(CV ) =< Dyn♯(CV ), �m1/2 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='29 Conf ♯(PV ) = Dyn♯(CV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='. We first prove that Dyn(CV )♯ ⊂ Conf ♯(PV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Dyn♯(CV ) is generated by g, Ty1, s1,  s2 and �m1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' From Proposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='26), since s−1  1  = g2,3(1) ◦ ty2(y2  2), Conf ♯(PV ) contains s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We  have:  �m = ty2(e−2  0 y4  2) =  �  ty2(e−1  0 ) ◦ ty2(y2  2)  �◦2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Hence, Conf ♯(PV ) also contains �m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' According to the relation g = s2 ◦ �m ◦ s1, s2 also belongs  to Conf ♯(PV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Finally since �m1/2 = ty2(e0) ◦ ty2(y2  2), we obtain the inclusion Dyn(CV )♯ ⊂  Conf ♯(PV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Now we prove that Conf ♯(PV ) ⊂ Dyn(CV )♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The relation  g2,3(κ) = s−1  1  ty2(y−2  2 ) ◦ ty2(κ2)  proves that g2,3(κ) belongs to Dyn(CV ) for any κ in C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Since g1,2 = g belongs to Dyn(CV ),  from the relation g1,2 ◦ g2,3(κ) ◦ g3,1(κ) = id, g3,1(κ) also belongs to Dyn(CV ) for any κ in C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Finally ty2(y2  2) = �m1/2 ◦ tt2(e−1  0 ) belongs to Dyn(CV )♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='✷  51  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='5  Comparison with the wild dynamics  The wild dynamics is a pseudogroup of non linear dynamics induced by the Painlev´e V foliation  PV (κ) in a neighborhood of each singular point of saddle-node type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We will recall here the  construction of its generators: the non linear stokes operators, non linear tori, and formal and  analytic non linear monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Through the Riemann-Hilbert map RHV it induces an (a priori)  local dynamics on CV (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The second author formulates the Wuhan conjecture which claims that  this dynamics extends into a symplectic rational dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This conjecture has been proved  for the Painlev´e V equation by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Klimes in [40] (there is also a very clear presentation of this  work in Klimes’s lecture [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' His method uses a description of the confluence on the foliations  of the Hamiltonian systems on the left side and on the linear isomonodromic systems and on  the associated character variety on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' An essential tool is the discretization of the  confluence as indicated in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1 on the baby model of confluence x(x − ε) → x2y′ + y = 0,  first introduced by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Zhang for the confluence for the hypergeometric equations [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We will  present here this result of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Klimes, and compare this dynamics with the canonical dynamics  Dyn(CV ) on the cubic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition of the wild dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The Painlev´e V foliation is given under its hamiltonian  form by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' From section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3, the irregular singular points si all appear in the Boutroux chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Around each irregular singular point of Painlev´e V of saddle-node type si the formal and sectoral  normal forms are given by [40]:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='30 Let α0 = 2α1 + α2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  1- In a neighborhood of a saddle-node si, the hamiltonian system (5) can be brought to a  formal normal form :  x2 du  dx = (1 − α0x + 4xu1u2)  \uf8eb  \uf8ec  \uf8ed  1  0  0  −1  \uf8f6  \uf8f7  \uf8f8 u,  u =  \uf8eb  \uf8ec  \uf8ed  u1  u2  \uf8f6  \uf8f7  \uf8f8  (13)  by means of a formal transversally symplectic change of coordinates :  \uf8eb  \uf8ec  \uf8ed  q  p  \uf8f6  \uf8f7  \uf8f8 := �Ψ(u, x) =  �  k≥0  ψ(k)(u)xk  where the ψ(k) are analytic on a polydisc P = {|u1|, |u2| < δ} (δ > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2- The formal normal form (13) is integrable in closed form :  �  u1(x, c1) = c1e−1/xx−α0+4c1c2  u2(x, c2) = c2e1/xxα0−4c1c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  (14)  and this local hamiltonian vector field (for du1 ∧ du2 and h = x−2(1 − α0x)u1u2, also admits the  analytic first integral u1u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3- The formal series �Ψ ∈ O(P)[[x]] is uniformly 1-summable with a pair of 1-sums Ψ↑(u, x),  Ψ↓(u, x), defined respectively above the sectors  U ↑ := {| arg x − π/2| < π − η, |x| < δ′} and U ↓ := {| arg x − π/2| < π − η, |x| < δ′}  for some 0 < η < π/2 arbitrary small and some δ′ > 0 (depending on η), and u ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  52  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='31 The formal Takano’s normal form [59] used here to define the non linear Stokes  operators will not suffice for the other Painlev´e equations in order to obtain overlapping open sec-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We will have to make use of the Bittman’s normal forms, which are no longer polynomials:  see [4], [5] and [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='32 On each sector U • (• =↑ or ↓),  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' the system (5) has a unique analytic bounded solution f • which is the 1-sum of the formal  solution �f : u1(x, 0) = u2(x, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We call these solutions the sectoral center solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  They are pole-free on the corresponding sector and they are characterized by this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' the system (5) has a 2-parameters family of solutions �fc1,c2:  (q•, p•)(x, c1, c2) = Ψ•(u(x, c1, c2), x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We consider the left and right intersection sectors of the overlapping sectors U ↑ and U ↓  V l := U ↑ ∩ U ↓ ∩ {ℜx < 0} and V r := U ↑ ∩ U ↓ ∩ {ℜx > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The non linear Stokes multipliers are defined by  S2 = (Ψ↑)−1 ◦ Ψ↓ on V r, S1 = (Ψ↑)−1 ◦ Ψ↓ on V l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The formal non linear monodromy is defined by the action induced by x �→ e2iπx on the space  of formal solutions �fc1,c2:  �  N : (c1, c2) �→ (e2iπ(−α0+4c1c2)c1, e2iπ(α0−4c1c2)c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The formal non linear exponential torus is defined by the analytic symplectic symmetries of the  formal normal forms (14):  tα : (c1, c2) �→  �  eα(c1c2)c1, e−α(c1c2)c2  �  ,  α ∈ O(C, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Using Ψ↑ and Ψ↓, the formal exponential torus induces two sectoral exponential tori T ↑ and T ↓  and the formal first integral h = c1c2 induces 2 sectoral first integrals h↑ and h↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='33 The wild dynamics Wild(PV , f ↑) based in the central solution f ↑ is the pseudo-  group generated by S1, S2, �  N, and T = {tα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Wild(PV , f ↑) also contains the actual monodromy N generated by a loop around x = ∞,  according to the relation S2 ◦ �  N ↑ ◦ S1 = �  N, where �  N ↑ = Ψ↑ ◦ �  N ◦ (Ψ↑)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The wild dynamics through the Riemann-Hilbert map RHV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We fix a value x0 in a  neighborhood of ∞, and we consider the Okamoto space of initial condition Vx0 over x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let  m• = (q•, p•)(x, 0, 0)  be the two points on Vx0 corresponding to the two central varieties in a neighborhood of a singular  point s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We denote by (c•  1, c•  2)(q, p) the inverse maps of (q•, p•)(x0, c1, c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The equations c•  2 = 0  define two germs of curves δ• in m•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The equations c•  1 = 0 define two germs of curves d• in m•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  On the rightside, we consider the 12 lines ∆·, Dr  , Dl· in the configuration of lines in χV : see  figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We can group them in four triples (∆i, Dri, Dli) such that ∆i∩Dri ̸= ∅ and ∆i∩Dli ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We set pri = ∆i ∩ Dri, pli = ∆i ∩ Dli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  53  Now we consider the confluent process between PV I and PV defined by  tV I = 1 + εtV ,  α3 = 1/ε,  α2,V I = ˜α2 = −1  ε + α2,V ,  x = 1  tV  + ε, α1,V I = α1,V  (15)  which sends the three fixed singularities to tV = −1/ε, 0, ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This change of variables transforms  εHV I into Hε  V I and the PV I Hamiltonian system into :  dq  dt = ∂Hε  V I  ∂p  ,  dp  dt = −∂Hε  V I  ∂q  ,  When ε → 0, the simple singular points −1/ε and ∞ merge into a double singular point, an  irregular singularity at infinity, and the limit of Hε  V I is HV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The four pairs of singularities over  −1/ε and ∞ merges to 4 saddle-nodes (the confluent saddle-nodes) among the five saddle nodes  si over ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In what follows, we suppose that s is one of these 4 singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We summarize the  results of Martin Klimes [40] by the following theorem:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='34 We consider the Riemann-Hilbert map RHV between an open set in the space  of initial condition Vx0 induced by a neighborhood of a confluent singularity s and the character  variety χV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The choice of s determines a triple of lines (∆, Dr, Dl) among the four triples  (∆i, Dr  i , Dl  i) such that  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' RHV (m↑) = ∆ ∩ Dr = pr, RHV (m↓) = ∆ ∩ Dl = pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' RHV (δ↑) = (∆, pr) (the germ of line at pr), RHV (δ↓) = (∆, pl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Furthermore, the germs δ↑  and δ↓ extend to a same analytic curve δ in Vx0 isomorphic to C, such that RHV (δ) = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' RHV (d↑) = (Dr, pr), RHV (d↓) = (Dl, pl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Furthermore the germs of curves d↑ and d↓  extends to 2 curves dr and dl isomorphic to C in Vx0, such that RHV (dr) = Dr and  RHV (dl) = Dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let (y1, z1, y2) be the triple of canonical coordinates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have:  RH∗  V y1 = y1(pr)eh↑, RH∗  V y2 = y2(pr)e−h↓, RH∗  V z1 = e0(e(h↑−h↓) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let g, s1, s2, be the generators of the canonical dynamics defined in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='18, let �m be the  canonical formal monodromy defined by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='20 and Ty1, Ty2 the canonical exponential tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have:  RH∗  V g = N, RH∗  V S1 = s1, RH∗  V S2 = s2, RH∗  V �m = �  N, RH∗  V T ↓ = Ty1, RH∗  V T ↑ = Ty2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Therefore the dynamics induced by Wild(PV , f ↑) through the Riemann-Hilbert correspondence  RHV extends to a rational symplectic dynamics Wild(PV ) on χV , and we have Wild(PV ) =  Dyn(CV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This proves the Wuhan conjecture of the second author for the PV foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The main ingredients of the proof of this result in [40] are:  an unfolded version of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='30 for the hamiltonian system related to Hε  V I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  a discretization of the confluent parameter by setting ε−1 = ε−1  0 +n, n ∈ Z+, along which the  monodromies of the unfolded system, a priori divergent, converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The choice of this sequence  depends on a parameter κ = e2iπ/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  a confluence on the rightside between χV I and χV which allows to compare these limits to  the wild dynamics Wild(PV , f ↑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  54  6  Conclusion and open questions  Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Several dynamics related to the Painlev´e V foliation through the Riemann-Hilbert  map RHV has been described here on the wild character variety χV (a) identified to the cubic  symplectic surface CV (θ) for generic parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The tame dynamics is induced here by only  one braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We first extended the tame dynamics by using a confluent dynamics obtained from  the Painlev´e VI dynamics by a birational confluent morphism between χV I and χV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This one,  first discovered by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Klimes, is obtained here by a confluent process between the corresponding  groupoids (after an extension with families of exponential loops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have constructed the canonical symplectic birational dynamics defined on the cubic  symplectic surface CV (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This one is the pullback of symplectic dynamics on C2 by a canonical  sequence of coordinates satisfying cluster-type relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This birational dynamics coincides with  the dynamics obtained by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Klimes from the wild dynamics defined by a confluent irregular  singular point of the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' After and extension by one element, confluent dynamics and  canonical dynamics also coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The cubic symplectic surface CV (θ) contains a skeleton generated by 18 lines and their im-  ages through the action of the tame dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have characterized this set by a condition of  reducibility of the corresponding linear representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Some intersections between these lines  correspond to particular solutions of the Painlev´e V equation : Kaneko solutions, or central so-  lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The restriction of the dynamics on this skeleton is an important tool in the comparisons  between these dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The equations of the character varieties χJ for the other Painlev´e equations  PJ, J = Vdeg, IV , III(D6), III(D7), III(D8), II(JM), II(FN), I (with the notations of [54])  are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We already know that each of them can be obtained as a quotient of a space of  linear representations of a wild groupoid, and that there exists canonical cluster sequences of  coordinates on each χJ, inducing a canonical birational symplectic dynamic DynJ which can be  explicitely computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We conjecture that for all J:  All the lines in χJ are a reducibility locus of some path in the groupoid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  the wild dynamics of all the Painlev´e equations (induced by non linear monodromies, non  linear Stokes operators and exponential tori) around an irregular singular point obtained by a  confluent process coincide with these canonical birational symplectic dynamics DynJ on χJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We already know that these dynamics coincide on the skeleton generated by the lines for PI and  PII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' With M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Klimes we also conjecture that  There is a diagram of families of birational symplectic confluences between the χJ, which par-  allels the diagram of confluence of Ohyama and Okumura [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We expect that we can prove this fact by a confluent process between the groupoids (extended  by families of exponential loops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Finally we mention here the initial motivation of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Cantat and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Loray  proved the irreducibility of PV I for generic parameters, by using the Malgrange closure of its  dynamics [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The aim of the Wuhan conjectures of the second author was to extend their  proof in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In the PV I case, the dynamics obviously belongs to the Malgrange  groupoid which is a closure of its holonomy pseudogroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' In the general case, we conjecture that  the wild dynamics is contained in the Malgrange pseudogroup, but now it is far from obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Modulo this conjecture, we would obtain a proof of the irreducibility of the Painlev´e V equation  for generic values of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Indeed since the extended canonical dynamic contains the  confluent dynamic, it contains a copy of the dynamics of a PV I and we can use Theorem D of  [14] in order to prove that its Malgrange closure is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  55  Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Structure of the symplectic Cremona group  The Cremona group Cr = Cr2 is the group of birational automorphisms of P2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It does not  depend on the representative X in the birational class of P2: C2, P1 × P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The symplectic Cremona group is the subgroup Symp of Cr ≃ Bir(P1(C) × P1(C)) of the  elements which preserve the differential form ω = du  u ∧ dv  v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' The Cremona group is an old topic  which appears at the end of the XIX-th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' One can find an introductive text in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For  a study of its subgroups, see [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' To the contrary, the study of its symplectic subgroup Symp  has been developed only since 2005, first by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Usnich [62, 63] and then J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Blanc [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We present  here without proofs the main results about subgroups of Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Some of them are new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Our terminology is inspired by the algebraic groups, due to some similarities in this non linear  infinite dimensional context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We denote  T = {t(λ, µ) : (u, v) �→ (λu, µv), (λ, µ) ∈ C∗ × C∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have T ⊂ Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We denote by Z(T) its centralizor, and N(T) its normalizor in Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We  consider the subgroup W of Symp of the monomial maps:  W = {(u, v) �→ (uavb, ucvd),  \uf8eb  \uf8ec  \uf8ed  a  b  c  d  \uf8f6  \uf8f7  \uf8f8 ∈ SL2(Z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='1 We have:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' T is an algebraic abelian maximal subgroup of Symp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Z(T) = T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' N(T) = W ⋉ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' T is an algebraic abelian maximal subgroup of Cr [56], which preserve ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The centralizor and normalizor of T in Cr has been computed by [20], since C is algebraically  closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have: NSymp(T) = NCr(T) ∩ Symp = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' ✷  Furthermore, according to a theorem of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Blanc [7], Symp is generated by N(T) and the order  five element p defined by:  (u, v) �→  �  v, 1 + v  u  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  The Cartan subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='2 A Cartan subgroup of Symp is an algebraic abelian maximal subgroup of rank  two in Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' T is the standard Cartan subgroup of Symp, and W is the Weyl group associated  to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We have T = T1 × T2, where  T1 = {t1(λ) = t(λ, 1), λ ∈ C∗}, T2 = {t2(µ) = t(1, µ), µ ∈ C∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  All the Cartan subgroups are conjugated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 8  8We have no reference for this fact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' the proof will be delivered in a forthcoming publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' This proof uses  a result of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Blanc [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  56  Borel subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' Let dJ1 be the subgroup of Cr ≃ Bir(P1 × P1) of the De Jonqui`eres maps,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' the rational maps which preserve the first projection:  dJ1 = {dj1 : (u, v) �→ (a(u), b(u, v)), a ∈ PGL2(C), b(·, v) ∈ PGL2(C(u))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Let dJS1 be the subgroup in Symp of the symplectic De Jonqui`eres maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' For any λ in C∗, any  r in C(X)∗, the maps:  dj1(λ, r) : (u, v) �→ (λu, r(u)v), σ : (u, v) �→ (u−1, v−1)  are examples of elements in dJS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We set  B1 = {dj1(λ, r), λ ∈ C∗, r ∈ C(u)∗} =< T1, T2(C(u)) >, B−  1 = {b1 ◦ σ, b1 ∈ B1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3 We have dJS1 = B1 ∪ B−  1 = B1 ⋊ Z/2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  We also define dJ2, dJS2, B2 with respect to the second projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We have B1 ∩ B2 = T, and  any pair (wB1w−1, w′B2w′−1), w, w′ ∈ W also satisfies this equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='4 (B1, B2) is the standard pair of Borel subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Any pair (wB1w−1, w′B2w′−1), w, w′ ∈ W is a pair of Borel subgroups related to the standard  Cartan subgroup T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Notice that the Borel subgroups are no longer algebraic subgroups since there are infinite di-  mensional groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' It can be proved that B1 and B2 generate Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='5 The subgroup U1 of B1 of the “unipotent” elements is defined by:  U1 = {dj1(1, r), r(0) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='6 We have:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' U1 is an abelian subgroup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' U1 is generated by the family dj1(1, 1 + νu), ν ∈ C∗, or by one element dj1(1, 1 + u) and  its conjugations with an element of T1 (we say that this element is a pseudo generator of  U1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' [B1, B1] = {dj1(1, r), r(0) = 1, deg(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='} Therefore, [B1, B1] ⊂ U1 and B1 is meta-  abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  In this non linear context, we clearly have some analogies with the structure of the algebraic  group SL2(C) but we also have some differences:  the inclusion [B1, B1] ⊂ U1 is a strict one, [B1, B1] is an invariant subgroup of U1 and we have  U1/[B1, B1] =< dj1(1, 1 + u) >≃ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  let B♮  1 =< U1, T1 >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' B♮  1 is also a meta-abelian group which contains U1, but it contains only a  maximal torus of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' We call B♮  1 a Borel of rank one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  57  References  [1] Atiyah M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
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+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
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+page_content=' Quasi-hamiltonian geometry of meromorphic connections, Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
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+page_content='  [11] Bolibruch A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
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+page_content=', Mitschi C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='3, (2006) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content=' 235–272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
+page_content='  [12] Brion M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9FAT4oBgHgl3EQfiB2Y/content/2301.08597v1.pdf'}
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+MITP-22-103
+Dark Matter Induced Nucleon Decay Signals in Mesogenesis
+Joshua Berger1, ∗ and Gilly Elor2, †
+1Colorado State University, Fort Collins, Colorado 80523, USA
+2PRISMA+ Cluster of Excellence & Mainz Institute for Theoretical Physics
+Johannes Gutenberg University, 55099 Mainz, Germany
+We introduce and study the first class of signals that can probe the dark matter in Mesogenesis
+which will be observable at current and upcoming large volume neutrino experiments. The well-
+motivated Mesogenesis scenario for generating the observed matter-anti-matter asymmetry neces-
+sarily has dark matter charged under baryon number. Interactions of these particles with nuclei can
+induce nucleon decay with kinematics differing from sponanteous nucleon decay. We calculate the
+rate for this process and develop a simulation of the signal that includes important distortions due to
+nuclear effects. We estimate the sensitivity of DUNE, Super-Kamiokande, and Hyper-Kamiokande
+to this striking signal.
+Mesogenesis mechanisms [1–4] utilize the Charge-
+Parity (CP) violation of Standard Model (SM) meson
+systems to generate the primordial baryon asymmetry
+and the dark matter (DM) abundance of the Universe.
+Excitingly, Mesogenesis is highly testable [5, 6] and ex-
+perimental searches are underway to probe signals di-
+rectly linked to the generated baryon asymmetry [6–10]
+(see overviews in [11–14]). However, a direct probe of the
+DM in Mesogenesis has remained elusive, until now. In
+this Letter we study DM induced nucleon decays (IND)
+in Mesogenesis which can produce a striking signal at cur-
+rent and upcoming neutrino detectors. While the Meso-
+genesis framework is the primary focus of this letter, the
+methods developed here can be more broadly applied to
+search for models containing dark baryons e.g. [15–17].
+The novel way in which Mesogenesis satisfies the
+Sakharov conditions [18], is as follows: mesons produced
+at late (MeV scale) times, having undergone CP violat-
+ing processes, decay out of thermal equilibrium into dark
+baryons. This process generates an equal and opposite
+baryon asymmetry between the dark and visible sector.
+For instance, the baryon asymmetry in B0-Mesogenesis
+[1] is generated from the late time production of B0
+s,d
+mesons which undergo CP violating particle-antiparticle
+oscillations and then quickly decay into a SM baryon and
+a dark Dirac fermion ψB carrying SM baryon number −1.
+To evade washing out the generated baryon asymmetry
+through e.g. ψB → p e ¯νe, the ψBs must rapidly decay
+into stable DM states.
+Mesogenesis DM consists of two components— a dark
+Majorana fermion ξ and a dark complex scalar φB which
+is charged under SM baryon number. These two stable
+particles will compose the entirety of the DM. As such the
+DM halo will consist of a mixture of ξ and φB particles
+which can scatter off target nuclei in neutrino detectors
+to produce mono-energetic pions or kaons and missing
+energy. This process of IND appears experimentally as
+nucleon decay but with different kinematics, and as such
+current limits are essentially not constraining.
+The cross section for IND, as it arises in Mesogene-
+sis, will be within reach of neutrino detectors and can be
+searched for at the Deep Underground Neutrino Experi-
+ment (DUNE) [22], Super-Kamiokande [23], and Hyper-
+Kamiokande [24]. Furthermore, given the mono-energetic
+(up to smearing effects) meson, such signals should be
+distinct over SM backgrounds which primarily consist of
+mesons produced in atmospheric neutrino processes.
+In order to study the details of the IND process, we
+have developed a Monte Carlo event generation tool
+within the GENIE [19, 20] software suite used to study
+accelerator and atmospheric neutrino scattering events
+[21]. Based on this tool, we are able to study the de-
+tailed kinematics of the outgoing mesons produced dur-
+ing DM scattering events and compare these with the
+dominant atmospheric neutrino scattering background.
+The event generation includes nuclear effects which smear
+out the spectrum of mesons from one that is nearly mono-
+energetic in the non-relativistic DM limit. Furthermore,
+these events are ready to be used for study in neutrino
+experiments, where they can be passed through detector-
+specific simulation software.
+In this Letter we first characterize the IND signal. We
+present the Mesogenesis specific parameter space which
+can be targeted by experiments.
+We then detail the
+simulation. Next, we apply this simulation to estimate
+the sensitvity of DUNE, Super-Kamiokande, and Hyper-
+Kamiokande to benchmark models.
+Characterizing the IND Signal. — Generating the
+baryon asymmetry in B-Mesogenesis [1, 3], requires the
+existence of a TeV scale colored scalar (which may be
+identified as a squark in a supersymmetric embedding
+[4]) which mediates the baryon number conserving decay
+of a B to ψB and a SM baryon through the (GeV scale)
+effective operator:
+Oab,c = Cab,cϵijk
+�
+ui
+adj
+b
+� �
+ψB dk
+c
+�
+,
+(1)
+where all fermions are right handed, though our results
+easily generalize to left handed operators.
+Cuadb,dc ≡
+yuadbyψdc/M 2
+Y , where MY is the mediator mass.
+ψB
+decays to the DM through a Yukawa coupling Ld ⊃
+yd ¯ψBξφB +h.c., which is allowed by a stabilizing Z2 sym-
+metry under which ψB is even and the DM particles odd.
+arXiv:2301.04165v1  [hep-ph]  10 Jan 2023
+
+2
+FIG. 1. Induced proton decay to a pion through Ou,dd.
+Since we remain agnostic about the dark sector, yd is
+a free parameter.
+However a motivated benchmark is
+yd ≲ O(0.1) which results in the correct DM abundance
+given an example UV embedding [4].
+Eq. 1 generates the IND signal in Mesogenesis: when
+kinematically allowed, an incoming ξ or φB scatters off a
+proton or neutron by exchanging a ψB and produces an
+energetic meson. Fig. 1 depicts an example process —
+incoming φB’s induce proton decay to π+ through Oud,d.
+Similarly induced decay to kaons arises through Oud,s
+and Ous,d. We consider searches at neutrino experiments
+for the following two processes:
+φB N → M ξ
+if
+mφB + mN > mM + mξ ,
+(2a)
+ξ N → M φ⋆
+B
+if
+mξ + mN > mM + mφB ,
+(2b)
+where N = n0, p+ and M is a SM meson. Recall that
+ξ is Majorana, allowing any of the DM states to partici-
+pate in this process when it is kinematically allowed. For
+decays induced by incoming φs, the kinetic energy of the
+outgoing meson, to O(vDM), is given by
+EM, kin
+φBN→ξM =
+m2
+M − m2
+ξ + (mN + mφB)2
+2(mN + mφB)
+− mM . (3)
+Swap mξ ↔ mφB in the above to obtain the meson energy
+from incoming ξs. The expected kinetic energy for each
+process that arises in Mesogenesis is given in Table I.
+If the struck nucleon is at rest, then the outgoing
+meson is mono-energetic with energy given in Eq (3).
+However, the nucleons are moving with a momentum of
+O(100 MeV) inside nuclei, smearing out of the meson sig-
+nal (except for the case of scattering off hydrogen in wa-
+ter Cherenkov detectors). We simulate the IND process,
+carefully accounting for this smearing.
+Note that the
+energies of these decays are shifted compared to sponta-
+neous nucleon decay, with higher energies when φB scat-
+ters and lower energies when ξ scatters. This alters the
+phenomenology of the Mesogenesis scenario compared
+with proton decay models such as grand unified theo-
+ries — the canonical targets of current nucleon decay
+searches [25–30]. As such, existing limits from nucleon
+Initial
+Final
+Mediating
+Meson
+Approx. ⟨σv⟩0
+DM
+Meson
+Operator
+EKin [GeV]
+�
+cm3/sec
+�
+φB
+π+/π0
+Oud,d
+0.6 - 1.2
+10−21.4 - 10−21.0
+ξ
+π+/π0
+Oud,d
+0.02 - 0.6
+10−22.5 - 10−21.9
+φB
+K+/K0 Ous,d, Oud,s
+0.3 - 0.9
+10−19.7 - 10−19.3
+ξ
+K+/K0 Ous,d, Oud,s
+0.04 - 0.3
+10−20.6 - 10−19.8.
+TABLE I. The induced nucleon decays allowed by kinemat-
+ics (2) and Mesogenesis considerations Eq. 5. We show the
+corresponding flavorful variation of the opeator Eq. 1 that
+generates the decay, the expected range of un-smeared ki-
+netic energy of the outgoing meson computed from Eq. (3),
+and the stripped cross section defined in Eq. (6).
+decay searches, with a few exceptions, do not constrain
+the Mesogeneis signal. Similar considerations were dis-
+cussed in the context of Hylogenesis [31].
+The cross section for IND is obtained from the matrix
+element:
+AφBN→ξM
+= ¯uξ(⃗pξ)
+yd Cab,c
+p2
+ψB −m2
+ψB
+�
+/pψB + mψB
+�
+(4)
+× PR
+�
+W RR
+0
+− i
+/pψB
+mN W RR
+1
+�
+uN(⃗pN) ,
+with ¯uξ → ¯vξ for ξ N → M φ⋆
+B. Here, pψB = pξ−pφB and
+PR is the right handed fermion projector. The Wilson
+Coefficients are constrained by a combination of LHC
+searches for the mediator and flavor observables: Cmax
+ud,d =
+0.07 TeV−2 and Cmax
+ud,s , Cmax
+us,d = 0.64 TeV−2 [6, 15]. Since
+INDs can lead to O(GeV) momentum transfer, we use
+high q2 extrapolated lattice results from [32] for the form
+factors W RR
+0,1 [33]. Including the momentum dependence
+of W1,2 negligibly affects the signal. The Ous,d and Oud,s
+require different form factors but lead to similar signals.
+Mesogenesis Parameter Space. — In addition to the
+kinematic constraints on IND Eq. (2), the allowed param-
+eter space is constrained by Mesogenesis-specific consid-
+erations i.e. generating the observed baryon asymmetry
+and DM abundance:
+mφB + mξ < mψB < mB − mp ≃ 4.34 GeV ,
+(5a)
+|mφB − mξ| < mp + me ≃ 938.8 MeV ,
+(5b)
+mψB , mφB > mp − me .
+(5c)
+mφB > mξ .
+(5d)
+The regions in {mξ, mφB} space excluded by these con-
+straints are shaded in Fig. 2, while the white region is
+allowed. Eq. (5a) ensured that ψB is light so that the
+baryon asymmetry can be generated through the decay
+B → BSM + ψB, while also being heavy enough to decay
+into the DM ξ and φB. Decreasing the value of mψB cor-
+responds to increasing the excluded green region.
+For
+mψB ≃ 1.1 GeV, there is no longer viable parameter
+space.
+Eq. (5b) is enforced to prevent ξ and φB from
+coalescing into SM baryons, which would washout the
+
+ΦB
+S
+山B
+u
+d
+p
+d
+u3
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+FIG. 2. Parameter space and kinetic energy contours for the eight different DM IND proceses arising in Mesogenesis. Colored
+regions are ruled out by kinematics Eq. (2) or mechanism considerations Eq. (5). Sold lines correspond to kinetic energy for
+scattering off protons N = p, and the dashed off nucleons N = n. In each panel, we indicate the location of the representative
+benchmark points: 1, 2, 3p, 4p for pions and 1, 2, 3k, 4k for kaons as summarized in Table II.
+baryon asymmetry.
+Proton decay through Eq. (1) to
+dark baryons is kinematically forbidden through Eq. (5c).
+A complete list of DM IND processes consistent with
+Eq. (2) and Eq. (5) is shown in Table I.
+There must exist dark sector interactions which de-
+plete the DM and ensure the correct abundance [1]. Since
+nφB − ¯nφB is related to the baryon asymmetry, φB must
+always constitute some, but not all, of the DM: if mφB <
+mξ, dark interactions could annihilate the entire ξ pop-
+ulation into ψBs. The measured ratio of DM to baryon
+densities [34] implies that mφB ≃ 5mp, violating Eq. (5a).
+This motivates multi-component DM where we enforce
+Eq. 5d so that the symmetric component of φBs annihi-
+late into ξs. Since the measured SM baryon asymmetry
+is always balanced by an asymmetry in φBs, the observed
+DM to baryon ratio ρDM ∼ 5ρB fixes the expected density
+of ξ and φB particles in the halo: ρξ/ρφB = 5mp/mφB −1,
+and ρtotal = ρφ + ρξ = 0.4 GeV/cm3. Given Eq. (5c),
+there will always be a substantial asymmetric compo-
+nent of DM, and so both INDs in Eq. (2) will be present
+if kinematically allowed.
+The scattering cross sections for the INDs are com-
+puted from Eq. (4). They scale roughly as ⟨σv⟩ ∝ m−2
+ψB,
+but the range of variation mψB ∼1-4.3 GeV, leads to a
+small effect. We parametrize the cross section as:
+⟨σv⟩M
+DM ≡ (yd × Cudi,dj)2
+m2
+ψB[GeV−4] ⟨σv⟩M,0
+DM
+(6)
+where M = π0, π+, K+, K0 and DM = φB, ξ. Val-
+ues of the coupling stripped cross section ⟨σv⟩M,0
+DM over
+the allowed parameter space of Fig. 2) are shown in Ta-
+ble I. For e.g. yd = 0.1 and Cmax
+udi,dj, the expected cross
+section can be as large as 10−38 − 10−36cm3/sec in the
+allowed parameter space for all channels. Meanwhile, the
+estimated sensitivity at DUNE and Super-Kamiokande is
+roughly 10−42 − 10−40cm2/sec .
+Benchmark
+mφB [GeV] mξ [GeV]
+1
+0.95
+0.92
+2
+2.45
+1.53
+3p
+2.38
+1.6
+3k
+2.2
+1.8
+4p
+0.95
+0.17
+4k
+0.95
+0.55
+TABLE II. Benchmarks highlighting the possible signal
+topologies.
+Benchmark 1 corresponds to Ekin,min
+pφB→Mξ
+∼
+Ekin,max
+pφB→Mξ for both πs and kaons. Benchmark 2 corresponds
+to Ekin,max
+pφB→Mξ. Benchmarks 3p and 3k correspond to the maxi-
+mal Ekin
+pφB→Mξ such that the incoming ξ process is still allowed
+for production of πs and kaons respectfully. Benchmarks 4p
+and 4k highlight a region of {mξ, mφB} which would still lead
+to a signal for small mψB (see also labels in Fig 2).
+Signal
+Monte
+Carlo
+and
+Benchmarks. —
+Non-
+relativistic DM striking a nucleon at rest and inducing
+decay would lead to a mono-energetic signal with kinetic
+energy given by Eq. (3) and shown in Fig. 2.
+We
+pick benchmark points to highlight the possible signal
+topologies; these are defined in Table II and labeled in
+Fig. 2.
+In the context of large nuclei, nuclear effects
+including nuclear motion of the nucleons and final
+state interactions of hadronic particles escaping the
+nuclear remnant smear the outgoing meson energy, can
+liberate additional hadrons, and can change the isospin
+characteristics of the meson.
+In order to account for
+these effects, as well as allow for future simulation of the
+detailed detector response at neutrino experiments, we
+have developed a Monte Carlo event generation tool for
+the IND process.
+Signal events are generated using a modified version
+of GENIE v3.0.2 [19, 20]. We employ the default tune
+(G18 02a) throughout, though we considered other nu-
+clear models. We found differences in the signal distri-
+butions of order 10% between hA and hN models of the
+intranuclear cascade. The current nucleon decay mod-
+
+4
+ule in GENIE was modified to allow for IND kinematics.
+The meson final states currently implemented are π0, π+,
+K0
+S/L, K+, and D0. This module propagates the outgo-
+ing mesons through the nuclear remnant to the edge of
+the nucleus. The kinematics of the IND process is fixed
+given masses for the two DM particles. The cross sec-
+tion is determined by Eq. (6). For non-relativistic DM,
+there is a small difference in the rate for interaction with
+a high speed nucleon compared with a nucleon at rest.
+We neglect this small difference of ∼10%.
+Signals at Neutrino Experiments. — The current best
+limits on spontaneous nucleon decays to pions are from a
+Super-Kamiokande [27], which applied a pion momen-
+tum cut of 1 GeV and is thus applicable to parts of
+Mesogenesis parameter space — here the experimental
+limit can be compared to an effective lifetime (τ Ind
+N )−1 =
+nDM⟨σv⟩DM [31, 35, 36].
+This yields an approximate,
+conservative, limit of yd × Cud,d/Cmax
+ud,d ≳ 0.03 − 0.1 for
+70 MeV ≲ Ekin
+π+ ≲ 870 MeV and Ekin
+π0 ≲ 870 MeV. Ex-
+isting searches in kaon channels [29, 30] placed narrow
+ranges of momentum cuts. Consequentially, the major-
+ity of parameter space of interest is unconstrained by ex-
+isting searches. Super-Kamiokande is expected to have
+sensitivity to IND signals given dedicated studies. Hyper-
+Kamiokande will improve on this with its larger exposure.
+Since DUNE is based on liquid argon time-projection
+chamber (LArTPC) technology, it could have particular
+sensitivity to certain models.
+We assume kinetic energy thresholds as indicated in
+Table III, based on the studies where available [37]
+and otherwise based on Cherenkov thresholds or de-
+tecting tracks/showers in LArTPCs.
+Determining re-
+construction efficiencies requires a dedicated study by
+the respective experimental collaborations.
+In Super-
+Kamiokande’s search for proton decay in the π+ν chan-
+nel, no muon/pion separation was attempted [27]. On
+the other hand, the LArTPC experiment MicroBooNE
+has demonstrated muon/pion separation at the 80%
+level [38], consistent with other particle identification ef-
+ficiencies. We therefore do not assume muon/pion sepa-
+ration for the Water Cherenkov experiments, while we as-
+sume perfect muon/pion separation for DUNE. Searches
+for similar signals of kaons in spontaneous nucleon decay
+at Super-Kamiokande indicate reconstruction efficiencies
+around or below 10% [29, 30]. The full 3D imaging of
+LArTPC detectors like DUNE may increase performance
+in reconstructing complex topologies arising from kaon
+decays, e.g. kaons above 40 MeV kinetic energy may be
+efficiently be reconstructed as tracks before decay [39].
+Sensitivity Estimate. — The dominant background for
+the channels we consider is expected to be inelastic scat-
+tering of atmospheric neutrinos off nuclei, leading to ad-
+ditional mesons. By looking for off beam timing events,
+beam-related backgrounds can be evaded.
+To study
+the atmospheric neutrino background, we generate at-
+mospheric neutrino events using GENIE v3.0.2, along
+Particle
+LArTPC
+Cherenkov
+Thresh. [MeV] Thresh. [MeV]
+e±
+30
+3.5
+γ
+30
+3.5
+µ±
+35
+55
+π±
+35
+72
+p
+80
+481
+TABLE III. Kinetic energy thresholds for particles assumed in
+LArTPC (DUNE) and Water Cherenkov (Super- and Hyper-
+Kamiokande) detectors.
+with the Bartol atmospheric neutrino flux model [40] at
+Soudan for DUNE and Kamioka.
+From samples of DM signals and atmospheric neutrino
+events, we look for events containing the relevant final
+state meson for the channel considered. Not all events
+with such a meson should be considered. Signal events
+contain a single meson and no other activity other than
+possible emissions from the nuclear remnant or byprod-
+ucts of final state interactions. Thus, it is highly benefi-
+cial to veto any activity beyond the expected meson. To
+do so, we first apply the thresholds in Table III to all final
+state particles. Of the remaining particles that can be de-
+tected, we veto on events that have anything other than a
+meson of the expected type. For the pion channels, order
+1% of atmospheric neutrino scattering events lead to an
+event matching these criteria. This leaves a search that
+is not entirely background free. For the kaon channels on
+the other hand, single kaon events are only possible with
+additional flavor-violating weak interactions. Searches in
+these channels may thus be background free if kaon re-
+construction is sufficiently good. To get a sense of the
+events after these vetoes we plot kinetic energy distribu-
+tions of the remaining signal and background events; a
+few illustrative distributions are shown in Fig. 3.
+We now estimate the sensitivity to yd×Cudi,dj/Cmax
+udi,dj.
+For pion channels, we apply the selection described
+above. To eliminate a majority of the background events,
+we also require that the selected pion has a kinetic energy
+within 100 MeV of its unsmeared value Eq. (3). For the
+kaon channels, no further selection is required as we have
+found that this channel is nearly background free, and we
+determine the coupling that would lead to 5 signal events
+over the assumed exposure of the experiment. In cases
+where there is background, we estimate the 2σ sensitivity
+to the signal. The sensitivity results are summarized in
+Table IV for the benchmarks listed in Table II.
+Discovering an IND signal would be a direct probe of
+the DM in Mesogenesis. In addition to providing exper-
+imentalists with the tools needed to search for DM IND
+signals at neutrino experiments, this letter paves the way
+to a signal driven model building effort of the dark sector.
+
+5
+FIG. 3. Kinetic energy distributions for sample benchmark models at DUNE and Hyper-Kamiokande. Super-Kamiokande, is
+simply a rescaling of the rate at Hyper-Kamiokande by a factor of 16. Dashed lines indicate the un-smeared energy Eq. 3.
+Green lines indicate, where relevant, the assumed threshold for the detector to see the meson. The top row correspond to
+Benchmark 1, while the bottom corresponds to Benchmark 3k. The Hyper-Kamiokande signal has a noticeable mono-energetic
+spike corresponding to scattering of hydrogen, while the smeared distribution corresponds to scatterings off oxygen.
+Benchmark Background yd(Cudi,dj/Cmax
+udi,dj)
+Background yd(Cudi,dj/Cmax
+udi,dj)
+Background yd(Cudi,dj/Cmax
+udi,dj)
+and Meson
+DUNE
+DUNE sensitivity
+Super-K
+Super-K sensitivity
+Hyper-K
+Hyper-K sensitivity
+1 π+
+118
+0.019
+1759
+0.030
+9452
+0.020
+2 π+
+14
+0.007
+432
+0.014
+2323
+0.0090
+3p π+
+584
+0.021
+2570
+0.023
+13835
+0.015
+4p π+
+600
+0.040
+2907
+0.045
+15653
+0.029
+1 π0
+140
+0.025
+125
+0.026
+672
+0.017
+2 π0
+26
+0.011
+17
+0.011
+94
+0.0069
+3p π0
+915
+0.080
+590
+0.080
+3135
+0.052
+4p π0
+923
+0.053
+608
+0.050
+3231
+0.033
+1 K+
+0
+0.0016
+0
+0.0014
+0
+0.00061
+2 K+
+0
+0.00038
+0
+0.00032
+0
+0.00014
+3k K+
+0
+0.00063
+0
+0.00054
+0
+0.00023
+4k K+
+0
+0.0010
+0
+0.00087
+0
+0.00038
+1 K0
+S
+0
+0.00047
+0
+0.00056
+0
+0.00024
+2 K0
+S
+0
+0.00034
+0
+0.00041
+0
+0.00018
+3k K0
+S
+0
+0.00037
+0
+0.00044
+0
+0.00019
+4k K0
+S
+0
+0.00049
+0
+0.00058
+0
+0.00025
+TABLE IV. Estimated coupling sensitivity at DUNE with 400 kton-years of exposure, at Super-Kamiokande with 350.8 kton-
+years of exposure, and at Hyper-Kamiokande with 1,900 kton-years of exposure. We apply the solar minimum flux model for
+all experiments. The solar maximum model gives slightly different sensitivity estimates.
+We thank Yun-Tse Tsai for comments on the draft and
+Yue Zhao for helpful discussions. The work of J.B. is sup-
+ported by the National Science Foundation under Grant
+No. 2112789. G.E. is supported by the Cluster of Excel-
+lence Precision Physics, Fundamental Interactions and
+Structure of Matter (PRISMA+ – EXC 2118/1) within
+the German Excellence Strategy (project ID 39083149).
+J.B. thanks the Mainz Institute for Theoretical Physics
+(MITP) of the Cluster of Excellence PRISMA+ (project
+ID 39083149) for their hospitality.
+
+200
+DUNE
+Atmospheric 
+m=0.92GeV
+150
+ΦB signal
+mg = 0.95GeV
+d = 0.05, Cud,d =
+$ signal
+max
+ud,d
+100
+Number
+50
+0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
++kinetic energy (GeV)5000
+Hyper-K
+Atmospheric v
+4000
+m=0.92GeV
+events
+ΦB signal
+ms=0.95 GeV
+$ signal
+3000
+ud,d
+Number of
+2000
+1000
+0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
++ kinetic energy (GeV)120
+DUNE
+ΦB signal
+100
+ofevents
+m = 1.8 GeV
+$ signal
+m= = 2.2 GeV
+80
+Yd = 0.01, Cud,s =
+max
+ud,s
+60
+40
+20 -
+0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+K+kinetic energy (GeV)Hyper-K
+Atmosphericv
+1500
+ms = 1.8 GeV
+events
+ΦB signal
+m = 2.2 GeV
+$ signal
+Yd = 0.01, Cud,s = Cmax
+ud,s
+JO
+1000
+Number
+500
+0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+K+ kinetic energy (GeV)6
+∗ Joshua.Berger@colostate.edu
+† gelor@uni-mainz.de
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diff --git a/v9E2T4oBgHgl3EQf2wjd/content/tmp_files/load_file.txt b/v9E2T4oBgHgl3EQf2wjd/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a0d65cdf89de21d25df758ff206beaf2ea99f96e
--- /dev/null
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@@ -0,0 +1,635 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf,len=634
+page_content='MITP-22-103  Dark Matter Induced Nucleon Decay Signals in Mesogenesis  Joshua Berger1, ∗ and Gilly Elor2, †  1Colorado State University, Fort Collins, Colorado 80523, USA  2PRISMA+ Cluster of Excellence & Mainz Institute for Theoretical Physics  Johannes Gutenberg University, 55099 Mainz, Germany  We introduce and study the first class of signals that can probe the dark matter in Mesogenesis  which will be observable at current and upcoming large volume neutrino experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The well-  motivated Mesogenesis scenario for generating the observed matter-anti-matter asymmetry neces-  sarily has dark matter charged under baryon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Interactions of these particles with nuclei can  induce nucleon decay with kinematics differing from sponanteous nucleon decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' We calculate the  rate for this process and develop a simulation of the signal that includes important distortions due to  nuclear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' We estimate the sensitivity of DUNE, Super-Kamiokande, and Hyper-Kamiokande  to this striking signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Mesogenesis mechanisms [1–4] utilize the Charge-  Parity (CP) violation of Standard Model (SM) meson  systems to generate the primordial baryon asymmetry  and the dark matter (DM) abundance of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Excitingly, Mesogenesis is highly testable [5, 6] and ex-  perimental searches are underway to probe signals di-  rectly linked to the generated baryon asymmetry [6–10]  (see overviews in [11–14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' However, a direct probe of the  DM in Mesogenesis has remained elusive, until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' In  this Letter we study DM induced nucleon decays (IND)  in Mesogenesis which can produce a striking signal at cur-  rent and upcoming neutrino detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' While the Meso-  genesis framework is the primary focus of this letter, the  methods developed here can be more broadly applied to  search for models containing dark baryons e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  The novel way in which Mesogenesis satisfies the  Sakharov conditions [18], is as follows: mesons produced  at late (MeV scale) times, having undergone CP violat-  ing processes, decay out of thermal equilibrium into dark  baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' This process generates an equal and opposite  baryon asymmetry between the dark and visible sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  For instance, the baryon asymmetry in B0-Mesogenesis  [1] is generated from the late time production of B0  s,d  mesons which undergo CP violating particle-antiparticle  oscillations and then quickly decay into a SM baryon and  a dark Dirac fermion ψB carrying SM baryon number −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  To evade washing out the generated baryon asymmetry  through e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' ψB → p e ¯νe, the ψBs must rapidly decay  into stable DM states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Mesogenesis DM consists of two components— a dark  Majorana fermion ξ and a dark complex scalar φB which  is charged under SM baryon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' These two stable  particles will compose the entirety of the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' As such the  DM halo will consist of a mixture of ξ and φB particles  which can scatter off target nuclei in neutrino detectors  to produce mono-energetic pions or kaons and missing  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' This process of IND appears experimentally as  nucleon decay but with different kinematics, and as such  current limits are essentially not constraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  The cross section for IND, as it arises in Mesogene-  sis, will be within reach of neutrino detectors and can be  searched for at the Deep Underground Neutrino Experi-  ment (DUNE) [22], Super-Kamiokande [23], and Hyper-  Kamiokande [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Furthermore, given the mono-energetic  (up to smearing effects) meson, such signals should be  distinct over SM backgrounds which primarily consist of  mesons produced in atmospheric neutrino processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  In order to study the details of the IND process, we  have developed a Monte Carlo event generation tool  within the GENIE [19, 20] software suite used to study  accelerator and atmospheric neutrino scattering events  [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Based on this tool, we are able to study the de-  tailed kinematics of the outgoing mesons produced dur-  ing DM scattering events and compare these with the  dominant atmospheric neutrino scattering background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  The event generation includes nuclear effects which smear  out the spectrum of mesons from one that is nearly mono-  energetic in the non-relativistic DM limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Furthermore,  these events are ready to be used for study in neutrino  experiments, where they can be passed through detector-  specific simulation software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  In this Letter we first characterize the IND signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' We  present the Mesogenesis specific parameter space which  can be targeted by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  We then detail the  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Next, we apply this simulation to estimate  the sensitvity of DUNE, Super-Kamiokande, and Hyper-  Kamiokande to benchmark models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Characterizing the IND Signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' — Generating the  baryon asymmetry in B-Mesogenesis [1, 3], requires the  existence of a TeV scale colored scalar (which may be  identified as a squark in a supersymmetric embedding  [4]) which mediates the baryon number conserving decay  of a B to ψB and a SM baryon through the (GeV scale)  effective operator:  Oab,c = Cab,cϵijk  �  ui  adj  b  � �  ψB dk  c  �  ,  (1)  where all fermions are right handed, though our results  easily generalize to left handed operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Cuadb,dc ≡  yuadbyψdc/M 2  Y , where MY is the mediator mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  ψB  decays to the DM through a Yukawa coupling Ld ⊃  yd ¯ψBξφB +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=', which is allowed by a stabilizing Z2 sym-  metry under which ψB is even and the DM particles odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='04165v1  [hep-ph]  10 Jan 2023  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Induced proton decay to a pion through Ou,dd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Since we remain agnostic about the dark sector, yd is  a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  However a motivated benchmark is  yd ≲ O(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='1) which results in the correct DM abundance  given an example UV embedding [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 1 generates the IND signal in Mesogenesis: when  kinematically allowed, an incoming ξ or φB scatters off a  proton or neutron by exchanging a ψB and produces an  energetic meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 1 depicts an example process —  incoming φB’s induce proton decay to π+ through Oud,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Similarly induced decay to kaons arises through Oud,s  and Ous,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' We consider searches at neutrino experiments  for the following two processes:  φB N → M ξ  if  mφB + mN > mM + mξ ,  (2a)  ξ N → M φ⋆  B  if  mξ + mN > mM + mφB ,  (2b)  where N = n0, p+ and M is a SM meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Recall that  ξ is Majorana, allowing any of the DM states to partici-  pate in this process when it is kinematically allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' For  decays induced by incoming φs, the kinetic energy of the  outgoing meson, to O(vDM), is given by  EM, kin  φBN→ξM =  m2  M − m2  ξ + (mN + mφB)2  2(mN + mφB)  − mM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (3)  Swap mξ ↔ mφB in the above to obtain the meson energy  from incoming ξs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The expected kinetic energy for each  process that arises in Mesogenesis is given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  If the struck nucleon is at rest, then the outgoing  meson is mono-energetic with energy given in Eq (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  However, the nucleons are moving with a momentum of  O(100 MeV) inside nuclei, smearing out of the meson sig-  nal (except for the case of scattering off hydrogen in wa-  ter Cherenkov detectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' We simulate the IND process,  carefully accounting for this smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Note that the  energies of these decays are shifted compared to sponta-  neous nucleon decay, with higher energies when φB scat-  ters and lower energies when ξ scatters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' This alters the  phenomenology of the Mesogenesis scenario compared  with proton decay models such as grand unified theo-  ries — the canonical targets of current nucleon decay  searches [25–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' As such, existing limits from nucleon  Initial  Final  Mediating  Meson  Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' ⟨σv⟩0  DM  Meson  Operator  EKin [GeV]  �  cm3/sec  �  φB  π+/π0  Oud,d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='6 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='2  10−21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='4 - 10−21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='0  ξ  π+/π0  Oud,d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='02 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='6  10−22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='5 - 10−21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='9  φB  K+/K0 Ous,d, Oud,s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='9  10−19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='7 - 10−19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='3  ξ  K+/K0 Ous,d, Oud,s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='04 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='3  10−20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='6 - 10−19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The induced nucleon decays allowed by kinemat-  ics (2) and Mesogenesis considerations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' We show the  corresponding flavorful variation of the opeator Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 1 that  generates the decay, the expected range of un-smeared ki-  netic energy of the outgoing meson computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (3),  and the stripped cross section defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  decay searches, with a few exceptions, do not constrain  the Mesogeneis signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Similar considerations were dis-  cussed in the context of Hylogenesis [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  The cross section for IND is obtained from the matrix  element:  AφBN→ξM  = ¯uξ(⃗pξ)  yd Cab,c  p2  ψB −m2  ψB  �  /pψB + mψB  �  (4)  × PR  �  W RR  0  − i  /pψB  mN W RR  1  �  uN(⃗pN) ,  with ¯uξ → ¯vξ for ξ N → M φ⋆  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Here, pψB = pξ−pφB and  PR is the right handed fermion projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The Wilson  Coefficients are constrained by a combination of LHC  searches for the mediator and flavor observables: Cmax  ud,d =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='07 TeV−2 and Cmax  ud,s , Cmax  us,d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='64 TeV−2 [6, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Since  INDs can lead to O(GeV) momentum transfer, we use  high q2 extrapolated lattice results from [32] for the form  factors W RR  0,1 [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Including the momentum dependence  of W1,2 negligibly affects the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The Ous,d and Oud,s  require different form factors but lead to similar signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Mesogenesis Parameter Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' — In addition to the  kinematic constraints on IND Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (2), the allowed param-  eter space is constrained by Mesogenesis-specific consid-  erations i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' generating the observed baryon asymmetry  and DM abundance:  mφB + mξ < mψB < mB − mp ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='34 GeV ,  (5a)  |mφB − mξ| < mp + me ≃ 938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='8 MeV ,  (5b)  mψB , mφB > mp − me .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  (5c)  mφB > mξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  (5d)  The regions in {mξ, mφB} space excluded by these con-  straints are shaded in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 2, while the white region is  allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (5a) ensured that ψB is light so that the  baryon asymmetry can be generated through the decay  B → BSM + ψB, while also being heavy enough to decay  into the DM ξ and φB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Decreasing the value of mψB cor-  responds to increasing the excluded green region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  For  mψB ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='1 GeV, there is no longer viable parameter  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (5b) is enforced to prevent ξ and φB from  coalescing into SM baryons, which would washout the  ΦB  S  山B  u  d  p  d  u3  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  ���  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Parameter space and kinetic energy contours for the eight different DM IND proceses arising in Mesogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Colored  regions are ruled out by kinematics Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (2) or mechanism considerations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Sold lines correspond to kinetic energy for  scattering off protons N = p, and the dashed off nucleons N = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' In each panel, we indicate the location of the representative  benchmark points: 1, 2, 3p, 4p for pions and 1, 2, 3k, 4k for kaons as summarized in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  baryon asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Proton decay through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (1) to  dark baryons is kinematically forbidden through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  A complete list of DM IND processes consistent with  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (2) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (5) is shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  There must exist dark sector interactions which de-  plete the DM and ensure the correct abundance [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Since  nφB − ¯nφB is related to the baryon asymmetry, φB must  always constitute some, but not all, of the DM: if mφB <  mξ, dark interactions could annihilate the entire ξ pop-  ulation into ψBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The measured ratio of DM to baryon  densities [34] implies that mφB ≃ 5mp, violating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  This motivates multi-component DM where we enforce  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 5d so that the symmetric component of φBs annihi-  late into ξs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Since the measured SM baryon asymmetry  is always balanced by an asymmetry in φBs, the observed  DM to baryon ratio ρDM ∼ 5ρB fixes the expected density  of ξ and φB particles in the halo: ρξ/ρφB = 5mp/mφB −1,  and ρtotal = ρφ + ρξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='4 GeV/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Given Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (5c),  there will always be a substantial asymmetric compo-  nent of DM, and so both INDs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (2) will be present  if kinematically allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  The scattering cross sections for the INDs are com-  puted from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' They scale roughly as ⟨σv⟩ ∝ m−2  ψB,  but the range of variation mψB ∼1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='3 GeV, leads to a  small effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' We parametrize the cross section as:  ⟨σv⟩M  DM ≡ (yd × Cudi,dj)2  m2  ψB[GeV−4] ⟨σv⟩M,0  DM  (6)  where M = π0, π+, K+, K0 and DM = φB, ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Val-  ues of the coupling stripped cross section ⟨σv⟩M,0  DM over  the allowed parameter space of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 2) are shown in Ta-  ble I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' For e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' yd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='1 and Cmax  udi,dj, the expected cross  section can be as large as 10−38 − 10−36cm3/sec in the  allowed parameter space for all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Meanwhile, the  estimated sensitivity at DUNE and Super-Kamiokande is  roughly 10−42 − 10−40cm2/sec .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Benchmark  mφB [GeV] mξ [GeV]  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='92  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='53  3p  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='38  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='6  3k  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='8  4p  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='17  4k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='55  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Benchmarks highlighting the possible signal  topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Benchmark 1 corresponds to Ekin,min  pφB→Mξ  ∼  Ekin,max  pφB→Mξ for both πs and kaons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Benchmark 2 corresponds  to Ekin,max  pφB→Mξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Benchmarks 3p and 3k correspond to the maxi-  mal Ekin  pφB→Mξ such that the incoming ξ process is still allowed  for production of πs and kaons respectfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Benchmarks 4p  and 4k highlight a region of {mξ, mφB} which would still lead  to a signal for small mψB (see also labels in Fig 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Signal  Monte  Carlo  and  Benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' —  Non-  relativistic DM striking a nucleon at rest and inducing  decay would lead to a mono-energetic signal with kinetic  energy given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (3) and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  We  pick benchmark points to highlight the possible signal  topologies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' these are defined in Table II and labeled in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  In the context of large nuclei, nuclear effects  including nuclear motion of the nucleons and final  state interactions of hadronic particles escaping the  nuclear remnant smear the outgoing meson energy, can  liberate additional hadrons, and can change the isospin  characteristics of the meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  In order to account for  these effects, as well as allow for future simulation of the  detailed detector response at neutrino experiments, we  have developed a Monte Carlo event generation tool for  the IND process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Signal events are generated using a modified version  of GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='2 [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' We employ the default tune  (G18 02a) throughout, though we considered other nu-  clear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' We found differences in the signal distri-  butions of order 10% between hA and hN models of the  intranuclear cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The current nucleon decay mod-  4  ule in GENIE was modified to allow for IND kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  The meson final states currently implemented are π0, π+,  K0  S/L, K+, and D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' This module propagates the outgo-  ing mesons through the nuclear remnant to the edge of  the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The kinematics of the IND process is fixed  given masses for the two DM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The cross sec-  tion is determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' For non-relativistic DM,  there is a small difference in the rate for interaction with  a high speed nucleon compared with a nucleon at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  We neglect this small difference of ∼10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Signals at Neutrino Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' — The current best  limits on spontaneous nucleon decays to pions are from a  Super-Kamiokande [27], which applied a pion momen-  tum cut of 1 GeV and is thus applicable to parts of  Mesogenesis parameter space — here the experimental  limit can be compared to an effective lifetime (τ Ind  N )−1 =  nDM⟨σv⟩DM [31, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  This yields an approximate,  conservative, limit of yd × Cud,d/Cmax  ud,d ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='03 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='1 for  70 MeV ≲ Ekin  π+ ≲ 870 MeV and Ekin  π0 ≲ 870 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Ex-  isting searches in kaon channels [29, 30] placed narrow  ranges of momentum cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Consequentially, the major-  ity of parameter space of interest is unconstrained by ex-  isting searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Super-Kamiokande is expected to have  sensitivity to IND signals given dedicated studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Hyper-  Kamiokande will improve on this with its larger exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Since DUNE is based on liquid argon time-projection  chamber (LArTPC) technology, it could have particular  sensitivity to certain models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  We assume kinetic energy thresholds as indicated in  Table III, based on the studies where available [37]  and otherwise based on Cherenkov thresholds or de-  tecting tracks/showers in LArTPCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Determining re-  construction efficiencies requires a dedicated study by  the respective experimental collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  In Super-  Kamiokande’s search for proton decay in the π+ν chan-  nel, no muon/pion separation was attempted [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' On  the other hand, the LArTPC experiment MicroBooNE  has demonstrated muon/pion separation at the 80%  level [38], consistent with other particle identification ef-  ficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' We therefore do not assume muon/pion sepa-  ration for the Water Cherenkov experiments, while we as-  sume perfect muon/pion separation for DUNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Searches  for similar signals of kaons in spontaneous nucleon decay  at Super-Kamiokande indicate reconstruction efficiencies  around or below 10% [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The full 3D imaging of  LArTPC detectors like DUNE may increase performance  in reconstructing complex topologies arising from kaon  decays, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' kaons above 40 MeV kinetic energy may be  efficiently be reconstructed as tracks before decay [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Sensitivity Estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' — The dominant background for  the channels we consider is expected to be inelastic scat-  tering of atmospheric neutrinos off nuclei, leading to ad-  ditional mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' By looking for off beam timing events,  beam-related backgrounds can be evaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  To study  the atmospheric neutrino background, we generate at-  mospheric neutrino events using GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='2, along  Particle  LArTPC  Cherenkov  Thresh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' [MeV] Thresh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' [MeV]  e±  30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='5  γ  30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='5  µ±  35  55  π±  35  72  p  80  481  TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Kinetic energy thresholds for particles assumed in  LArTPC (DUNE) and Water Cherenkov (Super- and Hyper-  Kamiokande) detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  with the Bartol atmospheric neutrino flux model [40] at  Soudan for DUNE and Kamioka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  From samples of DM signals and atmospheric neutrino  events, we look for events containing the relevant final  state meson for the channel considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Not all events  with such a meson should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Signal events  contain a single meson and no other activity other than  possible emissions from the nuclear remnant or byprod-  ucts of final state interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Thus, it is highly benefi-  cial to veto any activity beyond the expected meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' To  do so, we first apply the thresholds in Table III to all final  state particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Of the remaining particles that can be de-  tected, we veto on events that have anything other than a  meson of the expected type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' For the pion channels, order  1% of atmospheric neutrino scattering events lead to an  event matching these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' This leaves a search that  is not entirely background free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' For the kaon channels on  the other hand, single kaon events are only possible with  additional flavor-violating weak interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Searches in  these channels may thus be background free if kaon re-  construction is sufficiently good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' To get a sense of the  events after these vetoes we plot kinetic energy distribu-  tions of the remaining signal and background events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' a  few illustrative distributions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  We now estimate the sensitivity to yd×Cudi,dj/Cmax  udi,dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  For pion channels, we apply the selection described  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' To eliminate a majority of the background events,  we also require that the selected pion has a kinetic energy  within 100 MeV of its unsmeared value Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' For the  kaon channels, no further selection is required as we have  found that this channel is nearly background free, and we  determine the coupling that would lead to 5 signal events  over the assumed exposure of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' In cases  where there is background, we estimate the 2σ sensitivity  to the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The sensitivity results are summarized in  Table IV for the benchmarks listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Discovering an IND signal would be a direct probe of  the DM in Mesogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' In addition to providing exper-  imentalists with the tools needed to search for DM IND  signals at neutrino experiments, this letter paves the way  to a signal driven model building effort of the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Kinetic energy distributions for sample benchmark models at DUNE and Hyper-Kamiokande.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Super-Kamiokande, is  simply a rescaling of the rate at Hyper-Kamiokande by a factor of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Dashed lines indicate the un-smeared energy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Green lines indicate, where relevant, the assumed threshold for the detector to see the meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The top row correspond to  Benchmark 1, while the bottom corresponds to Benchmark 3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The Hyper-Kamiokande signal has a noticeable mono-energetic  spike corresponding to scattering of hydrogen, while the smeared distribution corresponds to scatterings off oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Benchmark Background yd(Cudi,dj/Cmax  udi,dj)  Background yd(Cudi,dj/Cmax  udi,dj)  Background yd(Cudi,dj/Cmax  udi,dj)  and Meson  DUNE  DUNE sensitivity  Super-K  Super-K sensitivity  Hyper-K  Hyper-K sensitivity  1 π+  118  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
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+page_content=' Estimated coupling sensitivity at DUNE with 400 kton-years of exposure, at Super-Kamiokande with 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='8 kton-  years of exposure, and at Hyper-Kamiokande with 1,900 kton-years of exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' We apply the solar minimum flux model for  all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The solar maximum model gives slightly different sensitivity estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  We thank Yun-Tse Tsai for comments on the draft and  Yue Zhao for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' The work of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' is sup-  ported by the National Science Foundation under Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' 2112789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' is supported by the Cluster of Excel-  lence Precision Physics, Fundamental Interactions and  Structure of Matter (PRISMA+ – EXC 2118/1) within  the German Excellence Strategy (project ID 39083149).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' thanks the Mainz Institute for Theoretical Physics  (MITP) of the Cluster of Excellence PRISMA+ (project  ID 39083149) for their hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
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+page_content='8 GeV  $ signal  m= = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='2 GeV  80  Yd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='01, Cud,s =  max  ud,s  60  40  20 -  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
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+page_content='8  K+ kinetic energy (GeV)6  ∗ Joshua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='Berger@colostate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='edu  † gelor@uni-mainz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='de  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Elor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Escudero, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Nelson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' D 99,  035031 (2019), arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='00880 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Elor and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' McGehee, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' D 103, 035005  (2021), arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='06115 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  [3] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Elahi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Elor, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' McGehee, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' D 105,  055024 (2022), arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='09751 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Alonso-´Alvarez, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Elor, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Nelson, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Xiao,  JHEP 03, 046 (2020), arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='10612 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Elor  and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Guerrera,  (2022),  arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='10553 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Alonso-´Alvarez, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Elor,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Escudero, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' D 104, 035028 (2021), arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='02706 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Hadjivasiliou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' (Belle), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' D 105, L051101  (2022), arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='14086 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Rodriguez, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Chobanova, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Cid Vidal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='  Soli no, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
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+page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
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+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
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+page_content='038 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
+page_content='05), and, through Oud,s,  for K+ are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E2T4oBgHgl3EQf2wjd/content/2301.04165v1.pdf'}
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diff --git a/w9FRT4oBgHgl3EQfhDe7/content/tmp_files/2301.13582v1.pdf.txt b/w9FRT4oBgHgl3EQfhDe7/content/tmp_files/2301.13582v1.pdf.txt
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@@ -0,0 +1,4650 @@
+arXiv:2301.13582v1  [math.AG]  31 Jan 2023
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES
+OVER FINITE FIELDS
+R´EGIS BLACHE AND EMMANUEL HALLOUIN
+Abstract. In this article, we consider weak del Pezzo surfaces defined over a
+finite field, and their associated, singular, anticanonical models.
+We first define arithmetic types for such surfaces, by considering the Frobe-
+nius actions on their Picard groups; this extends the classification of Swinnerton-
+Dyer and Manin for ordinary del Pezzo surfaces.
+We also show that some
+invariants of the surfaces only depend on the above type.
+Then we study an inverse Galois problem for singular del Pezzo surfaces
+having degree 3 ≤ d ≤ 6: we describe which types can occur over a given finite
+field (of odd characteristic when 3 ≤ d ≤ 4).
+Contents
+Introduction
+2
+1.
+Weak del Pezzo surfaces
+4
+1.1.
+A blow-up model
+5
+1.2.
+Roots, exceptional curves and geometric types
+6
+1.3.
+Arithmetic types
+8
+2.
+Singular del Pezzo surfaces
+9
+2.1.
+Anticanonical models: geometric aspects
+9
+2.2.
+Divisor class groups and zeta functions
+10
+3.
+Construction of weak del Pezzo surfaces of degree at least five
+13
+3.1.
+Degrees seven and eight
+13
+3.2.
+Degree six
+14
+3.3.
+Degree five
+15
+4.
+Construction of singular del Pezzo surfaces of degree four.
+16
+4.1.
+Pencils of quadrics, their Segre symbols, and geometric types
+16
+4.2.
+Morphisms to the projective line
+18
+4.3.
+Galois action on the singular quadrics, and arithmetic types
+23
+4.4.
+Quadratic module associated to a pencil of quadrics
+24
+4.5.
+Construction of degree four del Pezzo surfaces of any arithmetic type
+29
+5.
+Construction of singular del Pezzo surfaces of degree three.
+38
+5.1.
+Blowing up degree four surfaces
+38
+5.2.
+Other constructions
+39
+5.3.
+Type 36
+42
+Appendix A.
+Arithmetic types for degrees three to six
+44
+References
+48
+Date: February 1, 2023.
+2020 Mathematics Subject Classification. Primary 11G25, 14J26; Secondary 14G10, 11E12.
+Key words and phrases. del Pezzo surfaces over finite fields, zeta functions, quadratic modules.
+1
+
+2
+R. BLACHE AND E. HALLOUIN
+Introduction
+In this article, we study certain del Pezzo surfaces defined over a finite field.
+Recall that a smooth projective surface X is a weak del Pezzo surface when its
+anticanonical divisor −KX is big and nef; its degree is the self-intersection number
+d := K·2
+X. If moreover −KX is ample, we call X an ordinary del Pezzo surface.
+When a weak del Pezzo surface X is not ordinary, it contains absolutely irreducible
+curves with self intersection −2, that we shall call (−2)-curves in the following.
+The anticanonical model of a weak, non ordinary del Pezzo surface X is a singular
+surface, that we denote by Xs, and call a singular del Pezzo surface. Note that
+Xs has Du Val singularities, and is Gorenstein; for these reasons such surfaces are
+sometimes called Du Val del Pezzo or Gorenstein del Pezzo in the literature.
+The study of complex singular cubic surfaces (degree 3 del Pezzo surfaces over
+C) dates back to the nineteenth century [6, 23]. It has been generalized to del
+Pezzo surfaces along the twentieth century [13]. In particular we know all types of
+singularities (sometimes called the Dynkin types) that can occur in characteristic
+zero [12, Chapter 8]. Over an algebraically closed field of positive characteristic,
+some new types occur, but only in characteristic 2 [16].
+The interest on del Pezzo surfaces over finite fields is more recent. If X is an
+ordinary del Pezzo surface defined over the finite field Fq, where q = pm is a power
+of a prime, the Frobenius action σ∗ on Pic(X ⊗Fq) must preserve the anticanonical
+class and the intersection product. The group of automorphisms of Pic(X ⊗ Fq)
+with these properties is a (finite) Weyl group depending on the degree of the surface.
+Thus the image of the Galois group Gal(Fq/Fq) is a cyclic subgroup generated by
+the image of the Frobenius morphism σ; its conjugacy class is the arithmetic type
+of X. Swinnerton-Dyer [25] and Manin [21] construct tables of conjugacy classes in
+these Weyl groups in order to classify ordinary del Pezzo surfaces over finite fields
+(the table for degree 3 has been corrected in [2]).
+Many invariants of a del Pezzo surface only depend on its arithmetic type; this
+is the case for its zeta function [21, Theorem 27.1, Corollary 27.1.2]
+(0.1)
+Z(X, T )−1 = (1 − T )(1 − q2T ) det
+�
+I − qT σ∗| Pic(X ⊗ k)
+�
+The first aim of this paper is to extend the classification of Swinnerton-Dyer and
+Manin to weak del Pezzo surfaces, by defining their arithmetic type, and to give an
+expression for their zeta functions.
+We begin by classifying the possible singularities over the algebraic closure. Fol-
+lowing [7, 10], we define a geometric type. This is the configuration of negative
+curves on the surface X ⊗ k (corresponding to lines and singularities on the anti-
+canonical model), up to the action of the Weyl group. It is finer than the Dynkin
+type. The types are orbits of the Weyl action on the root bases in the lattice E9−d
+corresponding to the degree d.
+Coming back to the surface X, the Galois action must preserve the set Rirr of
+(−2)-curves. Thus the Frobenius maps to an element of its stabilizer Stab(Rirr).
+We define the arithmetic type of X as the conjugacy class of the Frobenius action
+in this last group.
+Here again, many properties of a weak del Pezzo surface only depend on its
+arithmetic type; this is true for its zeta function from (0.1), and we show that it
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+3
+remains true for its anticanonical model Xs (note that its geometric Picard lattice
+is the orthogonal of Rirr in the geometric Picard group of X).
+Theorem 1. We have the equality
+Z(Xs, T )−1 = (1 − T )(1 − q2T ) det
+�
+I − qT σ∗| Pic(Xs ⊗ k)
+�
+Our second aim is to construct such surfaces. When the finite field is small, not
+all are constructible [26]. For instance, over the field F2, there are no split ordinary
+del Pezzo surfaces of degree d ≤ 4 since there are at most four points in general
+position in P2(F2).
+This is the inverse Galois problem for singular del Pezzo surfaces over finite
+fields: in the ordinary case, it asks for which conjugacy classes of the Weyl group
+can arise as the conjugacy class of the Frobenius action. It has been solved for
+ordinary degree four surfaces in [26, Theorem 1.4], and for ordinary degree three
+and two surfaces in [20]. There is also a weaker version of this problem, asking for
+which integers can arise as the trace of the Frobenius action. It has been solved for
+ordinary del Pezzo surfaces, see [2].
+It should be easier for non ordinary del Pezzo surfaces since when we consider
+them as blowups of the projective plane, we relax the condition of blowing up points
+in general position to almost general position. But when the degree is at most four
+and the base field is not algebraically closed, some of the surfaces are no longer
+birational to the projective plane, and we have to provides other constructions.
+In this paper we solve the inverse Galois problem for singular del Pezzo surfaces
+of degree at least 5 over any finite field, and for surfaces of degree 3 or 4 when the
+characteristic is odd. We refer to Appendix A for the tables of arithmetic types for
+each degree 3 ≤ d ≤ 6.
+Theorem 2. There exists a weak, non ordinary del Pezzo surface of degree d and
+any arithmetic type T over the finite field Fq in the following cases
+1. we have d ≥ 5;
+2. we have d = 4 and Fq has odd characteristic.
+3. we have d = 3, Fq has odd characteristic and (q, T) /∈ {(3, 1), (3, 12)}.
+There does not exist any surface of degree 3 and arithm´etic type 1 or 12 over F3.
+The types corresponding to the degrees d ≥ 5 are not very difficult to construct.
+It turns out that these surfaces are birational to P2, and it is sufficient to blowup
+the projective plane at well chosen points, and to contract exceptional curves.
+In the case of degree four, we no longer consider a blowup model. We exploit an
+idea which is present in [3, 14, 24]. The anticanonical model of a del Pezzo surface
+a degree four is the base locus of a pencil of quadrics in P4. To such a pencil we
+associate a quadratic Fq|T ]-module (equivalently, a Frobenius algebra). Now fixing
+a geometric type, then an arithmetic type for a del Pezzo surface, gives a precise
+description of the quadratic module. We construct such modules with prescribed
+arithmetic properties, and their existence is sufficient to prove the second assertion
+of the above theorem. Note that these quadratic modules are rather different in
+even characteristic, since there a bilinear form over an odd dimensionnal vector
+space must be degenerate. This is why we do not consider that case here. However,
+nice normal forms have been described in this case [11], that should help solving
+this problem.
+
+4
+R. BLACHE AND E. HALLOUIN
+Note that we have used the mathematical software magma to construct explicit
+models (ie a couple of quadrics in P4) for all types of degree four singular del Pezzo
+surfaces over a given finite field. The code is freely available on the second author’s
+webpage.
+Finally, we construct degree three surfaces in different ways. Our main con-
+struction is by blowing up a point – not lying on any negative curve – on a well
+chosen degree four del Pezzo surface. We count the numbers of such points for each
+arithmetic type belonging to degree four; this allows us to show the existence of
+many degree three surfaces with given arithmetic type, but also – when there is
+no such point – the non existence result stated in the last sentence of the above
+Theorem. We also use other types of blow up (of the projective plane, or a degree
+four surface at a point lying on one or more exceptional curve) in order to construct
+the surfaces belonging to the remaining arithmetic types.
+The paper is organised as follows: in section 1, we describe weak del Pezzo
+surfaces, and recall their principal properties.
+This allows us to introduce the
+classification and to define the different types (geometric, then arithmetic) that
+we use in the rest of the article. Then we turn to the description of singular del
+Pezzo surfaces; we briefly describe their different groups of divisors, and we show
+Theorem 1 in section 2. The next section is devoted to the proof of the first assertion
+in Theorem 2. The fourth section is more technical: we treat the degree four del
+Pezzo surfaces over finite fields of odd characteristic; to such a variety, we associate
+a quadratic Fq|T ]-module. Then most of the work is devoted to describing the link
+between the arithmetic properties of the module and the arithmetic type of the
+surface; this allows us to prove the second assertion of Theorem 2. The last section
+mainly builds on the preceding one: we blow up the degree four surfaces at well
+chosen points in order to construct degree three surfaces. We also use some direct
+constructions; this allows us to prove the last two assertions of Theorem 2.
+We give the description of the different arithmetic types for degree d ≥ 3 in the
+Appendix.
+1. Weak del Pezzo surfaces
+We first define the smooth surfaces we shall consider in this article
+Definition 1.1. A smooth projective surface X defined over a field k is a weak del
+Pezzo surface when its anticanonical divisor −KX is
+(i) big, i.e. K·2
+X > 0;
+(ii) and nef, i.e. for any effective divisor D on X, (−KX) · D ≥ 0.
+It is an ordinary del Pezzo surface when −KX is ample. Its degree is d := K·2
+X.
+Note that since −KX is nef, the adjunction formula ensures that for any abso-
+lutely irreducible curve C on X, we have C · C ≥ C · (C + KX) = 2pa(C) − 2 ≥ −2.
+Moreover, if for such a curve this inequality is an equality, then we have C ·KX = 0,
+and from the Nakai-Moishezon criterion the anticanonical divisor is not ample: the
+surface X is a not an ordinary del Pezzo surface.
+It is well known [9, 7] that the geometry of del Pezzo surfaces depends to a
+large extent of its absolutely irreducible curves with negative self-intersection, the
+so-called negative curves. These are the generalization of the celebrated 27 lines on
+an smooth cubic hypersurface in P3.
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+5
+Definition 1.2. Let X denote a weak del Pezzo surface over the field k.
+An element D in the geometric Picard group Pic(X ⊗k) is an exceptional divisor
+when D·2 = D · KX = −1.
+An absolutely irreducible curve C on X whose class is an exceptional divisor is
+an exceptional curve.
+An element D in Pic(X ⊗k) is a root when it satisfies D·2 = −2 and D·KX = 0.
+We denote by R(X) the set of roots.
+When C is a curve on X whose class is a root, we say that C (or its class) is an
+effective root. If moreover C is absolutely irreducible, then it is a (−2)-curve.
+We denote by A the union of the (−2)-curves on X; this is a closed subscheme
+of X. We denote by U the complementary of A on X.
+Remark 1.3. First remark that the negative curves which are absolutely irreducible
+are isomorphic to P1 from the adjunction formula.
+The sets of exceptional divisors and roots are finite and depend up to isomor-
+phism only on the degree of the surface [9, II. Tables 2 et 3].
+Note also (contrary to the case of ordinary del Pezzo surfaces) that all exceptional
+divisors need not correspond to exceptional curves; see Lemma 1.11 for a numerical
+criterion.
+Finally, the sets of exceptional and (−2)-curves will be crucial in this article
+since they determine (up to an isomorphism) the geometric type.
+We first give a construction of weak del Pezzo surfaces as blow-ups of the pro-
+jective plane.
+1.1. A blow-up model. We assume k algebraically closed in this section.
+Definition 1.4. We denote by X(Σ) the surface obtained from the projective plane
+by successively blowing up the points in Σ := {p1, . . . , pr}
+π : X(Σ)
+Blpr
+−→ Xr → . . . → X2
+Blp1
+−→ X1 = P2
+where each pi, 1 ≤ i ≤ r is a closed point in the surface Xi.
+For 1 ≤ i ≤ r, we denote by Ei the total transform in X(Σ) of the exceptional
+divisor of the blowing-up Blpi : Xi+1 → Xi, and we write pi ≺ pi+1 when pi+1 is
+infinitely near to pi, ie when it lies on the exceptional divisor of Blpi in Xi+1.
+The Picard lattice of the surface X(Σ) is the group generated by E0 := π∗L, the
+total transform of the class L of a line in P2, and the Ei, 1 ≤ i ≤ r
+Pic(X(Σ)) = Zr+1 = ZE0 + ZE1 + · · · + ZEr
+endowed with the intersection product given by E·2
+0 = 1, E·2
+i
+= −1 for 1 ≤ i ≤ r
+and Ei · Ej = 0 for i ̸= j. The canonical class is given by
+KX(Σ) = −3E0 + E1 + · · · + Er
+From [9, III. Theorem 1], the surface X(Σ) is a weak del Pezzo surface of degree
+d = 9 − r if, and only if r ≤ 8, and the points in Σ are in almost general position:
+at each stage, the point pi does not lie on a (−2)-curve on Xi.
+The converse statement is almost true; actually we have the following description
+of weak del Pezzo surfaces [7, Proposition 0.4]
+Proposition 1.5. Let X denote a weak del Pezzo surface of degree d over an
+algebraically closed field k. Then we have 1 ≤ d ≤ 9, and if we set r = 9 − d, we
+must have one of the following
+
+6
+R. BLACHE AND E. HALLOUIN
+(i) r = 1 and X ≃ P1 × P1;
+(ii) r = 1 and X ≃ F2 the Hirzebruch surface;
+(iii) 0 ≤ r ≤ 8, and X ≃ X(Σ), where Σ := {p1, . . . , pr} consists of points in
+almost general position.
+1.2. Roots, exceptional curves and geometric types. In this section, we con-
+sider a weak del Pezzo surface X of degree d ≤ 7 over an algebraically closed field
+k, and we set r := 9 − d as above. Note that the description of del Pezzo surfaces
+of degrees 8 and 9 follows immediately from the preceding result.
+There exists some Σ := {p1, . . . , pr} consisting of points in almost general posi-
+tion such that X ≃ X(Σ); this choice allows us to identify Pic(X) ≃ Zr+1 as in the
+preceding section.
+Definition 1.6. Recall that R(X) is the set of roots in Pic(X)
+R(X) := {D ∈ Pic(X), D·2 = −2, D.KX = 0}
+We denote by Reff(X) ⊂ R(X) (resp. Rirr(X) ⊂ R(X)) the subset of effective
+roots (resp. of (−2)-curves) in Pic(X).
+Let R(X) denote the root module; it is the sub-Z-module of Pic(X) generated
+by Rirr(X).
+It is well known [9] that the set R(X) is a root system in the orthogonal (KX)⊥⊗
+Q of the canonical divisor KX. Under the identification of Pic(X) and Zr+1, it is
+sent on the root system Rd with basis {E0−E1 −E2−E3, E1 −E2, · · · , Er−1 −Er}.
+We denote by E9−d the intersection graph of this basis. It is the Dynkin diagram
+associated to the degree d, namely
+Degree d
+6
+5
+4
+3
+2
+1
+Dynkin diagram E9−d
+A2 × A1
+A4
+D5
+E6
+E7
+E8
+The group of automorphisms of the Picard group Pic(X) preserving the canonical
+divisor and the intersection product is isomorphic to the Weyl group associated to
+E9−d, which we denote by W(E9−d). It is generated by the reflections through the
+hyperplanes orthogonal to the roots, i.e. the sα : x �→ x + (x · α)α, α ∈ Rd.
+The following result [9, III Th´eor`eme 2] is fundamental for the geometric classi-
+fication of weak del Pezzo surfaces
+Proposition 1.7. Let X denote a weak del Pezzo surface of degree d ≤ 6. Then
+the set Reff(X) ∪ (−Reff(X)) is a closed and symmetric part of R(X).
+It is a root system in the space R(X)⊗Q, of which the set Rirr(X) forms a basis
+(and we call it a root basis).
+As a consequence, the free Z-module generated by Rirr(X) is equal to R(X).
+An immediate consequence is that since we have R(X) ⊂ K⊥
+X, and this last
+module has rank r = 9−d, there are at most r (−2)-curves on X. Their intersection
+graph has a strong geometric significance. It is sometimes called the Dynkin type
+of X.
+We are ready to define the first, geometric part of our classification
+Definition 1.8. The geometric type of X is the orbit of the image of its set of
+(−2)-curves Rirr(X) under the action of W(E9−d) on the set of bases for closed and
+symmetric parts of Rd.
+When X is ordinary, we say it has ordinary geometric type.
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+7
+Note that two isomorphisms between the lattices Pic(X) and Zr+1 differ by an
+element of the Weyl group. This is why we define the type as an orbit under the
+action of this group: this makes it independent of the choice of such an isomorphism,
+and of the blown-up points. Note also that the geometric type above is equivalent
+to the type from [10, Definition 3].
+The possible orbits can be deduced recursively from a theorem by Borel and
+de Siebenthal that classifies the closed symmetric parts of maximal rank in a root
+system up to the action of the Weyl group [22, p 29], [12, p 404]. They are given,
+degree by degree, in [12, Chapters 8 and 9]. We recall them in Appendix A as a
+column in the table of arithmetic types.
+For each orbit, one can choose a root basis in such a way that its intersection
+graph is a subgraph of the Dynkin diagram E9−d when d ≥ 5, and of the extended
+(or affine) Dynkin diagram �E9−d when 3 ≤ d ≤ 4; this is no longer true when
+d ∈ {1, 2}.
+Remark 1.9. The geometric type we have just defined is a finer invariant than
+the Dynkin type. For instance, if d = 6 there are two orbits of Dynkin type A1,
+depending on whether the (−2)-curve lies on the A1 or the A2 component of the
+root system: surfaces in the corresponding geometric types differ by the number of
+exceptional curves they contain. There are also two orbits for each of the Dynkin
+types 2A1 and A3 when d = 4.
+Note that for d ≥ 3, the possible geometric types of del Pezzo surfaces are in
+bijection with the orbits given by Borel-de Siebenthal theorem. This is no longer
+true when d ≤ 2: for instance, the orbits corresponding to certain types for d = 1
+or d = 2 only occur as geometric types of del Pezzo surfaces in characteristic 2 [16,
+Remark 1.5].
+A convenient way to represent the geometric type is to consider a new graph,
+containing the intersection graph of the (−2)-curves as a subgraph [7].
+Definition 1.10. We define the graph of negative curves associated to the surface
+X as the graph whose vertices are circles corresponding to (−2)-curves, and points
+corresponding to exceptional curves. Two vertices corresponding to curves C et C′
+are joined by n edges if we have C · C′ = n.
+Since the Weyl group preserves the intersection product, two surfaces sharing
+the same geometric type have the same graph of negative curves. The converse
+is true [10, Remark 4]; actually the geometric type of a weak del Pezzo surface
+is completely determined by its degree, its Dynkin type and the number of its
+exceptional curves. As a consequence, this is the way we will encode it in the tables
+of the Appendix.
+Here is a criterion for an exceptional divisor to be irreducible [9, Corollaire au
+Th´eor`eme III.2]
+Lemma 1.11. Let D denote an exceptional divisor of X. Then D is the class of
+an exceptional curve if, and only if we have D · R ≥ 0 for any effective root (or
+(−2)-curve) R.
+We end with [9, Lemme IV.2], that will be useful we we decribe the fibers of the
+desingularization of a songular del Pezzo surface
+
+8
+R. BLACHE AND E. HALLOUIN
+Lemma 1.12. Each connected component B of A is the support of a unique fun-
+damental cycle, which is the least effective root C such that for any irreducible
+component D of B we have C · D ≤ 0.
+The fundamental cycle depends only on the corresponding connected component
+of the Dynkin type. Actually it is the highest root of the root system associated to
+this component [12, Section 8.2.7].
+1.3. Arithmetic types. We assume here that k = Fq is a finite field. We denote
+by σ a generator of the absolute Galois group Γ := Gal(k/k).
+Let X denote a weak del Pezzo surface defined over k, having degree d. We
+denote by Rirr(X) the set of the (−2)-curves of X ⊗ k, and by Rirr a representative
+of the geometric type of X.
+We fix an isomorphism between Pic(X ⊗ k) and
+ZE0 + . . . + ZE9−d that send Rirr(X) to Rirr.
+The automorphism σ induces the automorphism Id ×σ of the surface X ⊗ k,
+and an automorphism (Id ×σ)∗ of the group Pic(X ⊗ k), that we denote by σ∗ in
+the following; it preserves the intersection pairing, and the canonical class since X
+is defined over k. Under the action of the isomorphism between Pic(X ⊗ k) and
+ZE0 + . . .+ ZE9−d, the image of σ∗ is an element of the Weyl group W(E9−d), and
+we get a morphism from Γ to W(E9−d), whose image is a finite cyclic group.
+Finally, σ∗ preserves the set of (−2)-curves, and its image in W(E9−d) must lie
+in Stab(Rirr), the stabilizer of Rirr in W(E9−d) which is the subgroup consisting of
+the θ such that θRirr = Rirr.
+This motivates the following
+Definition 1.13. Let X denote a weak del Pezzo surface defined over k, and Rirr
+the image of the set of its (−2)-curves described above.
+The arithmetic type T of X is the conjugacy class of the image of σ∗ in Stab(Rirr).
+Two isomorphisms between Pic(X ⊗ k) and ZE0 + . . . + ZE9−d sending the
+(−2)-curves of X to Rirr differ by an element of the above stabilizer, which is a
+subgroup of W(E9−d). Defining the type as a conjugacy class in this group makes
+it independent of the choice of such an isomorphism.
+Remark 1.14. We will see below that two singular del Pezzo surfaces such that the
+Frobenius actions on the Picard groups of their associated weak del Pezzo surfaces
+lie in the same conjugacy class in W(E9−d) (not in the stabilizer) can have different
+arithmetic properties, in particular different zeta functions. This is easily seen on
+the tables given in Appendix A that describe the different arithmetic types for
+degrees 3 ≤ d ≤ 6, in particular in degree 4 where we precise the corresponding
+conjugacy classes in W(D5).
+To end this section, we remark that two weak del Pezzo surfaces sharing the
+same arithmetic type have the same zeta function. Actually, since a weak del Pezzo
+surface is rational, we have the following [21, Theorem 27.1, Corollary 27.1.2]
+Proposition 1.15. For any n ≥ 1, the number of rational points of the weak del
+Pezzo surface X over the finite field Fqn is
+#X(Fqn) = q2n + qn Tr(σ∗n) + 1
+As a consequence, the zeta function of the weak del Pezzo surface X is
+Z(X, T )−1 = (1 − T )(1 − q2T ) det
+�
+I − qT σ∗| Pic(X ⊗ k)
+�
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+9
+2. Singular del Pezzo surfaces
+In this section, we consider a weak del Pezzo surface X of degree d defined over
+the finite field k = Fq.
+If X is not ordinary, its anticanonical divisor is no longer ample, and the mor-
+phism it (or one of its multiples) defines is no longer an embedding; its image Xs
+is singular.
+In this section, we first describe the geometry of this image, then we determine
+the divisor class groups of X and the associated singular variety in order to prove
+Theorem 1.
+2.1. Anticanonical models: geometric aspects.
+Definition 2.1. The anticanonical model of the surface X is the variety
+Xs := Proj
+∞
+�
+n=0
+H0(X, −nKX)
+and we denote by ϕ : X → Xs the associated morphism.
+The variety Xs just defined is the singular del Pezzo surface associated to X.
+Remark 2.2. One can also consider for i ≥ 1, the plurianticanonical linear system
+| − iKX| and the image X(i) of the morphism it defines. This variety is isomorphic
+to Xs as long as di ≥ 3 [9, V Th´eor`eme 1]. As a consequence, the variety Xs can
+be identified with the anticanonical image of X when d ≥ 3.
+The anticanonical model of a degree d del Pezzo surface is [17, Theorem 3.5]
+(i) if d = 4, the complete intersection of two quadrics in P4;
+(ii) if d = 3, a cubic in P3;
+(iii) if d = 2, a degree four hypersurface in weighted projective space P(1, 1, 1, 2);
+(iv) if d = 1, a degree six hypersurface in P(1, 1, 2, 3).
+We list below some properties of the surface Xs, and of the morphism ϕ [9, IV
+Th´eor`eme 1, V Proposition 1, V Th´eor`emes 1 et 2]. Recall that A is the union of
+the (−2)-curves on X, and U is its complementary
+Theorem 2.3. 1) The schematic fibers of the morphism ϕ are the points of U and
+the fundamental cycles. As a consequence, ϕ is birational, and we have ϕ∗OX =
+OXs.
+2) for all n ∈ Z, i > 0, we have Riϕ∗O(nKX) = 0.
+3) The surface Xs is normal. The singular points of Xs are the images of the
+fundamental cycles; they are rational double points.
+4) If we set O(KXs) := ϕ∗O(KX), then O(KXs) is locally free of rank 1, and
+for every integer n we have canonical isomorphisms
+O(nKXs) = ϕ∗O(nKX), O(nKX) = ϕ∗O(nKXs)
+In other words, ϕ is an isomorphism from U on its image Us, and each connected
+component of A is sent to a point which is a rational double point on Xs. We obtain
+all singular points of Xs in this way. The map ϕ is a minimal resolution of the
+singularities of Xs, and the Dynkin type of X is the dual graph to this resolution.
+Thus the singularity type of Xs is exactly the Dynkin type of X.
+The type of a singularity x of Xs is the type of the connected component corre-
+sponding to x in the intersection graph of (−2)-curves of X. Since all singularities
+are rational double points, their types fall in the ADE classification.
+
+10
+R. BLACHE AND E. HALLOUIN
+Note also that since O(KXs) is locally free of rank 1, this invertible sheaf corre-
+sponds to a Cartier divisor KXs.
+We end by recalling the following result [9, V Corollaire 2].
+Theorem 2.4. Let F denote a locally free sheaf on Xs. Then, for any i ≥ 0, we
+have
+Hi(Xs, F) = Hi(X, ϕ∗F)
+Moreover, we have Serre duality
+Hi(Xs, F) = H2−i(Xs, O(KXs) ⊗ F∨)∨
+From this result, we can describe the global sections of the Cartier divisors on
+Xs. Moreover we see that the sheaf O(KXs) is dualizing; since it is locally free, the
+surface Xs is Gorenstein.
+2.2. Divisor class groups and zeta functions. Let X denote a weak non ordi-
+nary del Pezzo surface defined over k = Fq, and Xs its anticanonical model. Then
+ϕ and Xs are defined over k since the anticanonical divisor is. In the same way,
+the varieties Sing(Xs) of dimension zero, and A of dimension 1 are defined over k.
+Recall that the geometric Picard group (the group of classes of Cartier divisors)
+Pic(X ⊗ k) identifies to the free Z-module generated by E0 and E1, . . . , Er, with
+r = 9 − d. It is equal to the group of classes of Weil divisors Cl(X ⊗ k) since X is
+smooth. Using this identification, we will no longer mention the dependance on X
+of some objects such as the root modules.
+We first describe the groups Cl(Xs ⊗ k) and Pic(Xs ⊗ k). Note that since Xs is
+normal, the groups of Cartier divisors and of invertible sheaves coincide.
+The restriction to U⊗k of Weil divisors X⊗k is surjective [15, Proposition II.6.5];
+its kernel consists of divisors whose support is contained in the complementary of
+U ⊗ k, i.e. in A ⊗ k. This last group is R, since it is generated by the irreducible
+components of A ⊗ k, and these are exactly the (−2)-curves.
+No principal divisor on X ⊗ k has support contained in A ⊗ k [19, p 225]. Thus
+R remains the kernel of the restriction of classes of Weil divisors from Cl(X ⊗ k)
+to Cl(U ⊗ k).
+The morphism ϕ induces an isomorphism from U to Us, and we have Cl(Us⊗k) =
+Cl(U ⊗ k). Now Xs ⊗ k \ Us ⊗ k has codimension 2 in Xs ⊗ k, and we deduce [15,
+Proposition II.6.5] that Cl(Us ⊗ k) = Cl(Xs ⊗ k). We get
+(2.1)
+0
+� R
+� Cl(X ⊗ k) = Pic(X ⊗ k)
+� Cl(Xs ⊗ k)
+� 0
+We come to the Picard group. The pull-back ϕ∗ : Pic(Xs ⊗ k) → Pic(X ⊗ k)
+gives rise to the exact sequence [4, Proposition 1]
+(2.2)
+0
+� Pic(Xs ⊗ k)
+ϕ∗
+� Pic(X ⊗ k)
+θ
+� R∨
+� Br(Xs ⊗ k)
+ϕ∗
+� Br(X ⊗ k)
+where we have set R∨ := Hom(R, Z), and θ comes from the intersection product:
+for any D ∈ Pic(X ⊗ k), θ(D) : R → Z is defined by θ(D)(R) = D · R.
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+11
+In other words, the group Pic(Xs ⊗ k) identifies to the following subgroup of
+Pic(X ⊗ k)
+Pic(Xs ⊗ k)
+=
+{D ∈ Pic(X ⊗ k), ∀R ∈ R, D · R = 0}
+=
+{D ∈ Pic(X ⊗ k), ∀R ∈ Rirr, D · R = 0}
+Since X ⊗ k is a rational surface, its Brauer group is trivial. We deduce the
+equality Coker θ = Br(Xs ⊗ k), and we have the following explicit description of
+this last group [4, Proposition 4]. The root module R is a subgroup of K⊥
+X, and
+Br(Xs ⊗ k) is the torsion subgroup of the quotient
+Coker θ = Br(Xs ⊗ k) =
+�
+K⊥
+X/R
+�
+tors
+It depends on the geometric type defined above (not only on the Dynkin type in
+general: for instance there are two orbits for the Dynkin type 4A1 when d = 2,
+one gives a trivial cokernel, the other an order 2 cokernel), and the different cases
+are described in [4, Theorem 6]. Note that θ is often surjective (this is always true
+when d ≥ 5).
+We turn to the study of the zeta function. We first determine the zeta function
+of the union A of the (−2)-curves in X.
+Since the set Rirr is stable under the action of Γ, the action of σ∗ on Pic(X ⊗ k)
+restricts to an action on R. Moreover this action preserves the intersection product,
+and it induces an automorphism on the singular graph; as a consequence it permutes
+the (−2)-curves. The matrix of the action of σ∗ over R with respect to the basis
+Rirr is a permutation matrix.
+We can express the zeta function of A in terms of the characteristic polynomial
+of this action
+Lemma 2.5. We have the equality
+Z(A, T ) = Z(Sing(Xs), T ) det(I − qT σ∗|R)−1
+Proof. Write A as the disjoint union of its connected components Ax, where for
+any x ∈ Sing(Xs)(Fq), Ax is the fiber (seen as a set) ϕ−1({x}).
+Let n ≥ 1 an integer; since ϕ is defined over k, we have Axσn = Aσn
+x . If we have
+x /∈ Sing(Xs)(Fqn), then Aσn
+x ∩ Ax = ∅, and we deduce that Ax(Fqn) = ∅.
+Now assume x ∈ Sing(Xs)(Fqn), and let us denote by Cx1, . . . , Cxk the absolutely
+irreducible components of Ax, whuch form the Coxeter-Dynkin graph associated to
+x.
+For any two curves Cxi and Cxj defined over Fqn, there is a unique (since the
+graph has no cycle) chain with minimal length in the graph between them. As the
+extremities of this chain are fixed by σn, the whole chain is fixed. We deduce from
+this fact that the subgraph of the Cxi, 1 ≤ i ≤ k defined over Fqn is connected; let
+us denote Nnx the number of its vertices. As the (−2)-curves have normal crossings,
+we deduce
+
+12
+R. BLACHE AND E. HALLOUIN
+#Ax(Fqn)
+=
+Nnx
+�
+i=1
+#Cxi(Fqn) −
+�
+1≤i<j≤Nn
+#(Cxi ∩ Cxj)(Fqn)
+=
+Nnx(1 + qn) −
+�
+1≤i<j≤Nnx
+Cxi · Cxj
+=
+1 + Nnxqn
+where the last equality comes from the fact that the sum of the intersection
+products is the number of edges in the subgraph whose vertices are the Cxi defined
+over Fqn. Since this graph is connected, has Nnx vertices and type A, D, E, it
+contains exactly Nnx − 1 edges.
+Finally, if none of the absolutely irreducible components of Ax is defined over
+Fqn, then σn acts without fixed points over the Coxeter-Dynkin graph associated
+to x. The only graphs admitting such an action are the A2k, and the action is the
+symmetry around the center of the graph, of order 2. In this case, the unique point
+defined over Fqn is the intersection of the effective roots Cxk and Cxk+1.
+Summing up, we have
+#Ax(Fqn) =
+� 1 + Nnxqn
+if x ∈ Sing(Xs)(Fqn)
+0
+else
+Thus, for any n ≥ 1 we have
+#A(Fqn) = # Sing(Xs)(Fqn) + Nnqn
+where Nn is the number of (−2)-curves on X defined over Fqn.
+Replacing this expression in the usual expansion of zeta functions, we get
+Z(A, T ) = exp
+
+�
+n≥1
+#A(Fqn)T n
+n
+
+ = Z(Sing(Xs), T ) exp
+
+�
+n≥1
+Nn
+(qT )n
+n
+
+
+Since the matrix of σ∗n with respect to the basis Rirr(X) is a permutation matrix,
+its trace is exactly the number of elements in the basis which remain invariant under
+the action of σ∗n. We deduce that Nn = Tr(σ∗n|R), and the result comes from
+the expansion of the characteristic polynomial of an endomorphism in terms of the
+traces of its iterates.
+□
+We are ready to prove Theorem 1.
+Proof. From the above results, the following equalities hold for any n ≥ 1
+#Xs(Fqn)
+=
+#Us(Fqn) + # Sing(Xs)(Fqn)
+excision for Xs
+=
+#U(Fqn) + # Sing(Xs)(Fqn)
+isomorphim U ≃ Us
+=
+#U(Fqn) + #A(Fqn) − qn Tr(σ∗n|R)
+Lemma 2.5
+=
+#X(Fqn) − qn Tr(σ∗n|R)
+excision for X
+Now from Proposition 1.15, we have #X(Fqn) = q2n + qn Tr(σ∗n) + 1, and
+#Xs(Fqn) = q2n + qn (Tr(σ∗n) − Tr(σ∗n|R)) + 1
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+13
+If we pass to zeta functions as in the proof of Proposition 1.15, we get
+Z(Xs, T )−1 = (1 − T )(1 − q2T ) det
+�
+I − qT σ∗| Pic(X ⊗ k)
+�
+det(I − qT σ∗|R)−1
+Tensoring the exact sequence of Z-modules (2.2) with C, the torsion disappears
+and we get the exact sequence of vector spaces
+0
+� Pic(Xs ⊗ k) ⊗ C
+� Pic(X ⊗ k) ⊗ C
+� R∨ ⊗ C
+� 0
+Finally, the action of σ∗ on R∨ ⊗C is adjoint to that of σ∗−1 over R ⊗C, and they
+share the same characteristic polynomial. Finally the matrix of σ∗ in the basis Rirr
+of the the Z-module R is a permutation matrix; thus the actions of σ∗−1 and σ∗
+also share the same characteristic polynomial. We get the equality
+det
+�
+I − qT σ∗| Pic(X ⊗ k)
+�
+= det(I − qT σ∗|R) det
+�
+I − qT σ∗| Pic(Xs ⊗ k)
+�
+This ends the proof.
+□
+3. Construction of weak del Pezzo surfaces of degree at least five
+The aim of this section is to show that for any finite field Fq and arithmetic
+type belonging to a degree d ≥ 5, there exists a weak del Pezzo surface of this type
+defined over Fq. In order to do this, we give explicit constructions.
+These surfaces are birational to the projective plane P2 [7, Lemma 9.3], and our
+constructions are sequences of blowups and contractions. The fundamental remark
+is that the configurations of points to be blown-up exist over any finite field. For
+instance, we will often need three collinear points, but this is possible over any
+finite field since a line defined over Fq contains q + 1 rational points.
+We will often use the graphs of negative curve from Definition 1.10 in order to
+represent the geometric type.
+3.1. Degrees seven and eight. First note that in this case, there is only one
+geometric type (of Dynkin type A1), and one arithmetic type.
+From Proposition 1.5, the only degree 8 weak, non ordinary del Pezzo surface is
+the Hirzebruch surface F2; the only degree 8 singular del Pezzo surface is its image
+by the contraction of its negative section. We construct these surfaces from degree
+7 surfaces below.
+Again from Proposition 1.5, all the degree 7 del Pezzo surfaces over an alge-
+braically closed field are obtained from the projective plane P2 by two successive
+blowups at points p1, p2. Such a surface is not ordinary if and only if the center of
+the second blowup is a point p2 ≻ p1 of the exceptional divisor of the first one. In
+this case, the effective root is the strict transform of this exceptional divisor, whose
+class in the Picard group is E1 − E2. We get the following graph of negative curves
+E1 − E2
+E2
+E0 − E1 − E2
+In order to get a non ordinary del Pezzo surface of degree 7 defined over the finite
+field Fq, we have to choose rational points p1, p2 for the centers of the blowups.
+The Hirzebruch surface can be obtained from the degree 7 non ordinary del Pezzo
+surface by contracting the strict transform of the line (p1p2) (the unique line in P2
+passing trough p1 and whose strict transform after the first blowup passes trough
+p2).
+
+14
+R. BLACHE AND E. HALLOUIN
+3.2. Degree six. Recall that the Weyl group is generated by the reflections sE1−E2,
+sE2−E3 and sE0−E1−E2−E3. It is isomorphic to the dihedral group of order 12.
+The possible geometric types are given in [12, Section 8.4.2] or [7, Proposition
+8.3]. Except in one case, we follow the choices made in this last article for the set of
+roots representing the corresponding orbit. As a consequence, most of the graphs
+of negative curve are already described in the above mentioned article, and we do
+not recall them.
+We refer to the corresponding table in the appendix for the different arithmetic
+types.
+A1(4) to obtain the split surface (arithmetic type 1) we blowup three non collinear
+points rational points p1 ≺ p2 and p3 (in other words, the strict transform
+of the line (p1p3) after the first blowup does not pass through p2).
+The stabilizer of the root E1−E2 is the subgroup of order 2 generated by
+the reflection sE0−E1−E2−E3. Its elements fall into two conjugacy classes.
+In order to construct a surface of arithmetic type 2, we will contract
+a line on a weak degree 5 del Pezzo surface X(Σ) of geometric type A1,
+obtained by blowing-up four points p1, . . . , p4 in P2, the first three being
+collinear (see the degree 5 surface of geometric type A1 below).
+If we contract the exceptional curve E0 − E3 − E4 on such a surface, the
+vertices E3 and E4 disappear, and we obtain a degree 6 del Pezzo surface
+with the graph of negative curves
+E0 − E1 − E2 − E3
+E1
+E0 − E1 − E4
+E0 − E2 − E4
+E2
+Assume that the first two points p1, p2 ∈ P2(Fq2) are conjugate over
+Fq, and p3, p4 are defined over Fq. The exceptional curve E0 − E3 − E4
+is defined over Fq. Contracting it, we get a surface X defined over Fq of
+degree 6, of the geometric type under consideration, which is not split: this
+is a surface of arithmetic type 2.
+A1(3) we blow-up three collinear points in P2; we get the three exceptional curves
+E1, E2, E3, and the effective root E0 − E1 − E2 − E3 which corresponds to
+the strict transform of the line through the pi.
+The stabilizer of the root in the Weyl group W(E6) is the subgroup
+generated by the reflections sE2−E1 and sE2−E3, which is isomorphic to the
+symmetric group S3.
+There are three conjugacy classes in this group, corresponding to the
+three arithmetic types 3, 4, 5. We construct a surface of each type in the
+following way
+3 : the points p1, p2, p3 are defined over Fq;
+4 : the points p1, p2 = pσ
+1 are defined over Fq2, and p3 over Fq;
+5 : the points p1, p2 = pσ
+1, p3 = pσ2
+1
+are defined over Fq3.
+2A1 we blowup three collinear points p1 ≺ p2 and p3 (the point p2 is the in-
+tersection of the strict transform of the line (p1p3) with the exceptional
+curve of the first blowup). The exceptional curves on the resulting surface
+are E2, E3, and the (−2)-curves correspond to the classes E1 − E2 and
+E0 − E1 − E2 − E3.
+The stabilizer of the set {E1 −E2, E0 −E1 −E2 −E3} in the Weyl group
+is trivial, and there is exactly one arithmetic type, with number 6.
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+15
+A2 we blowup three non collinear points p1 ≺ p2 ≺ p3 (the point p3 is not the
+intersection of the strict transform of (p1p2) with the exceptional divisor
+of the second blowup). The exceptional curves correspond to the classes
+E0 − E1 − E2, E3, and the (−2)-curves to E1 − E2, E2 − E3.
+The stabilizer of the set {E1−E2, E2−E3} in the Weyl group is generated
+by the reflection sE0−E1−E2−E3, it has order 2. As a consequence, there are
+two arithmetic types, one split that can be constructed by choosing rational
+points above, the other non-split.
+In order to construct the non-split one, we start with a degree 5 del
+Pezzo surface of geometric type A2 and arithmetic type 7 (see below). If
+we contract the exceptional curve with class E0 −E1 −E2 (which is defined
+over Fq), we get a degree 6 del Pezzo surface of arithmetic type 8.
+A1A2 we blowup three collinear points p1 ≺ p2 ≺ p3, where the point p3 is the
+intersection of the strict transform of (p1p2) with the exceptional divisor
+of the second blowup. We get an exceptional curve of class E3, and (−2)-
+curves corresponding to the classes E1 − E2, E2 − E3, E0 − E1 − E2 − E3.
+Once again, the stabilizer is trivial and the only arithmetic type (number
+9) is split.
+3.3. Degree five. Here the Weyl group is generated by the reflections sE1−E2,
+sE2−E3, sE3−E4 and sE0−E1−E2−E3. It is isomorphic to the symmetric group S5.
+In this section, we shall not systematically follow the choices made in [7, Propo-
+sition 8.5] for the classes of (−2)-curves representing each geometric type (see also
+[12, Section 8.5.1] for a list of these types). The reason is that there are many ways
+to construct a del Pezzo surface of a given geometric type by playing on the con-
+figuration of the points we blow up. Here we adopt the configuration that seems
+best adapted to the description of the arithmetic types (in some sense the most
+“symmetric ” one), and it need not coincide with the one described in the above
+article. We rewrite the graphs when they differ from [7].
+We refer to the corresponding table in the Appendix for the different arithmetic
+types.
+A1 We blowup four points p1, . . . , p4 in P2 with the first three collinear. The
+only (−2)-curve is the strict transform of the line through these points, its
+class is E0 − E1 − E2 − E3; moreover there exist seven exceptional curves,
+and the graph of negative curves is
+E0 − E1 − E2 − E3
+E1
+E2
+E3
+E0 − E1 − E4
+E0 − E2 − E4
+E0 − E3 − E4
+E4
+The stabilizer of the root E0−E1−E2−E3 is generated by the reflections
+sE1−E2, sE2−E3, and is isomorphic to the symmetric group S3. There are
+three conjugacy classes, giving rise to three arithmetic types.
+The corresponding surfaces are easily constructed, playing on the defi-
+nition fields of the points p1, p2, p3 (we always choose p4 ∈ P2(Fq))
+1 : the three points are in P2(Fq);
+2 : there are two conjugate points in P2(Fq2), and one in P2(Fq);
+
+16
+R. BLACHE AND E. HALLOUIN
+3 : the three points are conjugate in P2(Fq3).
+2A1 Here we follow [7, Proposition 8.5 2.]; we blowup p1 ≺ p2, p3 ≺ p4 such
+that the strict transform of the line (p1p3) does not meet the exceptional
+divisor of the first blowup at p2 nor of the third at p4. We get two disjoint
+(−2)-curves with classes E1 − E2 and E3 − E4, and five exceptional curves.
+The stabilizer of the set of negative curves is the order two subgroup of
+W(A4) generated by sE1−E3 ◦ sE2−E4.
+We get the two arithmetic types in the following way
+4 : if p1 ≺ p2, p3 ≺ p4 are defined over Fq, the surface is split;
+5 : we choose couples of conjugate points p1 ≺ p2, p3 = pσ
+1 ≺ p4 = pσ
+2
+defined over Fq2.
+A2 We blow up p1, p3, p4 collinear, then p2 ≻ p1 which does not lie on the strict
+transform of the line (p1p3); we obtain the following graph of negative curves
+E1 − E2
+E0 − E1 − E3 − E4
+E0 − E1 − E2
+E2
+E3
+E4
+The stabilizer of the set of (−2)-curves is the subgroup of the Weyl group
+generated by sE3−E4; it has order 2 and we get two arithmetic types. These
+are easily described
+6 : if the pi are defined over Fq, the surface is split;
+7 : if we choose p4 = pσ
+3 defined over Fq2, it is not.
+A1A2 We blowup p1 ≺ p2 ≺ p3 collinear and p4 ∈ P2 outside the line (p1p2).
+The stabilizer is trivial, and the only arithmetic type is split.
+A3 We blowup p1 ≺ p2 ≺ p3 ≺ p4 where p1, p2, p3 are not collinear.
+The
+stabilizer is trivial, and the only arithmetic type is split.
+A4 We blowup p1 ≺ p2 ≺ p3 ≺ p4 where p1, p2, p3 are collinear. The stabilizer
+is trivial, and the only arithmetic type is split.
+4. Construction of singular del Pezzo surfaces of degree four.
+We adopt a different point of view in this section. Actually not all weak degree
+four del Pezzo surfaces are birational to the projective plane over a finite field, and
+the preceding constructions, which use sequences of blowups and contractions, no
+longer suffice to describe all types.
+We shall mostly use a well known property of these surfaces: their anticanonical
+model is the base locus of a pencil of quadrics in projective space P4. The classi-
+fication of these objects is different in characteristic two, and we assume that the
+characteristic of our base field is odd in this section.
+We shall show that singular del Pezzo surfaces of every arithmetic type exist
+over any finite field with odd cardinality.
+4.1. Pencils of quadrics, their Segre symbols, and geometric types. In
+this section, we work over the field k, an algebraic closure of a finite field of odd
+characteristic.
+We recall [12, Section 8.6.1]. The results there are described over the field of com-
+plex numbers, and our aim is to show that they remain valid over any algebraically
+closed field of odd characteristic.
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+17
+The anticanonical model of a weak del Pezzo surface of degree 4 is the base locus
+of a pencil of quadrics in projective space P4 [17, Theorem 3.5]. We denote by Q0
+and Q∞ two distinct quadrics of this pencil. This gives us two quadratic forms over
+the vector space k5, and two symmetric 5×5 matrices; we denote the characteristic
+polynomial of the pencil corresponding to this basis by F(λ, µ) := det(λQ0+µQ∞).
+A theorem of Kronecker [27, Theorem 3.1] tells us that the vector space k5 can
+be written as an orthogonal direct sum of subspaces of two types:
+• a non singular space, say of dimension d, over which the determinant has
+degree d;
+• some basic singular spaces; here the equations of the quadrics take a very
+special form with respect to a well chosen basis.
+A direct calculation from such equations shows that the base locus of a pencil of
+quadrics has one-dimensional singular locus as long as the space contains a basic
+singular space. Thus the space k5 must be non-singular for the pair of quadratic
+forms. Note that this is no longer true in characteristic two, even for ordinary
+degree 4 del Pezzo surfaces [11], since a quadratic form over an odd dimensional
+vector space must be degenerate.
+We assume in the following that the quadric Q∞ is non-singular; thus the poly-
+nomial P(t) := F(1, t) has degree 5. This is a harmless assumption since we assume
+k algebraically closed.
+We follow Waterhouse [27, Section 1] and associate to our pair of quadratic forms
+a quadratic form Φ on a k[T ]-module of finite length. If ϕ0, ϕ∞ denote the bilinear
+forms on k5 associated to Q0 and Q∞, then ϕ∞ is non degenerate, and there is
+an unique endomorphism u of k5 such that ϕ∞(u(x), y) = ϕ0(x, y) for any vectors
+x, y. Moreover u is symmetric with respect to ϕ∞.
+The vector space k5, endowed with the action of u, becomes a k[T ]-module of
+finite length that we denote by M. The non degenerate form ϕ∞ defines an isomor-
+phism Φ of k[T ]-modules between M and its dual M ∗ := Homk[T ](M, k(T )/k[T ]).
+Consider the factorisation of P over k, P(T ) = �t
+i=1(T − θi)mi; we get an
+isomorphism of k[T ]-modules
+M = ⊕t
+i=1 ⊕ei
+j=1 (k[T ]/((T − θi)mij))nij
+As a consequence, we can associate to M the following Segre symbol
+(4.1)
+[(m11 . . . m11
+�
+��
+�
+n11 ×
+. . . m1e1 . . . m1e1
+�
+��
+�
+n1e1 ×
+) . . . (mt1 . . . mt1
+�
+��
+�
+nt1×
+. . . mtet . . . mtet
+�
+��
+�
+ntet ×
+)]
+Note that we remove the parentheses around the ith term when it contains exactly
+one integer, ie when we have ei = ni1 = 1.
+As in [27, Section 2], one can write the restrictions of the quadratic forms to a
+submodule of the form k[T ]/(T − θ)m as (here res denotes the residue in the sense
+of Laurent series)
+Q0(x) = res(T (T − θ)−mx2), Q∞(x) = res((T − θ)−mx2)
+since every element in (k[T ]/(T − θ)m)× is a square.
+In the basis ((T −θ)m−1, . . . , T −θ, 1), the expressions of these forms are exactly
+those given in [12, Equation (8.21)]. Now the analytic expressions for the rational
+double points of type An or Dn (the only ones occuring for degree 4 singular del
+Pezzo surfaces) are exactly the same in characteristic 0 or odd [1], and we deduce
+
+18
+R. BLACHE AND E. HALLOUIN
+that the following one to one correspondance between the geometric types and
+(some of the) Segre symbols [12, Table 8.6] remains true in our setting
+Type
+Number
+Segre
+of singularity
+of lines
+symbol
+Ordinary
+16
+[11111]
+A1
+12
+[2111]
+2A1
+9
+[221]
+2A1
+8
+[(11)111]
+A2
+8
+[311]
+3A1
+6
+[(11)21]
+A1A2
+6
+[32]
+A3
+5
+[41]
+Type
+Number
+Segre
+of singularity
+of lines
+symbol
+A3
+4
+[(21)11]
+4A1
+4
+[(11)(11)1]
+2A1A2
+4
+[3(11)]
+A1A3
+3
+[(21)2]
+A4
+3
+[5]
+D4
+2
+[(31)1]
+2A1A3
+2
+[(21)(11)]
+D5
+1
+[(41)]
+Some Segre symbols do not give rise to singular degree 4 del Pezzo surfaces. For
+instance the symbol [(11111)] gives rise to two collinear quadratic forms over k5,
+and the base locus of the pencil is a quadric in P4.
+We shall often use the following remark along this section
+Remark 4.1. These symbols carry the following informations: the number of differ-
+ent terms (ie the number t in (4.1), each couple of parentheses counts for one term)
+is equal to the number of singular quadrics in the pencil, and for such a quadric,
+its co-rank is the number of terms in the corresponding parenthesis. Actually we
+remark that the possible ranks for the singular quadrics in the pencil are 3 or 4.
+In the following, we shall often use the Segre symbol instead of the geometric
+type for our contructions. Moreover, we always choose the same set of (−2)-curves
+as in [7, Proposition 6.1] to represent the orbit.
+4.2. Morphisms to the projective line. We continue to work over an alge-
+braically closed field of odd characteristic in this section. We keep on with the
+notations of the preceding section.
+A convenient way to classify the arithmetic types for an ordinary del Pezzo
+surface is to visualize the Galois action on a graph consisting of its conic bundles
+[18, pages 8 and 9]. A non ordinary del Pezzo surface no longer has conic bundles,
+due to the presence of (−2)-curves; the aim of this section is to define a new graph
+for each geometric type, which is very close to the original one, whose vertices
+represent
+• the fibers of rational maps or morphisms from X to the projective line,
+• the classes of (−2)-curves.
+We will use these graphs in the next sections in order to control the arithmetic
+types.
+Let ϕ : X → P1 denote a surjective morphism. Since X is a rational surface, the
+generic fiber of ϕ is isomorphic to P1. Moreover all its fibers are linearly equivalent.
+We deduce from the adjunction formula that their class C satisfies C·2 = 0 and
+C · KX = −2; moreover we have ϕ = ϕ|C|.
+Recall [3, Theorem 2]: there exist exactly ten classes in Pic(X) satisfying the
+above equations (note that they form complementary couples in the sense that we
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+19
+have Ci + C′
+i = −KX for each i)
+Ci = E0 − Ei, C′
+i = 2E0 −
+5
+�
+j=1
+Ej + Ei, 1 ≤ i ≤ 5
+Definition 4.2. We set C := ∪5
+i=1{Ci, C′
+i} in the following. We represent these
+classes in the following graph, in which the couples are materialized by vertical
+dashed lines.
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+In the case of singular del Pezzo surfaces, the elements of C no longer correspond
+to conic bundles. But they still correspond to rational maps from X to the projec-
+tive line, as we shall see below. We will adapt the graph according to the geometric
+type, by adding vertices corresponding to (−2)-curves, and an edge between a class
+in C and such a curve when they intersect positively.
+From the description of the elements in C on one hand, and of the (−2)-curves
+(recall that their classes have the form Ei − Ej or E0 − Ei − Ej − Ek) on the other,
+we see that we must have C · R ∈ {−1, 0, 1}.
+Let C ∈ C be such that we have C · R = −1 for some (−2)-curve R; we deduce
+that C − R ∈ C . Thus R is a fixed component of the linear system |C|, and ϕ|C| is
+a rational map, but not a morphism.
+We deduce that there exists at most
+NX := #{C ∈ C , ∀R ∈ Rirr(X), C · R ≥ 0}
+morphisms from X to P1.
+Our strategy is to construct exactly NX such morphisms by considering the
+singular quadrics in the pencil defining the anticanonical model Xs. Recall that
+P denotes the characteristic polynomial of the pencil, and that θ1, . . . , θt are its
+(pairwise distinct) roots.
+Let θi denote a simple root of P, if any; the ith part of the Segre symbol (4.1) is
+a 1. From remark 4.1, the quadric Qi := Q0 − θiQ∞ is a singular quadric in P4 of
+rank 4. Thus it is a cone whose vertex is a point vi and base a non singular quadric
+qi := Qi ∩ Hi for some hyperplane Hi not containing vi. Moreover the vertex vi is
+not contained in Xs since θi is a simple root of P.
+The projection P4 → Hi from vi restricts to a double covering Xs → qi. Now qi
+is non singular, thus it is isomorphic to P1 × P1, and the projections on the factors
+give rise to two morphisms from Xs to P1. Composing them with the anticanonical
+morphism gives two morphisms ϕi1, ϕi2 : X → P1.
+Note that we have morphisms from Xs to P1, thus their fibers are Cartier divisors
+on Xs, and their classes in Pic(X) satisfy C · R = 0 for all (−2)-curves R on X.
+Finally, the fibers of the above morphisms from Xs to P1 are of the form Π∩Q∞,
+where Π is some plane containing one of the lines of qi and passing through vi.
+These are plane conics. For any point p ∈ Xs, the hyperplane H of P4 tangent
+to Qi at p intersects Qi along the union of two planes Π1 and Π2, and we have
+H ∩ Xs = H ∩ Qi ∩ Q∞ = (Π1 ∩ Q∞) ∪ (Π2 ∩ Q∞). We deduce that the sum of the
+
+20
+R. BLACHE AND E. HALLOUIN
+classes of the fibers of these two morphisms is exactly the class of the anticanonical
+divisor: they are complementary.
+Now assume that θi is a multiple root of P, and that Qi has rank 4 (this means
+that the ith part of the Segre symbol (4.1) satisfies ei = ni1 = 1 and mi1 = mi > 1).
+The same construction as in the preceding case yields two morphisms from Xs\{vi}
+to P1 (note that the vertex vi of Qi is a singular point of Xs here).
+Consider the blowup Blvi(P4) → P4; then Blvi(Xs) is the strict transform of
+Xs [15, Corollary II.7.15], and the two morphisms above become morphisms from
+Blvi(Xs) to P1. The fiber of vi in the blowup Blvi(Xs) → Xs contains only (−2)-
+curves, and we must have a morphism X → Blvi(Xs) since X is the minimal
+desingularization of Xs. Composing, we get two morphisms ϕi1, ϕi2 : X → P1.
+The last case to be considered is when θi is a multiple root of P, and Qi has rank
+3; here the ith part of the Segre symbol (4.1) contains exactly two terms between
+parentheses. In this case Qi is a cone with vertex ℓi ≃ P1 and base a smooth
+plane conic ci. The projection with vertex ℓi from P4 \ ℓi to this plane induces a
+morphism from Xs \ ℓi ∩ Xs to ci ≃ P1 (a rational map Xs ��� P1).
+The blowup Blℓi(P4) → P4 restricts to Blℓi∩Xs(Xs) → Xs as above, and we
+get a morphism Blℓi∩Xs(Xs) → P1 from the above rational map.
+Once again,
+the fibers above the points in ℓi ∩ Xs in the morphism Blℓi∩Xs(Xs) → Xs only
+contain (−2)-curves, and we get a morphism X → Blℓi∩Xs(Xs) from the minimal
+desingularization, from which we get a morphism ϕi1 : X → P1.
+Summing up, we have constructed 2a + 2b + c morphisms from X to P1, where
+a is the number of simple roots of P, b the number of multiple roots corresponding
+to a rank 4 quadric in the pencil, and c the number of rank 3 quadrics in the pencil.
+The following table contains the graphs we mentioned at the beginning of the
+section. It is sufficient to prove the next proposition for each geometric type.
+Note that for each type we use an empty square to denote the classes in C
+that intersect negatively a (−2)-curve, and a full square for the other ones; we
+also denote by ri the curve with class Ei − Ei+1, and by rijk the curve with class
+E0 − Ei − Ej − Ek
+Table 1: Conics-(−2)-curves graphs
+Geom. Type
+Conics-(−2) Curves Graph
+N
+(a, b, c)
+(−2)-curves
+A1 [2111]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R
+8
+(3, 1, 0)
+R = r45
+2A1(9) [221]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R1
+R2
+6
+(1, 2, 0)
+R1 = r23
+R2 = r45
+2A1(8) [111(11)]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R1
+R2
+7
+(3, 0, 1)
+R1 = r123
+R2 = r45
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+21
+Table 1: Conics-(−2)-curves graphs
+Geom. Type
+Conics-(−2) Curves Graph
+N
+(a, b, c)
+(−2)-curves
+A2 [311]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R1
+R2
+6
+(2, 1, 0)
+R1 = r34
+R2 = r45
+3A1 [(11)21]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R1
+R2
+R3
+5
+(1, 1, 1)
+R1 = r23
+R2 = r45
+R3 = r123
+A1A2 [32]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R1
+R2
+R3
+4
+(0, 2, 0)
+R1 = r12
+R2 = r34
+R3 = r45
+A3(5) [41]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R1
+R2
+R3
+4
+(1, 1, 0)
+R1 = r23
+R2 = r34
+R3 = r45
+A3(4) [(21)11]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R3
+R2
+R1
+5
+(2, 0, 1)
+R1 = r45
+R2 = r34
+R3 = r123
+4A1 [(11)(11)1]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R4
+R1
+R2
+R3
+4
+(1, 0, 2)
+R1 = r12
+R2 = r345
+R3 = r45
+R4 = r123
+2A1A2 [3(11)]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R1
+R2
+R4
+R3
+3
+(0, 1, 1)
+R1 = r12
+R2 = r23
+R3 = r45
+R4 = r123
+A1A3 [(21)2]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R1
+R2
+R4
+R3
+3
+(0, 1, 1)
+R1 = r12
+R2 = r34
+R3 = r45
+R4 = r123
+A4 [5]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R1
+R2
+R3
+R4
+2
+(0, 1, 0)
+R1 = r12
+R2 = r23
+R3 = r34
+R4 = r45
+
+22
+R. BLACHE AND E. HALLOUIN
+Table 1: Conics-(−2)-curves graphs
+Geom. Type
+Conics-(−2) Curves Graph
+N
+(a, b, c)
+(−2)-curves
+D4 [(31)1]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R4
+R1
+R2
+R3
+3
+(1, 0, 1)
+R1 = r23
+R2 = r34
+R3 = r45
+R4 = r123
+2A1A3 [(21)(11)]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R5
+R1
+R3
+R4
+R2
+2
+(0, 0, 2)
+R1 = r12
+R2 = r345
+R3 = r34
+R4 = r45
+R5 = r123
+D5 [(41)]
+C1
+C2
+C3
+C4
+C5
+C′
+1
+C′
+2
+C′
+3
+C′
+4
+C′
+5
+R1
+R2
+R3
+R4
+R5
+1
+(0, 0, 1)
+R1 = r12
+R2 = r23
+R3 = r34
+R4 = r45
+R5 = r123
+We summarize the results of this section in the following
+Proposition 4.3. Let X denote a degree four weak del Pezzo surface, and P a
+characteristic polynomial for the pencil of quadrics defining its anticanonical model
+in P4. Denote by aX the number of simple roots of P, bX ( resp. cX) the number
+of multiple roots corresponding to a rank 4 ( resp. rank 3) quadric in the pencil.
+If NX is the number of classes in C not intersecting any (−2)-curve negatively,
+we have NX = 2aX + 2bX + cX, and this is the number of surjective morphisms
+from X to the projective line.
+For each of these classes C, the morphism ϕ|C| : X → P1 has fibers linearly
+equivalent to C, and the type of map ϕ|C| defines on Xs depends on the following
+numerical criterion
+(a) 2aX of them satisfy C · R = 0 for all (−2)-curves R ∈ Rirr(X); for those
+ones ϕ|C| factors through the anticanonical morphism and yields a mor-
+phism from Xs to P1; moreover these classes lie in Pic(Xs), and they form
+aX couples of complementary classes;
+(b) 2bX + cX satisfy C · R ≥ 0 for all (−2)-curves R and C · R > 0 for at least
+one, and for those ϕ|C| does not define a morphism from Xs to P1.
+Remark 4.4. Note that in the ordinary case, we have aX = 5 and bX = cX = 0.
+Then the fibers of the 10 morphisms form the conic bundles, and we recover the
+original graph.
+Remark 4.5. Another consequence of the discussion above is that, for a fixed class
+C ∈ C which intersects all the (−2)-curves nonnegatively, we have two possibilities
+for the relative position of a (−2)-curve with class R with the fibers of the morphism
+ϕ|C|
+(i) when we have C · R = 1, R is transverse to the fibration, and ϕ|C| restricts
+to an isomorphism from R to P1. The images in Xs of its fibers all pass
+through the singular point which is the image of R;
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+23
+(ii) when we have C · R = 0, R is contained in a fiber of the morphism ϕ|C|.
+4.3. Galois action on the singular quadrics, and arithmetic types. In this
+section we work over the finite field k = Fq, q a power of an odd prime.
+It remains to provide information on the arithmetic type of a degree four del
+Pezzo surface X from arithmetic information on the rank four singular quadrics
+in the pencil defining Xs. We begin by using the results of the preceding section
+to describe the Galois action on the morphisms to the projective line from this
+information.
+Later, we will transpose this action to an action on the graph of
+conics and (−2)-curves, which will naturally lead to a (“part” of a) conjugacy class
+in the Weyl group W(D5), that we describe below, and that is given in the fourth
+column of the table for degree four surfaces in the Appendix.
+We begin by a definition (note that it is independant of the choice of the sup-
+plementary)
+Definition 4.6. Let Q denote a rank four quadric in P4, defined over Fqd; its
+restricted discriminant is the discriminant of its base in F×
+qd/F×2
+qd . In other word,
+it is the discriminant of the restriction of quadratic form corresponding to Q to a
+supplementary of its kernel.
+The Weyl group W(D5) can be identified with the group H5 ⋊ S5, where H5 is
+the subgroup of (Z/2Z)5 which is the kernel of the map ε = (ε1, . . . , ε5) �→ �5
+i=1 εi,
+and S5 acts on H5 by σ · ε = (εσ−1(1), . . . , εσ−1(5)).
+Recall from [5, Proposition 25] that the conjugacy classes in W(D5) are even
+signed cycle-types; they form a partition of 5 where each integer in the partition is
+overlined or not, and there is an even number of overlines. The conjugacy class of
+an element (ε, σ) has partition corresponding to the conjugacy class of σ in S5, and
+a number in this partition is overlined when the sum of the εi where i describes the
+support of the corresponding cycle is 1. Note that if σ is a d-cycle, the conjugacy
+class of an element of the form (ε, σ) is d when its d-th power is trivial, and d else.
+Note that W(D5) is a subgroup of W(B5) = (Z/2Z)5 ⋊ S5, whose conjugacy
+classes can be represented by (even or odd) signed cycle-types.
+Note also that an arithmetic type is a conjugacy class in some subgroup of
+W(D5). As a consequence, different arithmetic types can be contained in the same
+conjugacy class in W(D5), and we will have to be more precise in the last section.
+With these notations (and the ones used in the preceding sections) at hand, we
+begin by describing the Galois action on the part of the graph coming from the
+morphisms associated to the simple roots of P
+Lemma 4.7. Let X be a weak degree 4 del Pezzo surface over Fq whose anti-
+canonical model is the intersection of two quadrics Q0 and Q∞ in P4, both defined
+over Fq. Let G ∈ Fq[T ] be a simple irreducible divisor of degree d of the character-
+istic polynomial P(T ) and let θ ∈ Fqd be a root of G.
+We denote by ∆(θ) ∈ F×
+qd the restricted discriminant of Q = Q0 − θQ∞, ie the
+discriminant of its base q.
+To G, one can associate d couples of complementary classes in C , and the action
+of the Frobenius on those classes induces a permutation of signed sub-type d or d
+depending on whether ∆(θ) is or is not a square in F×
+qd (note that this does not
+depend on the choice of the root θ).
+
+24
+R. BLACHE AND E. HALLOUIN
+Proof. Let θ1 = θ, θ2 = θq, . . . , θd = θqd−1 ∈ Fqd denote the roots of G. From the
+results in the preceding section, we obtain d rank 4 quadrics Qi := Q0 − θiQ∞,
+and 2d morphisms ϕij from Xs to P1, 1 ≤ i ≤ d, 1 ≤ j ≤ 2. It follows from
+Proposition 4.3 that the classes fij of the fibers of these morphisms form d couples
+of complementary classes in C .
+The action of Frobenius sends Qi on Qi+1 (the indices are read modulo d), and
+the couple {fi1, fi2} to the couple {fi+1,1, fi+1,2}. Thus the permutation associated
+to the Frobenius (seen as an element in W(D5)) is the cycle (1 . . . d) of length d.
+The base q1 of the quadric Q1 is defined over Fqd; it is well known that it is either
+hyperbolic (isomorphic to P1 × P1 over Fqd) or elliptic (it becomes isomorphic to
+P1 × P1 only after a quadratic extension of the base field) depending on whether
+its discriminant is or is not a square in Fqd.
+In the hyperbolic case (then any of the qi is hyperbolic), the d-th power of the
+Frobenius stabilizes each one of the two P1, and the fibers f11 and f12; the Galois
+action has order d, and type d. Else the d-th power of the Frobenius exchanges the
+two P1 and the fibers f11, f12; the Galois action has order 2d, and type d.
+□
+We now describe the Galois action on the morphisms associated to the rank 4
+quadrics in the pencil coming from multiple roots of its characteristic polynomial.
+Lemma 4.8. Let θ ∈ Fq be a multiple root of P, say of multiplicity m, such that
+the associated quadric Q := Q0 − θQ∞ has rank 4.
+Denote by G the minimal
+polynomial of θ over Fq, by d its degree, and by ∆(θ) the restricted discriminant of
+Q.
+The d vertices of Q and its conjugates Qσ, . . . , Qσd−1 are singular points on Xs,
+and these singularities have Dynkin type dAm−1. To these correspond 2d morphisms
+from X to P1, whose fibers f1, f2, . . . , f σd−1
+1
+, f σd−1
+2
+form 2d classes in C , and the
+following cases occur
+(a) d = 1, 2 ≤ m ≤ 4: the Galois action fixes f1 and f2 if ∆(θ) is a square in
+F×
+q , and exchanges them otherwise;
+(b) d = 2, m = 2 and the Galois action acts on the fibers as the bitransposition
+(f1f σ
+1 )(f2f σ
+2 ) if ∆(θ) is a square in F×
+q2, and as the four cycle (f1f σ
+1 f2f σ
+2 )
+else.
+Proof. Since P has degree 5 and Gm is a divisor, we must have md ≤ 5, which
+leaves us with the listed possibilities, and d = 1, m = 5. But we see from the
+Segre symbols that this case corresponds to a singularity of Dynkin type A4. For
+this geometric type, there is only one arithmetic type, and the Galois action is not
+relevant: it will be sufficient to construct any del Pezzo surface over Fq with this
+singularity.
+The assertion on the Dynkin type of the singularities comes from the Segre sym-
+bols: the number m appearing as a part of it corresponds to an Am−1 singularity.
+The assertion on the fibers comes from Proposition 4.3.
+Finally the arithmetic assertions on the Galois action are easily deduced from
+the proof of the preceding lemma.
+□
+4.4. Quadratic module associated to a pencil of quadrics. We continue to
+work over the finite field k = Fq.
+The anticanonical model of a degree four del Pezzo surface is the base locus of a
+pencil of quadrics in P4, thus it is defined by the vanishing of a pair of quadratic
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+25
+forms. We follow [27] in this section, and define a quadratic module (here a k[T ]-
+module of finite length endowed with a non degenerate bilinear form) from the
+surface.
+We give a normal form for such a module and we deduce arithmetic
+information on the singular quadrics of the pencil from this normal form.
+As usual we denote by {Q0, Q∞} a basis for the pencil of quadratic forms defining
+X. We assume Q∞ is non-degenerate. Note that this requires the existence of a
+non-degenerate quadric of the pencil which is defined over k. We will have to drop
+this assumption farther in some very particular cases when q = 3.
+If ϕ0, ϕ∞ denote the bilinear forms on k5 associated respectively to Q0 and Q∞,
+then ϕ∞ is non degenerate, and there is an unique endomorphism u of k5 such that
+ϕ∞(u(x), y) = ϕ0(x, y) for any vectors x, y. Moreover u is symmetric with respect
+to ϕ∞.
+The vector space V = k5, endowed with the action of u becomes a k[T ]-module
+of finite length that we denote by Vu, and ϕ∞ is a nondegenerate symmetric k[T ]-
+bilinear form on Vu.
+Definition 4.9. We call the pair (Vu, ϕ∞) a quadratic k[T ]-module in the following.
+Since k[T ] is a principal ideal domain, the module Vu decomposes as a direct
+sum of primary cyclic components:
+Vu =
+t
+�
+i=1
+
+
+ni
+�
+j=1
+k[T ].xij
+
+ ,
+ann (xij) = P mij
+i
+k[T ],
+(4.2)
+where the xij are elements of Vu, the polynomials P1, . . . , Pt ∈ k[T ] are irreducible
+and pairwise distinct, and the exponents mi1, . . . miti are (not necessarily distinct)
+positive integers. Note that when k is algebraically closed we recover the Segre
+symbol (4.1).
+Waterhouse has proved that the cyclic primary components of the decomposition
+(4.2) can be chosen in such a way that they are two-by-two ϕ∞-orthogonal.
+Moreover, it turns out that each cyclic component can be explicitly described.
+To this end, we need to introduce some notations. Let F ∈ k[T ] be a non constant,
+unitary polynomial and let us consider the quotient algebra k[T ]/(F).
+We put
+t = T mod F, we choose some δ ∈ k[T ]/(F) and we define λF , ΦF , δ ·λF and δ ·ΦF
+as
+• the linear form λF : k[T ]/(F) → k defined as the last vector of the dual
+basis of (1, t, . . . , tdeg(F )−1);
+• the symmetric bilinear form ΦF : k[T ]/(F) × k[T ]/(F) → k defined by
+ΦF (x, y) = λF (xy);
+• the linear form defined by δ · λF (x) = λF (δx) and
+• the bilinear form defined by δ · ΦF (x, y) = λF (δxy).
+Then the linear form λF is a dualizing form and the algebra k[T ]/(F) with its du-
+alizing form is called a Frobenius algebra; by definition, this means that the bilinear
+form ΦF is non degenerate or equivalently that for every λ ∈ Homk (k[T ]/(F), k),
+there exists α ∈ k[T ]/(F) such that λ(x) = α · λF (x) = λF (αx).
+Definition 4.10. The quadratic module [[F, δ]] is the k[T ]-module k[T ]/(F) en-
+dowed with the bilinear form ϕ = δ · ΦF .
+
+26
+R. BLACHE AND E. HALLOUIN
+This is an example of quadratic module with cyclic underlying k[T ]-module.
+Note that the pair of quadratic forms on the corresponding k-vector space is given
+by Q∞(x) = λF (δx2) and Q0(x) = λF (tδx2).
+In fact this is a generic example.
+Proposition 4.11. Let (Vu, ϕ) be k[T ]-quadratic module. Suppose that the k[T ]-
+module Vu is cyclic, let x ∈ Vu be such that Vu = k[T ] · x, and let F ∈ k[T ] be a
+generator of its annihilator. Then there exists δ ∈ k[T ]/(F) such that for all p, q in
+k[T ]/(F) we have ϕ(p · x, q · x) = δ · ΦF (p, q).
+Proof. By definition of F, the map ι defined by p �→ p · x is an isomorphism of
+k[T ]-modules from k[T ]/(F) to Vu = k[T ]·x. The map ψ defined by ψ(p, q) = ϕ(p·
+x, q · x) for every p, q ∈ k[T ]/(F) is bilinear symmetric. Since u is ϕ-symmetric, the
+multiplication by t endomorphism is ψ-symmetric, and we have ψ(tp, q) = ψ(p, tq).
+In particular, we get ψ(p, q) = ψ(1, pq). But we know that there exists δ ∈ k[T ]/(F)
+such that ψ(1, q) = δ · λF (q) for every q ∈ k[T ]/(F). We deduce that
+ϕ(p · x, q · x) = ψ(p, q) = ψ(1, pq) = δ · λF (pq) = δ · ΦF (p, q)
+and the result follows.
+□
+We can use the quadratic modules we have just defined to give a normal form
+for all quadratic modules. From [27, Theorem 1.1 and Section 2], we have
+Theorem 4.12 (Waterhouse). Let (V, ϕ) be a non-degenerate k[T ]-quadratic mod-
+ule of finite length.
+Let
+�
+P eij
+i
+�
+i,j, 1 ≤ i ≤ r, 1 ≤ j ≤ nij, be the elementary divisors of u. Then there
+exists δij ∈ (k[T ]/(P eij
+i
+))× such that the pair (V, ϕ) is isometric to the orthogonal
+sum �
+i,j[[P eij
+i
+, δij]].
+Thus we can associate to the anticanonical model of any degree 4 del Pezzo
+surface (with at least one non-singular quadric defined over Fq in the pencil) a
+k[T ]-quadratic module of the form �
+i[[Fi, δi]].
+Conversely, to such a module we associate the del Pezzo surface defined by the
+vanishing of the following two quadratic forms defined for x = (xi) by
+(4.3)
+Q∞(x) =
+�
+i
+λFi(δix2
+i ), Q0(x) =
+�
+i
+λFi(T δix2
+i )
+Remark 4.13. Note that this association is in no way an application: we have arbi-
+trarily chosen the basis {Q0, Q∞}, and one could replace some δi by x2δi without
+changing the resulting del Pezzo surface.
+Also replacing all δi by aδi for some
+a ∈ F×
+q gives the quadratic forms aQ∞, aQ0 and does not change the del Pezzo
+surface.
+For this reason, we will frequently be able to put additional restrictions on the δi
+in the last section, in order to ease the computation of the restricted discriminants
+of the singular quadrics of the pencil.
+Moreover we have chosen to use the elementary divisors of the k[T ]-module in
+the above presentation, but we could have chosen the invariant factors instead, or
+any other possible decomposition in cyclic submodules.
+We have seen that one of the key tools to compute the arithmetic type of a Del
+Pezzo surface of degree 4 is to control the discriminants of the bases of the rank 4
+singular quadrics in the pencil. From the Frobenius algebras point of view, we have
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+27
+Lemma 4.14. Let F ∈ k[T ] be a unitary polynomial and let δ ∈ k[T ]/(F).
+We denote by NF : k[T ]/(F) → k the norm.
+(1) The discriminant of the bilinear form δ · ΦF equals
+(−1)
+deg(F )(deg(F )−1)
+2
+NF (δ)
+(2) Let θ ∈ k be a root of F. Then the bilinear form (θ − T )δ ·ΦF is degenerate
+of rank deg F − 1 and its restricted discriminant equals
+(−1)
+deg(F )(deg(F )−1)
+2
+NF (δ)δ(θ)
+Proof. (1) Put n = deg(F). Let C = (1, . . . , tn−1) be the canonical basis of k[T ]/(F),
+and let D = (f1, . . . , fn) be its dual basis with respect to λF , i.e.
+such that
+λF (ti−1fj) = δij, 1 ≤ i, j ≤ n.
+The coordinates of the elements of D in the
+basis C are given by the columns of the matrix:
+P =
+
+
+
+
+
+
+a1
+· · ·
+an−1
+1
+...
+...
+1
+an−1
+...
+1
+
+
+
+
+
+
+where the ai’s are the coefficients of F = T n+an−1T n−1+· · ·+a1T +a0. Moreover,
+for any x ∈ k[T ]/(F), one has x = �n
+i=1 λF (xfi)ei. Let us compare the matrices
+of the bilinear form δ · φF and of the linear map mδ (multiplication by δ) in the
+basis C; by definition they are equal to
+Mat (δ · ΦF , C) = (λF (δeiej))1≤i,j≤n
+and
+Mat (mδ, C) = (λF (δfiej))1≤i,j≤n .
+Thus they are related by the formula Mat (mδ, C) = tP Mat (δ · ΦF , C) and the
+result follows from the equality of the determinants.
+(2) Let e denote the multiplicity of the root θ for F, and set F = (T − θ)eG;
+there exist U, V ∈ k[T ] satisfying U(T − θ)e + V G = 1. Then for any x ∈ k[T ]/(F),
+we have
+λF (x)
+=
+λF (xU(T − θ)e) + λF (xV G)
+=
+λF ((xU mod G)(T − θ)e) + λF ((xV mod (T − θ)e)G)
+=
+λG(xU) + λ(T −θ)e(xV )
+where the last equality holds since G and (T − θ)e are unitary. We deduce the
+following orthogonal decomposition
+(4.4)
+(θ − T )δ · ΦF = (θ − T )V δ · Φ(T −θ)e ⊕ (θ − T )Uδ · ΦG.
+In order to compute the restricted discriminant of the first component, we note
+that (θ − T )V δ · λ(T −θ)e((T − θ)a) = −λ(T −θ)e((T − θ)a+1δV ) is equal to 0 as long
+as a ≥ e − 1, and to −δ(θ)V (θ) when a = e − 1. We get
+Mat
+�
+(θ − T )V δ · Φ(T −θ)e,
+�
+(T − θ)e−1, . . . , 1
+��
+=
+
+
+
+
+
+
+0
+...
+−V (θ)δ(θ)
+...
+...
+...
+0
+−V (θ)δ(θ)
+· · ·
+⋆
+
+
+
+
+
+
+.
+
+28
+R. BLACHE AND E. HALLOUIN
+This form has rank e − 1; its kernel is generated by (T − θ)e−1, and its restricted
+discriminant equals:
+(−1)
+(e−1)(e−2)
+2
+× (−1)e−1 × V (θ)e−1δ(θ)e−1 = (−1)
+e(e−1)
+2
+δ(θ)e−1
+G(θ)e−1
+because V (θ)G(θ) = 1.
+For the second component of the decomposition (4.4), putting d = deg(G), and
+using (1), its discriminant equals
+(−1)d(d−1)NG ((θ − T )Uδ) = (−1)
+d(d−1)
+2
++ed
+NG(δ)
+NG(θ − T )e−1
+since U(θ − T ) × (−1)e(θ − T )e−1 = 1 mod G.
+We have NG(θ − T ) = G(θ), and the product of these two discriminants gives
+the result as NF (δ) = NG(δ)N(T −θ)e(δ) = NG(δ)NT −θ(δ)e = NG(δ)δ(θ)e.
+□
+Remark 4.15. At this point, we make the following observation: when the underly-
+ing k[T ]-module is cyclic, all singular quadrics in the correspondong pencil (defined
+by equations (4.3)) have corank one.
+The converse is true: when the underlying k[T ]-module is not cyclic, it admits
+at least two invariant factors, and for all roots of the one of smallest degree the
+above result shows that the corresponding singular quadric has corank equal to the
+number of invariant factors.
+We end this section with a criterion that determines the field of definition of
+certain singular points of the del Pezzo surface. We will use it in the next section in
+order to determine the arithmetic types when the pencil contains a rank 3 quadric.
+Lemma 4.16. Let X be a degree four del Pezzo surface over Fq such that for some
+θ ∈ Fq, the non-degenerate quadratic k[T ]-module associated to X can be written
+as an orthogonal sum [[T − θ, δ1]] ⊕ [[T − θ, δ2]] ⊕ M, where the annihilator P of the
+k[T ]-module M satisfies P(θ) ̸= 0.
+Then the quadric of the pencil with equation Q0 − θQ∞ = 0 has rank 3, and
+if ℓ ≃ P1 is its vertex, the singular del Pezzo variety Xs meets ℓ at two distinct
+points, which are defined over Fq if −δ1δ2 is a square in F×
+q , and conjugate over
+Fq else.
+Proof. We choose a basis (e1, . . . , e5) of k5 which is adapted to the orthogonal
+decomposition. In this basis, from the expressions (4.3) of the forms Q0, Q∞, their
+matrices can be written in block diagonal form as respectively
+Mat(Q∞) =
+
+
+δ1
+0
+0
+0
+δ2
+0
+0
+0
+A∞
+
+ ., Mat(Q0) =
+
+
+θδ1
+0
+0
+0
+θδ2
+0
+0
+0
+A0
+
+ .
+Since the annihilator P of the k[T ]-module M satisfies P(θ) ̸= 0, the matrix A0 −
+θA∞ is invertible, and the quadric with equation Q0 − θQ∞ has rank 3.
+Its vertex is the line (x1 : x2 : 0 : 0 : 0) in P4, the intersection of which with the
+quadric Q∞ = 0 is defined by the equation δ1x2
+1 + δ2x2
+2 = 0. Now we have δ1δ2 ̸= 0
+since the form Q∞ is assumed non degenerate, and we get two solutions since the
+characteristic is odd. The last assertion is classical.
+□
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+29
+4.5. Construction of degree four del Pezzo surfaces of any arithmetic
+type. We are ready to show Theorem 2 2. Our strategy is the following : in order
+to construct a surface of a given arithmetic type, we construct a quadratic module
+as a sum of cyclic modules in normal form. We proceed geometric type by geometric
+type, and for each one we do the following
+1. the Segre symbol gives the factorisation of the characteristic polynomial
+over Fq, and we deduce from it a decomposition of a quadratic k[T ]-module
+associated to X. We choose it cyclic when this is possible, but in general
+we choose the decomposition that have seemed us best adapted to the com-
+putation of the restricted discriminants from lemma 4.14, and of the infor-
+mation needed in lemma 4.16. We sometimes put additional restrictions on
+the δi in order to ease this computation.
+2. we describe all possible Galois actions on the graph of conics and (−2)-
+curves given in Table 4.2; they give the possible arithmetic types. For each
+one, we use lemmas 4.7, 4.8 and 4.16 to describe the factorization type of
+the characteristic polynomial over Fq, the restricted discriminants of rank
+4 singular quadrics and the squareness of certain invariants;
+3. we choose some δi in the cyclic quadratic submodules such that the invari-
+ants associated to the singular quadrics in the pencil satisfy the conditions
+listed at the second point above.
+The existence of a polynomial with this factorization, and of some δi with the
+claimed properties, is sufficient to ensure that the del Pezzo surface associated to
+the quadratic k[T ]-module we have constructed has the arithmetic type we need.
+Remark 4.17. As already mentioned in the introduction, we have used the math-
+ematical software magma to construct a couple of quadrics in P4 for each type of
+degree four singular del Pezzo surfaces over a given finite field. There is a slight
+difference between the procedure used in this program and the description of the
+quadratic modules presented below: here we decompose the modules using a biggest
+possible cyclic module. In the program we use another decomposition, with more
+terms. There is a Chinese remainder theorem for quadratic modules –not presented
+here– that gives an equivalence between the two descriptions.
+4.5.1. Ordinary geometric type. The factorization of P over Fq is P(T ) = �5
+i=1(T −
+θi) with distinct θi, 1 ≤ i ≤ 5.
+All singular quadrics in the pencil have rank four, and we can associate to the
+del Pezzo surface X a cyclic quadratic module [[P, δ]]. We will construct δ such
+that NP (δ) is a square in F×
+q ; then from lemma 4.14, the restricted discriminants
+of the four singular quadrics are the ∆(θi) = NP (δ)δ(θi) ∈ Fq(θi)× and we have
+∆(θi) ≡ δ(θi) mod Fq(θi)×2.
+Applying lemma 4.7, we see that the signed types representing the conjugacy
+classes in W(D5) (which are the arithmetic types of ordinary degree four del Pezzo
+surfaces) give
+• the degrees of the irreducible factors of P in Fq[T ] just by removing the
+bars;
+• the reduced discriminants ∆(θi) mod Fq(θi)×2, ie the classes δ(θi) mod
+Fq(θi)×2 from our convention on NP (δ);
+
+30
+R. BLACHE AND E. HALLOUIN
+For each geometric type, we present the correspondance between signed types
+and properties of P and δ in a table. In the last column, we write {□, · · · , □
+�
+��
+�
+d times
+} when
+the corresponding d roots θi, . . . , θi+d−1 are conjugate over Fq and δ(θi) is a square
+in F×
+qd, and {⊠, · · · , ⊠
+�
+��
+�
+d times
+} when δ(θi) is not a square in F×
+qd.
+Signed type
+(δ(θ1), . . . , δ(θ5))
+11111
+(□, □, □, □, □)
+11111
+(□, □, □, ⊠, ⊠)
+11111
+(□, ⊠, ⊠, ⊠, ⊠)
+2111
+({□, □}, □, □, □)
+2111
+({□, □}, □, ⊠, ⊠)
+221
+({□, □}, {□, □}, □)
+311
+({□, □, □}, □, □)
+221
+({□, □}, {⊠, ⊠}, ⊠)
+41
+({□, □, □, □}, □)
+2111
+({⊠, ⊠}, □, □, ⊠)
+2111
+({⊠, ⊠}, ⊠, ⊠, ⊠)
+221
+({⊠, ⊠}, {⊠, ⊠}, □)
+5
+({□, □, □, □, □})
+32
+({□, □, □}, {□, □})
+311
+({□, □, □}, ⊠, ⊠)
+311
+({⊠, ⊠, ⊠}, □, ⊠)
+41
+({⊠, ⊠, ⊠, ⊠}, ⊠)
+32
+({⊠, ⊠, ⊠}, {⊠, ⊠})
+Note that a polynomial P and an element δ with the desired properties exist as
+long as we have q ≥ 5. But when we have q = 3, one cannot construct a polynomial
+with five roots over Fq, and some types do not exist [26].
+4.5.2. Geometric type A1. The factorization of P over Fq is P(T ) = �3
+i=1(T −
+θi)(T − θ4)2 with distinct θi, 1 ≤ i ≤ 4, and θ4 ∈ Fq.
+All singular quadrics in the pencil have rank four, and we can associate to the
+del Pezzo surface X a cyclic quadratic module [[P, δ]]. For each arithmetic type, we
+will choose some δ such that NP (δ) is a square in F×
+q ; then from lemma 4.14, the
+restricted discriminants of the four singular quadrics are the ∆(θi) = NP (δ)δ(θi) ∈
+Fq(θi)× and we have ∆(θi) ≡ δ(θi) mod Fq(θi)×2.
+An element of Stab(Rirr) must act on the last two couples of complementary
+conics as the identity (of signed type 11) or the bitransposition (C4C′
+5)(C5C′
+4) (of
+signed type 2); note that both have even signed types. Since it is an element of
+W(D5), it must act on the first three couples as an element of even signed type, ie
+as an element of H3 ⋊ S3, where H3 is the subgroup of Z/2Z3 which is the kernel
+of the map ε = (ε1, . . . , ε3) �→ �3
+i=1 εi. We get the following ten conjugacy classes
+in Stab(Rirr)
+111 · 11
+111 · 11
+21 11
+21 · 11
+3 · 11
+111 · 2
+111 · 2
+21 · 2
+21 · 2
+3 · 2.
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+31
+Write P(T ) = F(T )(T − θ4)2; applying lemma 4.7, we see that the first part of
+the above signed types give
+• the degrees of the irreducible factors of F in Fq[T ] just by removing the
+bars;
+• for 1 ≤ i ≤ 3, the reduced discriminants ∆(θi) mod Fq(θi)×2, ie the classes
+δ(θi) mod Fq(θi)×2 from our convention on NP (δ);
+Then, from lemma 4.8, the classes C4 and C′
+5, which correspond to the fibers of the
+morphisms defined by the last singular quadric Q0 − θ4Q∞, are fixed by Frobenius
+when δ(θ4) is a square in F×
+q , and exchanged else. This gives us the second part of
+the signed type. We obtain the following table
+N ◦
+Cl. Stab.
+Cl. Weyl
+(δ(θ1), . . . , δ(θ4))
+1
+111 · 11
+11111
+(□, □, □, □)
+2
+111 · 2
+2111
+(□, □, □, ⊠)
+3
+21 · 11
+2111
+({□, □}, □, □)
+4
+111 · 11
+11111
+(⊠, ⊠, □, □)
+5
+21 · 2
+221
+({□, □}, □, ⊠)
+6
+111 · 2
+2111
+(⊠, ⊠, □, ⊠)
+7
+3 · 11
+311
+({□, □, □}, □)
+8
+21 · 2
+221
+({⊠, ⊠}, ⊠, ⊠)
+9
+21 · 11
+2111
+({⊠, ⊠}, ⊠, □)
+10
+3 · 2
+32
+({□, □, □}, ⊠)
+Note that we can always construct a polynomial with the claimed factorization,
+except when q = 3 in cases 1, 2, 4 and 6 since then P needs to have four distinct
+roots in Fq.
+Elements δ ∈ k[T ]/(P) with the required properties always exist.
+Finally note that we have NP (δ) ∈ F×2
+q
+in all cases, as required.
+It remains to construct surfaces of types 1,2,4 and 6 over F3. We mimic the above
+construction: we consider the quadratic forms with the following block diagonal
+matrices
+Mat(Q0) =
+
+
+
+
+
+
+δ∞
+1
+1
+0
+0
+0
+δ0
+
+
+
+
+
+
+, Mat(Q∞) =
+
+
+
+
+
+
+0
+1
+−1
+0
+δ0
+δ0
+0
+
+
+
+
+
+
+The four quadrics of the pencil which are defined over F3 (namely Qt = Q0 +
+tQ∞, t ∈ F3 and Q∞) are singular of rank four, and the vertex of Q0 –here (0 : 0 :
+0 : 1 : 0)– is the unique singular point of the base of the pencil. We deduce that
+Xs has geometric type A1.
+The restricted discriminants are, modulo squares
+∆(Q0) ≡ δ0δ∞, ∆(Q1) ≡ ∆(Q2) ≡ δ∞, ∆(Q∞) ≡ 1
+and we get the types 1, 2, 4 and 6 respectively choosing (δ0, δ∞) of the form (□, □),
+(⊠, □), (⊠, ⊠) and (□, ⊠).
+4.5.3. Geometric type 2A1 with 9 lines. The factorization of P over Fq is P(T ) =
+(T − θ1)(T − θ2)2(T − θ3)2 with distinct θi, 1 ≤ i ≤ 3, and θ1 ∈ Fq.
+
+32
+R. BLACHE AND E. HALLOUIN
+All singular quadrics in the pencil have rank four, and we can associate to the
+del Pezzo surface X a cyclic quadratic module [[P, δ]]. We will always choose some
+δ such that NP (δ) (or equivalently δ(θ1)) is a square in F×
+q ; then we have the
+congruence ∆(θi) ≡ δ(θi) mod Fq(θi)×2 for the restricted discriminants.
+An element of Stab(Rirr) can
+• fix the (−2)-curves, and the sets {C2, C′
+2, C3, C′
+3} and {C4, C′
+4, C5, C′
+5}; in
+this case it acts on the last four couples of complementary conics as
+– the identity, of signed type 11 · 11 or
+– a bitransposition such as (C2C′
+3)(C2C′
+3), of signed type 2 · 11 or
+– four transpositions such as (C2C′
+3)(C2C′
+3)(C4C′
+5)(C4C′
+5), of signed type
+2 · 2 ;
+• exchange the (−2)-curves, and the sets {C2, C′
+2, C3, C′
+3} and {C4, C′
+4, C5, C′
+5};
+in this case it acts on the last four couples of complementary conics as
+– four transpositions such as (C2C4)(C′
+2C′
+4)(C3C5)(C′
+3C′
+5), of signed type
+we note here {2 · 2} or
+– a permutation such as (C3C5C′
+2C′
+4)(C′
+3C′
+5C2C4), of signed type 4.
+Note that all have even signed types, and the action must be trivial on the remaining
+couple of conics. We get the following five conjugacy classes in Stab(Rirr)
+1 · 11 · 11,
+1 · 2 · 11,
+1 · 2 · 2,
+1 · {2 · 2},
+1 · 4.
+Applying lemma 4.8, we get the following table
+N ◦
+Cl. Stab.
+Cl. Weyl
+(δ(θ1), δ(θ2), δ(θ3))
+11
+1 · 11 · 11
+11111
+(□, □, □)
+12
+1 · 2 · 11
+2111
+(□, ⊠, □)
+13
+1 · 2 · 2
+221
+(□, ⊠, ⊠)
+14
+1 · {2 · 2}
+221
+(□, {□, □})
+15
+1 · 4
+41
+(□, {⊠, ⊠})
+4.5.4. Geometric type 2A1 with 8 lines. The factorization of P over Fq is P(T ) =
+�3
+i=1(T − θi)(T − θ4)2 with distinct θi, 1 ≤ i ≤ 4, and θ4 ∈ Fq.
+There is a rank three singular quadric in the pencil, corresponding to the root
+θ4, and we can associate to the del Pezzo surface X a quadratic module of the form
+[[F, δ]] ⊕ [[T − θ4, δ1]] ⊕ [[T − θ4, δ2]], where F(T ) = �3
+i=1(T − θi).
+With the help of lemma 4.14, and using a block diagonal form as in the proof of
+lemma 4.16, we get that the rank four singular quadrics in the pencil have restricted
+discriminants
+∆(θi) = (θi − θ4)2δ1δ2(−1)3NF (δ)δ(θi) ≡ −δ1δ2NF (δ)δ(θi) mod Fq(θi)×2
+for 1 ≤ i ≤ 3.
+We shall construct quadratic modules satisfying δ2 = −1 and
+δ1NF (δ) ∈ F×2
+q , so that ∆(θi) ≡ δ(θi) mod Fq(θi)×2 for 1 ≤ i ≤ 3.
+An element of Stab(Rirr) must act on the last two couples of complementary
+conics as the identity or as (C5C′
+5) of signed type 11. In the first case it acts on
+the first three couples as an even signed type permutation, ie as an element of
+H3 ⋊ S3, and in the second as an odd signed type permutation, ie as an element
+of Z/2Z3 ⋊ S3 \ H3 ⋊ S3. We get the following conjugacy classes in Stab(Rirr)
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+33
+111 · 11, 111 · 11, 21 · 11, 21 · 11, 3 · 11, 111 · 11, 111 · 11, 21 · 11, 21 · 11, 3 · 11.
+The second part of the signed type is 11 exactly when the singular points are
+defined over Fq; lemma 4.16, applied with our convention δ2 = −1, tells us that
+this happens exactly when δ1 is a square in F×
+q .
+It remains to use lemma 4.7 to determine the factorization pattern of F and the
+δ(θi) mod Fq(θi)×2 from the signed type of the action on the first three couples.
+We get the following table
+N ◦
+Cl. Stab.
+Cl. Weyl
+(δ(θ1), δ(θ2), δ(θ3), δ1)
+16
+111 · 11
+11111
+(□, □, □, □)
+17
+21 · 11
+2111
+({□, □}, □, □)
+18
+111 · 11
+11111
+(⊠, ⊠, □, □)
+19
+111 · 11
+11111
+(⊠, □, □, ⊠)
+20
+21 · 11
+2111
+({□, □}, ⊠, ⊠)
+21
+111 · 11
+11111
+(⊠, ⊠, ⊠, ⊠)
+22
+3 · 11
+311
+({□, □, □}, □)
+23
+21 · 11
+2111
+({⊠, ⊠}, □, ⊠)
+24
+21 · 11
+2111
+({⊠, ⊠}, ⊠, □)
+25
+3 · 11
+311
+({⊠, ⊠, ⊠}, ⊠)
+Note that we can always construct a polynomial over Fq with the claimed factor-
+ization, except when q = 3 in cases 16, 18, 19 and 21. Elements δ ∈ k[T ]/(F) with
+the required properties always exist. Finally note that we have δ1NF (δ) ∈ F×2
+q
+in
+all cases.
+It remains to construct surfaces of types 16, 18, 19 and 21 over F3. We mimic
+the above construction: we consider the quadratic forms with the following block
+diagonal matrices
+Mat(Q0) =
+
+
+
+
+
+
+δ∞
+δ1
+1
+0
+0
+
+
+
+
+
+
+, Mat(Q∞) =
+
+
+
+
+
+
+0
+δ1
+−1
+δ0
+−1
+
+
+
+
+
+
+The quadrics with equations Q∞, Q0 + Q∞ and Q0 − Q∞ have rank 4, and the
+quadric Q0 has rank 3. The singular points of Xs are the intersections of the vertex
+of Q0 with Q∞, that is the points (0 : 0 : 0 : a : b) with δ0a2 − b2 = 0. We get
+a surface of geometric type 2A1 with 8 lines as above, since the line joining the
+singularities is not contained in Xs.
+Computing the restricted discriminants, and applying lemma 4.16, we see that
+we get the types 16, 18, 19 and 21 respectively choosing (δ0, δ1, δ∞) of the form
+(□, □, □), (□, ⊠, ⊠), (⊠, ⊠, ⊠) and (⊠, □, □).
+4.5.5. Geometric type A2. The factorization of P over Fq is P(T ) = (T − θ1)(T −
+θ2)(T − θ3)3 with distinct θi, 1 ≤ i ≤ 3, and θ3 ∈ Fq.
+All singular quadrics in the pencil have rank four, and we can associate to the
+del Pezzo surface X a cyclic quadratic module [[P, δ]]. We will construct δ such that
+NP (δ) is a square in F×
+q ; then we have the congruence ∆(θi) ≡ δ(θi) mod Fq(θi)×2
+for the restricted discriminants.
+
+34
+R. BLACHE AND E. HALLOUIN
+An element of Stab(Rirr) must act on the last three couples of complementary
+conics as the identity (of signed type 111) or the permutation (C5C′
+3)(C3C′
+5)(C4C′
+4)
+(of signed type 21). It will act on the first two couples as en even signed permutation
+in the first case, as an odd signed one in the second. We get the following five
+conjugacy classes in Stab(Rirr)
+11 · 111,
+11 · 111,
+2 · 111,
+11 · 21,
+2 · 21.
+Note that the signed type is completely determined by its first part, and we just
+have to describe the action on it via lemma 4.7. We get the following table, where
+we have chosen δ(θ3) so that the convention NP (δ) = δ(θ1)δ(θ2)δ(θ3)3 ∈ F×2
+q
+holds
+N ◦
+Cl. Stab.
+Cl. Weyl
+(δ(θ1), δ(θ2), δ(θ3))
+26
+11 · 111
+11111
+(□, □, □)
+27
+2 · 111
+2111
+({□, □}, □)
+28
+11 · 111
+11111
+(⊠, ⊠, □)
+29
+11 · 21
+2111
+(⊠, □, ⊠)
+30
+2 · 21
+221
+({⊠, ⊠}, ⊠)
+4.5.6. Geometric type 3A1. The factorization of P over Fq is P(T ) = (T − θ1)(T −
+θ2)2(T − θ3)2 with distinct θi, 1 ≤ i ≤ 3. We assume that the rank three conic in
+the pencil corresponds to θ3. We can write the quadratic k[T ]-module [[F, δ]]⊕[[T −
+θ3, δ1]] ⊕ [[T − θ3, δ2]] where F(T ) := (T − θ1)(T − θ2)2.
+With the help of lemma 4.14, and using a block diagonal form as in the proof of
+lemma 4.16, we get that the rank four singular quadrics in the pencil have restricted
+discriminants
+∆(θi) = (θi − θ3)2δ1δ2(−1)3NF (δ)δ(θi) ≡ −δ1δ2δ(θ1)δ(θi) mod F×2
+q , 1 ≤ i ≤ 2
+We shall construct quadratic modules satisfying δ2 = −1 and δ1δ(θ1) ∈ F×2
+q , so
+that ∆(θi) ≡ δ(θi) mod F×2
+q
+for 1 ≤ i ≤ 2.
+An element of Stab(Rirr) must act on the last two couples of complementary
+conics as the identity (of signed type 11) or the permutation (C5C′
+5) (of signed type
+11). On the second and third couples, it acts as the identity (of signed type 11) or
+as the bitransposition (C2C′
+3)(C3C′
+2) of signed type 2. We then have to “complete”
+the action by fixing C1 and C′
+1 if C5 and C′
+5 are fixed, and by transposing them
+else. We get the following four conjugacy classes in Stab(Rirr)
+1 · 11 · 11,
+1 · 11 · 11,
+1 · 2 · 11,
+1 · 2 · 11.
+As above, we use lemma 4.8 (a) to read the restricted discriminant ∆(θ2) from
+the action on the second and third couples, and lemma 4.16 to get δ1 from the
+action on the last couple. We get
+N ◦
+Cl. Stab.
+Cl. Weyl
+(δ(θ1), δ(θ2), δ1)
+31
+1 · 11 · 11
+11111
+(□, □, □)
+32
+1 · 2 · 11
+2111
+(□, ⊠, □)
+33
+1 · 11 · 11
+11111
+(⊠, □, ⊠)
+34
+1 · 2 · 11
+2111
+(⊠, ⊠, ⊠)
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+35
+4.5.7. Geometric type A1A2. The factorization of P over Fq is P(T ) = (T −
+θ1)2(T − θ2)3 with distinct θi, 1 ≤ i ≤ 2. There is no rank three conic in the
+pencil, and we can write the quadratic k[T ]-module in cyclic form [[P, δ]].
+Lemma 4.14 tells us that the restricted discriminants are ∆(θi) = NP (δ)δ(θi) =
+δ(θ1)2δ(θ2)3δ(θi) ≡ δ(θ2)δ(θi) mod F×2
+q
+for 1 ≤ i ≤ 2. We shall construct quadratic
+modules satisfying δ(θ2) ∈ F×2
+q , so that ∆(θi) ≡ δ(θi) mod F×2
+q
+for 1 ≤ i ≤ 2.
+An element of Stab(Rirr) must act on the first two couples of complementary
+conics as the identity (of signed type 11) or a bitransposition (of signed type 2).
+On the last three couples, it acts as the identity in order to have en even signed
+type (recall that the non trivial action on these three couples has signed type 21).
+We get two conjugacy classes in Stab(Rirr), namely 11 · 111 and 2 · 111, and using
+lemma 4.8 (a), we get the table
+N ◦
+Cl. Stab
+Cl. Weyl
+(δ(θ1), δ(θ2))
+35
+11 · 111
+11111
+(□, □)
+36
+2 · 111
+2111
+(⊠, □)
+4.5.8. Geometric type A3 with five lines. The factorization of P over Fq is P(T ) =
+(T − θ1)(T − θ2)4 with distinct θi, 1 ≤ i ≤ 2. There is no rank three conic in the
+pencil, and we can write the quadratic k[T ]-module in cyclic form [[P, δ]].
+Lemma 4.14 tells us that the restricted discriminants are ∆(θi) = NP (δ)δ(θi) ≡
+δ(θ1)δ(θi) mod F×2
+q
+for 1 ≤ i ≤ 2. We shall construct quadratic modules satisfying
+δ(θ1) ∈ F×2
+q , so that ∆(θi) ≡ δ(θi) mod F×2
+q
+for 1 ≤ i ≤ 2.
+An element of Stab(Rirr) must act on the last four couples of complementary
+conics as the identity (of signed type 1111) or as (C2C′
+5)(C′
+2C5)(C3C′
+4)(C′
+3C4) (of
+signed type 22). Thus it must act as the identity on the first couple in order to have
+en even signed type. We get two conjugacy classes in Stab(Rirr), namely 1 · 1111
+and 1 · 22, and using lemma 4.8 (a), we get the table
+N ◦
+Cl. Stab.
+Cl. Weyl
+(δ(θ1), δ(θ2))
+37
+1 · 1111
+11111
+(□, □)
+38
+1 · 22
+221
+(□, ⊠)
+4.5.9. Geometric type A3 with four lines. The factorization of P over Fq is P(T ) =
+(T − θ1)(T − θ2)(T − θ3)3 with distinct θi, 1 ≤ i ≤ 3. We can write the quadratic
+k[T ]-module [[F, δ]] ⊕ [[T − θ3, δ1]] ⊕ [[(T − θ3)2, δ2]] where F(T ) := (T − θ1)(T − θ2).
+With the help of lemma 4.14, and using a block diagonal form as in the proof of
+lemma 4.16, we get that the rank four singular quadrics in the pencil have restricted
+discriminants
+∆(θi) = (θi − θ3)3δ1δ2
+2NF (δ)δ(θi) ≡ (θi − θ3)δ1δ(θ1)δ(θ2)δ(θi) mod F×2
+q , 1 ≤ i ≤ 2
+We shall construct quadratic modules satisfying δ1 = (θ1 − θ3)(θ2 − θ3), so that
+∆(θi) ≡ (θj − θ3)δ(θj) mod F×2
+q
+for 1 ≤ i ̸= j ≤ 2.
+An element of Stab(Rirr) must act on the last three couples of complementary
+conics as the identity (of signed type 111) or the transposition (C5C′
+5) (of signed
+type 111). We deduce the possible actions on the first two couples, and the following
+five conjugacy classes in Stab(Rirr)
+11 · 111,
+11 · 111,
+2 · 111,
+11 · 111,
+2 · 111.
+
+36
+R. BLACHE AND E. HALLOUIN
+The type is completely determined by the action on the first two couples: it is
+sufficient to use lemma 4.7 to obtain the table
+N ◦
+Cl. Stab.
+Cl. Weyl
+((θ1 − θ3)δ(θ1), (θ2 − θ3)δ(θ2))
+39
+11 · 111
+11111
+(□, □)
+40
+2 · 111
+2111
+({□, □})
+41
+11 · 111
+11111
+(⊠, ⊠)
+42
+11 · 111
+11111
+(□, ⊠)
+43
+2 · 111
+2111
+({⊠, ⊠})
+4.5.10. Geometric type 4A1. The factorization of P over Fq is P(T ) = (T −θ1)(T −
+θ2)2(T − θ3)2 with distinct θi, 1 ≤ i ≤ 3. We can write the quadratic k[T ]-module
+[[T − θ1, δ]] ⊕ [[F, δ1]] ⊕ [[F, δ2]] where F(T ) := (T − θ2)(T − θ3). We will construct
+such a module satisfying the additional assumption δ2 = −1.
+An element in Stab(Rirr) acts on the four (−2)-curves, and we deduce from this
+its action on the couples {Ci, C′
+i}, i ∈ {1, 2, 4, 5}
+(a) it fixes the four (−2)-curves, and the eight conics classes; then its signed
+type is 1111;
+(b) it fixes two (−2)-curves (necessarily corresponding to the same rank 3
+quadric of the pencil, thus in the same “component” of the graph), and
+permutes the other ones; for instance the transposition (R3R4) corresponds
+to (C5C′
+5), of signed type 1111;
+(c) it acts as the bitransposition (R1R2)(R3R4) on (−2)-curves, and on conic
+classes as (C2C′
+2)(C5C′
+5), its signed type is 1111
+(d) it acts as the bitransposition (R1R3)(R2R4) on (−2)-curves, and on conic
+classes as (C1C4)(C2C5)(C′
+1C′
+4)(C′
+2C′
+5) of signed type 22;
+(e) it acts as the four-cycle (R1R3R2R4) on (−2)-curves and on conic classes
+as (C1C4)(C′
+1C′
+4)(C2C5C′
+2C′
+5) of signed type 22;
+Then we complete with the correct action on {C3, C′
+3} in order to get an even signed
+permutation. We get five conjugacy classes
+1 · 1111,
+1 · 1111,
+1 · 1111,
+1 · 22,
+1 · 22.
+The two rank three conics can be defined over Fq (we have θ2, θ3 ∈ Fq), or over
+Fq2, and conjugate over Fq. Applying lemma 4.16 to X in the first case, and to
+X ⊗ Fq2 in the second, we get the following table under the assumption δ2 = −1
+N ◦
+Cl. Stab.
+Cl. Weyl
+(δ1(θ2), δ1(θ3))
+44
+1 · 1111
+11111
+(□, □)
+45
+1 · 1111
+11111
+(⊠, ⊠)
+46
+1 · 22
+221
+({□, □})
+47
+1 · 1111
+11111
+(⊠, □)
+48
+1 · 22
+221
+({⊠, ⊠})
+4.5.11. Geometric type 2A1A2. The factorization of P over Fq is P(T ) = (T −
+θ1)3(T − θ2)2 with distinct θi, 1 ≤ i ≤ 2. We can write the quadratic k[T ]-module
+[[(T −θ1)3, δ]]⊕[[T −θ2, δ1]]⊕[[T −θ2, δ2]]. We will construct such a module satisfying
+the additional assumptions δ = 1, δ2 = −1.
+We have already seen that an element in Stab(Rirr) can act on the first three
+couples with a signed type 111 or 21, and on the last two with a signed type 11 or
+11. This leaves us with two conjugacy classes, namely 11 · 111 and 21 · 11, which
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+37
+are determined by the action on the last couple, ie on the two A1-singularities.
+Applying lemma 4.16, and from our assumption δ2 = −1, we get the table
+N ◦
+Cl. Stab.
+Cl. Weyl
+δ1
+49
+111 · 11
+11111
+□
+50
+21 · 11
+2111
+⊠
+4.5.12. Geometric type A1A3. The factorization of P over Fq is P(T ) = (T −
+θ1)2(T − θ2)3 with distinct θi, 1 ≤ i ≤ 2. We can write the quadratic k[T ]-module
+[[(T − θ1)2, δ]] ⊕ [[(T − θ2)2, δ1]] ⊕ [[T − θ2, δ2]]. We will construct such a module
+satisfying the additional assumptions δ1 = δ2 = 1. The restricted discriminant of
+the rank four quadric in the pencil (corresponding to root θ1) is
+∆(θ1) = δ(θ1)(θ1 − θ2)3δ2
+1δ2 ≡ δ(θ1)(θ1 − θ2) mod F×2
+q
+An element in Stab(Rirr) acts on the first two couples with a signed type 11 or
+2, and on the last three with a signed type 111 or 111. This leaves us with two
+conjugacy classes, namely 111·11 and 2·111, and they are determined by the action
+on the first two couples. Applying lemma 4.8 (a), and from our assumptions, we
+get the table
+N ◦
+Cl. Stab
+Cl. Weyl
+(θ1 − θ2)δ(θ1)
+51
+111 · 11
+11111
+□
+52
+2 · 111
+2111
+⊠
+4.5.13. Geometric type A4. The factorization of P over Fq is P(T ) = (T − θ1)5,
+and we can consider the cyclic quadratic k[T ]-module [[(T − θ1)5, δ]].
+An element in Stab(Rirr) can only act on the graph trivially (the other automor-
+phism of this graph is (C1C′
+5)(C′
+1C5)(C2C′
+4)(C′
+2C4)(C3C′
+3) which has odd signed
+type 221). Thus for any choice of δ we get the arithmetic type 53.
+4.5.14. Geometric type D4. The factorization of P over Fq is P(T ) = (T − θ1)(T −
+θ2)4 with distinct θi, 1 ≤ i ≤ 2. We can write the quadratic k[T ]-module [[T −
+θ1, δ]] ⊕ [[(T − θ2)3, δ1]] ⊕ [[T − θ2, δ2]]. We will construct such a module satisfying
+the additional assumptions δ = δ2 = 1. In this case the restricted discriminant of
+the rank four quadric in the pencil (corresponding to root θ1) is
+∆(θ1) = −(θ2 − θ1)4δ3
+1δ2 ≡ δ1 mod F×2
+q
+An element in Stab(Rirr) acts on the last four couples with a signed type 1111
+or 1111, and on the first one with a signed type 1 or 1. This leaves us with two
+conjugacy classes, namely 1 · 1111 and 1 · 1111, and they are determined by the
+action on the first couple. Applying lemma 4.7 to the value of ∆(θ1) above, we get
+the table
+N ◦
+Cl. Stab
+Cl. Weyl
+δ1
+54
+1 · 1111
+11111
+□
+55
+1 · 1111
+11111
+⊠
+
+38
+R. BLACHE AND E. HALLOUIN
+4.5.15. Geometric type 2A1A3. The factorization of P over Fq is P(T ) = (T −
+θ1)2(T − θ2)3 with distinct θi, 1 ≤ i ≤ 2. We can write the quadratic k[T ]-module
+[[T − θ1, δ1]] ⊕ [[T − θ1, δ2]] ⊕ [[(T − θ2)2, η1]] ⊕ [[T − θ2, η2]]. We will construct such a
+module satisfying the additional assumptions δ2 = −1 and η1 = η2 = 1.
+An element in Stab(Rirr) acts on the first two couples with a signed type 11
+or 11, and on the last three with a signed type 111 or 111. This leaves us with
+two conjugacy classes, namely 11 · 111 and 11 · 111, and they are determined by
+the action on the first two couples, ie by the action on the two A1-singularities.
+Applying lemma 4.16 with δ2 = −1, we get the table
+N ◦
+Cl. Stab.
+Cl. Weyl
+δ1
+56
+11 · 111
+11111
+□
+57
+11 · 111
+11111
+⊠
+4.5.16. Geometric type D5. The factorization of P over Fq is P(T ) = (T −θ)5. We
+can write the quadratic k[T ]-module [[(T − θ)4, δ1]] ⊕ [[T − θ, δ2]].
+An element in Stab(Rirr) can only act on the graph trivially (the other auto-
+morphism of this graph is (C5C′
+5) which has odd signed type 11111). Thus for any
+choice of δi we get the arithmetic type 58.
+5. Construction of singular del Pezzo surfaces of degree three.
+The aim of this section is to prove the last two assertions of Theorem 2.
+We blow up the degree four surfaces from the preceding section at well chosen
+rational points in order to construct degree three surfaces, but we also use some
+direct constructions by blowing up some well chosen configurations of points in the
+projective plane.
+5.1. Blowing up degree four surfaces. Our first construction of degree three
+del Pezzo surfaces is by blowing up a rational point not lying on any of the negative
+curves of a degree four del Pezzo surface. In this way we get a surface of the same
+Dynkin type (note that for degree three, the geometric type coincides with the
+Dynkin type), whose zeta function is the one of the degree four surface, divided
+by 1 − qT . This (and the Galois action on the (−2)-curves) makes it very easy to
+control the arithmetic type of the new surface.
+In order to do this, we count the number of rational points not lying on any
+negative curve on a degree four weak del Pezzo surface of a given arithmetic type.
+If a rational point lies on a negative curve, then the curve must be defined over Fq,
+or the point is the intersection of two Galois conjugate curves (thus defined over
+Fq2). We deduce that the number we are looking for is (note that there is no 3-cycle
+in any graph of negative curves, and such three curves cannot be concurrent)
+N
+=
+#X(Fq) − (#N(Fq)(q + 1) − I1) − I2
+=
+q2 − tq + 1 − (#N(Fq)(q + 1) − I1) − I2
+where t is the trace of the action of the Frobenius operator on Pic(X⊗Fq), N(Fq) is
+the number of negative curves defined over Fq, I1 is the number of their intersection
+points (which is readily computed as an intersection number from the N(Fq) curves
+above), and I2 is the number of couples of conjugate negative curves intersecting
+themselves.
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+39
+We compute these data and present them in table 5.1, which is constructed as
+follows. For each geometric type of degree four del Pezzo surfaces, we consider the
+graph of negative curves given in [7, Proposition 6.1]. We denote the curves in the
+same way, except for the (−2)-curves : recall that we denote by ri the curve with
+class Ei − Ei+1, and by rijk the curve with class E0 − Ei − Ej − Ek.
+For each arithmetic type in degree four, we give in the second column an element
+of the Weyl group in the conjugacy class corresponding to the Frobenius action.
+We present it as a composite of reflections, where we denote by sij (resp. sijk) the
+reflection around the root Ei − Ej (resp. E0 − Ei − Ej − Ek). In the third column
+we present the action of this element on the negative curves, as a product of cycles.
+As explained above, this is sufficient to determine the numbers t, N(Fq), I1, I2 and
+N; they are given in the following columns.
+The last column gives the type of the degree three surface obtained by blowing
+up a rational point not lying on any negative curve, if any.
+Note that this is sufficient to prove the existence of a degree three del Pezzo
+surface of the type given in the last column over the finite field Fq as long as the
+number given in the last but one column is positive for the given q.
+Assume that there exists a weak del Pezzo surface of degree three and arithmetic
+type 1, defined over Fq; the Galois action on the Picard group is trivial, and all its
+negative curves are defined over Fq. It contains an exceptional curve that does not
+meet the (−2)-curve. Contracting this curve gives a degree three del Pezzo surface
+of arithmetic type 1, and the image of the curve is a point that does not lie on any
+negative curve. But such a point does not exist when q = 3 from the first line of
+the above table.
+One shows in the same way that a weak del Pezzo surface of degree three and
+arithmetic type 12 does not exist over F3. The only difference is that contracting
+an exceptional curve that does not meet the (−2)-curve gives a point outside the
+negative curves on a weak del Pezzo surface of degree four and arithmetic type 11
+or 16; such a point does not exist when q = 3.
+5.2. Other constructions. There remains 77 − 48 = 29 types to be constructed
+over any finite field with odd characteristic.
+For those we give an alternative contruction: we often present them as blow ups
+of the projective plane at well-chosen rational points, and sometimes as blow ups of
+a degree four del Pezzo surface at some point lying on one or two negative curve(s).
+First remark that we get all arithmetic types for the geometric type A1 over Fq
+by blowing up the points of a degree 6 zero-dimensional subscheme of a smooth
+conic, everything being defined over Fq, when q is large enough. Note that the only
+(−2)-curve is the strict transform of the conic. Playing on the fields of definition
+of the points in the subscheme, we get all partitions of the integer 6, and all types
+(note the stabilizer here is S6). In this way we construct a surface for each of
+the remaining types. We get type 5 by blowing up two points defined over Fq3,
+type 10 by blowing up one point defined over Fq and one over Fq5, and type 5 by
+blowing up one point defined over Fq6. Since a smooth conic defined over Fq has
+qk + 1 points over Fqk, such configurations exist over any finite field (also of even
+characteristic).
+
+40
+R. BLACHE AND E. HALLOUIN
+Type
+Frobenius in W(D5)
+Galois action on neg. curves
+(t, #N(Fq), I1, I2)
+N
+Type
+1
+A1(12)
+Id
+(6, 13, 24, 0)
+q2 − 7q + 12
+1
+2
+s12
+(ℓ1ℓ2)(ℓ13ℓ23)(ℓ14ℓ24)
+(4, 7, 6, 0)
+q2 − 3q
+2
+3
+s123
+(ℓ45q)(ℓ1ℓ23)(ℓ2ℓ13)(ℓ3ℓ12)
+(4, 5, 0, 0)
+q2 − q − 4
+2
+4
+s12 ◦ s345
+(ℓ12q)(ℓ1ℓ2)(ℓ3ℓ45)(ℓ5ℓ34)(ℓ14ℓ24)(ℓ13ℓ23)
+(2, 1, 4, 0)
+q2 + q + 4
+3
+5
+s12 ◦ s123
+(ℓ3q)(ℓ2ℓ13)(ℓ1ℓ23)(ℓ5ℓ34)(ℓ12ℓ45)
+(2, 3, 2, 2)
+q2 − q − 2
+3
+6
+s12 ◦ s345 ◦ s123
+(ℓ1ℓ13)(ℓ3q)(ℓ2ℓ23)(ℓ5ℓ34)(ℓ12ℓ45)(ℓ14ℓ24)
+(0, 1, 0, 4)
+q2 − q − 4
+4
+7
+s12 ◦ s23
+(ℓ1ℓ2ℓ3)(ℓ13ℓ12ℓ23)(ℓ14ℓ24ℓ34)
+(3, 4, 3, 0)
+q2 − q
+6
+8
+s13 ◦ s23 ◦ s145 ◦ s123
+(ℓ1qℓ3ℓ13)(ℓ2ℓ23ℓ45ℓ12)(ℓ14ℓ5ℓ34ℓ24)
+(0, 1, 0, 0)
+q2 − q
+8
+9
+s345 ◦ s23 ◦ s12
+(ℓ23qℓ12ℓ13)(ℓ2ℓ1ℓ45ℓ3)(ℓ14ℓ5ℓ34ℓ24)
+(2, 1, 0, 0)
+q2 + q
+9
+10
+s12 ◦ s23 ◦ s123
+(ℓ14ℓ24ℓ34)(ℓ45q)(ℓ1ℓ13ℓ3ℓ23ℓ2ℓ12)
+(1, 2, 1, 0)
+q2 − q
+7
+11
+2A1(9)
+Id
+(6, 11, 16, 0)
+q2 − 5q + 6
+12
+12
+s123
+(qℓ45)(ℓ1ℓ23)(ℓ3ℓ12)
+(4, 5, 4, 0)
+q2 − q − 2
+13
+13
+s123 ◦ s145
+(qℓ1)(ℓ3ℓ12)(ℓ5ℓ14)(ℓ45ℓ23)
+(2, 3, 2, 2)
+q2 − q − 2
+15
+14
+s24 ◦ s35
+(ℓ3ℓ5)(ℓ45ℓ23)(ℓ14ℓ12)(r2r4)
+(2, 3, 1, 1)
+q2 − q − 2
+14
+15
+s145 ◦ s24 ◦ s35
+(qℓ23ℓ1ℓ45)(ℓ3ℓ14ℓ12ℓ5)(r2r4)
+(2, 1, 0, 0)
+q2 + q
+19
+16
+2A1(8)
+Id
+(6, 10, 12, 0)
+q2 − 4q + 3
+12
+17
+s12
+(ℓ1ℓ2)(ℓ14ℓ24)
+(4, 6, 6, 0)
+q2 − 2q + 1
+13
+18
+s345 ◦ s12
+(ℓ1ℓ2)(ℓ14ℓ24)(ℓ3ℓ45)(ℓ34ℓ5)
+(2, 2, 0, 0)
+q2 − 1
+15
+19
+s15 ◦ s234
+(ℓ1ℓ5)(ℓ2ℓ34)(ℓ3ℓ24)(ℓ45ℓ14)(r123r4)
+(2, 0, 0, 0)
+q2 + q + 1
+14
+20
+s134 ◦ s245 ◦ s25
+(ℓ1ℓ34)(ℓ2ℓ24)(ℓ3ℓ14)(ℓ45ℓ5)(r123r4)
+(0, 0, 0, 2)
+q2 − 1
+16
+21
+s145 ◦ s234 ◦ s15 ◦ s23
+(ℓ1ℓ14)(ℓ2ℓ24)(ℓ3ℓ34)(ℓ45ℓ5)(r123r4)
+(−2, 0, 0, 4)
+q2 − 2q − 3
+17
+22
+s12 ◦ s23
+(ℓ1ℓ2ℓ3)(ℓ14ℓ24ℓ34)
+(3, 4, 3, 0)
+q2 − q
+18
+23
+s134 ◦ s15 ◦ s25
+(ℓ1ℓ5ℓ2ℓ34)(ℓ3ℓ14ℓ45ℓ24)(r123r4)
+(2, 0, 0, 0)
+q2 + 2q + 1
+19
+24
+s145 ◦ s12 ◦ s23
+(ℓ1ℓ2ℓ3ℓ45)(ℓ5ℓ14ℓ24ℓ34)
+(2, 2, 0, 0)
+q2 − 1
+20
+25
+s124 ◦ s15 ◦ s35 ◦ s23
+(ℓ1ℓ5ℓ3ℓ14ℓ45ℓ34)(ℓ2ℓ24)(r123r4)
+(1, 0, 0, 1)
+q2 + q
+21
+26
+A2(8)
+Id
+(6, 10, 13, 0)
+q2 − 4q + 4
+22
+27
+s12
+(ℓ1ℓ2)(ℓ13ℓ23)
+(4, 6, 5, 0)
+q2 − 2q
+23
+28
+s345 ◦ s12
+(ℓ1ℓ2)(ℓ13ℓ23)(ℓ5ℓ34)(ℓ12q)
+(2, 2, 1, 0)
+q2
+24
+29
+s235 ◦ s124 ◦ s14
+(ℓ1ℓ12)(ℓ2q)(ℓ5ℓ23)(ℓ13ℓ34)(r3r4)
+(0, 0, 0, 3)
+q2 − 2
+25
+30
+s135 ◦ s124 ◦ s14 ◦ s24
+(ℓ1ℓ12ℓ2q)(ℓ5ℓ13ℓ34ℓ23)(r3r4)
+(0, 0, 0, 1)
+q2
+28
+31
+3A1(6)
+Id
+(6, 9, 10, 0)
+q2 − 3q + 2
+31
+32
+s145
+(ℓ1ℓ45)(ℓ5ℓ14)
+(4, 5, 4, 0)
+q2 − q
+32
+33
+s234 ◦ s15
+(ℓ5ℓ1)(ℓ3ℓ24)(ℓ45ℓ14)(r4r123)
+(2, 1, 0, 0)
+q2 + q
+33
+34
+s234 ◦ s145 ◦ s15
+(ℓ1ℓ14)(ℓ5ℓ45)(ℓ3ℓ24)(r4r123)
+(0, 1, 0, 2)
+q2 − q − 2
+34
+35
+A1A2(6)
+Id
+(6, 9, 10, 0)
+q2 − 3q + 2
+37
+36
+s345
+(ℓ5ℓ34)(qℓ12)
+(4, 5, 4, 0)
+q2 − q
+38
+37
+A3(5)
+Id
+(6, 8, 8, 0)
+q2 − 2q + 1
+40
+38
+s125 ◦ s134
+(ℓ5ℓ12)(qℓ1)(r2r4)
+(2, 2, 1, 1)
+q2 − 1
+42
+39
+A3(4)
+Id
+(6, 7, 6, 0)
+q2 − q
+40
+40
+s345
+(ℓ5ℓ34)
+(4, 5, 4, 0)
+q2 − q
+41
+41
+s345 ◦ s12
+(ℓ1ℓ2)(ℓ5ℓ34)
+(2, 3, 2, 0)
+q2 − q
+43
+42
+s234 ◦ s15
+(ℓ1ℓ5)(ℓ2ℓ34)(r123r4)
+(2, 1, 0, 0)
+q2 + q
+42
+43
+s134 ◦ s15 ◦ s25
+(ℓ1ℓ5ℓ2ℓ34)(r123r4)
+(2, 1, 0, 0)
+q2 + q
+44
+44
+4A1(4)
+Id
+(6, 8, 8, 0)
+q2 − 2q + 1
+45
+45
+s134 ◦ s25
+(ℓ2ℓ5)(ℓ3ℓ14)(r1r345)(r4r123)
+(2, 0, 0, 0)
+q2 + 2q + 1
+46
+46
+s14 ◦ s25
+(ℓ2ℓ5)(r1r4)(r123r345)
+(2, 2, 0, 0)
+q2 − 1
+46
+47
+s124 ◦ s35
+(ℓ2ℓ14)(ℓ3ℓ5)(r4r123)
+(2, 2, 0, 0)
+q2 − 1
+47
+48
+s145 ◦ s14 ◦ s25 ◦ s35
+(ℓ2ℓ14ℓ5ℓ3)(r1r4r345r123)
+(0, 0, 0, 0)
+q2 − 1
+49
+49
+2A1A2(4)
+Id
+(6, 8, 8, 0)
+q2 − 2q + 1
+50
+50
+s134 ◦ s245 ◦ s25
+(ℓ3ℓ14)(ℓ5ℓ45)(r1r2)(r123r4)
+(0, 0, 0, 2)
+q2 − 1
+51
+51
+A1A3(3)
+Id
+(6, 7, 6, 0)
+q2 − q
+52
+52
+s345
+(ℓ34ℓ5)
+(4, 5, 4, 0)
+q2 − q
+53
+53
+A4(3)
+Id
+(6, 7, 6, 0)
+q2 − q
+60
+54
+D4(2)
+Id
+(6, 6, 5, 0)
+q2
+62
+55
+s234 ◦ s15
+(ℓ5ℓ2)(r4r123)
+(2, 2, 1, 0)
+q2
+63
+56
+2A1A3(2)
+Id
+(6, 7, 6, 0)
+q2 − q
+67
+57
+s134 ◦ s25
+(ℓ5ℓ2)(r1r345)(r123r4)
+(2, 1, 0, 0)
+q2 + q
+68
+58
+D5(1)
+Id
+(6, 6, 5, 0)
+q2
+72
+Table 4. Galois action on negative curves in degree 4
+We get all arithmetic types for the geometric type A2 over Fq by blowing up
+the points of two degree 3 zero-dimensional subschemes lying on two different lines,
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+41
+but not containing their intersection point, everything being defined over Fq for q
+large enough. Note that the (−2)-curves are the strict transforms of the two lines.
+As above, the fields of definitions of the points, and the lines, give the arithmetic
+type. We get types
+26 when we choose the two lines to be defined over Fq, with three points
+defined over Fq on one, and three conjugate points defined over Fq3 on the
+other line.
+27 when we choose the two lines to be defined over Fq, with three conjugate
+points defined over Fq3 on the both lines.
+29 when we choose the two lines to be defined over Fq, with one point defined
+over Fq and two defined over Fq2 on one, and three conjugate points defined
+over Fq3 on the other line.
+30 when we choose the two lines to be conjugate, defined over Fq2, with one
+point defined over Fq6 on one, and all its conjugates points.
+We get type 35 by blowing up three conjugate points defined over Fq3 in the
+projective plane, then three infinitely near conjugate points. Note that the (−2)-
+curves are the strict tranforms of the exceptional divisors of the first blow up.
+In order to construct a surface of type 39, we start from a degree four del Pezzo
+surface of arithmetic type 22. Such a surface contains two exceptional curves defined
+over Fq, that meet together and each one meets exactly one (−2)-curve. We blow
+up a point defined over Fq on one of these exceptional curve, different from their
+intersection point and not lying on a (−2)-curve. This is possible for any q, and we
+get the desired surface.
+In order to get a degree three surface of geometric type 4A1, we start with four
+lines in general position in the projective plane, their union being defined over
+Fq. Then we blow up their six intersection points; the (−2)-curves are the strict
+transforms of the lines. Then we get the arithmetic types by playing on the fields
+of definition of the lines. For instance we get type 48 by choosing three conjugates
+lines defined over Fq3 and the last one defined over Fq.
+If we start from a degree four surface of geometric type 2A1 defined over Fq,
+and we blow up a rational point which lies at the intersection of two exceptional
+curve (ie corresponding to a “middle” edge of the graph of negative curves), we
+get a degree three surface of geometric type 2A2 over Fq.
+In this way we get
+types 54, 55, 56, 57, 58 59 in degree three respectively from types 16, 17, 20, 21, 22
+et 25 in degree four. Note that in any case these degree four surfaces contain two
+intersecting exceptional curves which are either defined over Fq, or defined over
+Fq2 and conjugate over Fq: their intersection point is always a rational point.
+We get type 61 by blowing up p1 ≺ p2 ≺ p3 ≺ p4 four infinitely near points with
+the first three not collinear, and a point of degree 2 on a line passing through p1
+but whose strict transform by the first blow up does not contain p2.
+We get a degree three surface of type 64 by blowing up three collinear points
+p1, p2, p3 defined over Fq3 and conjugate over Fq then three conjugate points
+p4, p5, p6 with pi ≺ pi+3.
+We get a degree three del Pezzo surface of geometric type A12A2 when we blow
+up the point ℓ1 ∩ ℓ14 on a degree four del Pezzo surface of type 3A1. This point is
+defined over Fq when the degree four surface has one of the types 31 or 34, and we
+get a surface of type 65 or 66 in this way.
+
+42
+R. BLACHE AND E. HALLOUIN
+Most of the remaining constructions are completely geometric and can be found
+in [12, pp 493-494]. Namely we get type
+69 by blowing up p1 ≺ p2 ≺ p3 ≺ p4 ≺ p5 with p1, p2, p3 not collinear and p6
+on the (smooth) conic defined by the first five ones;
+70 by blowing up p1 ≺ p2 ≺ p3 ≺ p4 ≺ p5 ≺ p6 with p1, p2, p3 not collinear;
+73 by blowing up p1 ≺ p2 ≺ p3 ≺ p4 ≺ p5 ≺ p6 with p1, p2, p3 not collinear
+and p6 on the conic defined by the first five ones;
+77 by blowing up p1 ≺ p2 ≺ p3 ≺ p4 ≺ p5 ≺ p6 with p1, p2, p3 collinear.
+Type 71 in degree three is obtained by blowing up a rational point lying on the
+exceptional curve ℓ2 (but not on any (−2)-curve) in a degree four del Pezzo surface
+of arithmetic type 52.
+We end with the geometric type 3A2; we blow up three points p1, p2, p3 in
+general position in the projective plane. Then, on the resulting degree 6 ordinary
+del Pezzo surface, we blow up the intersection points ℓ1 ∩ ℓ12, ℓ2 ∩ ℓ23 and ℓ3 ∩ ℓ13.
+The (−2)-curves on the corresponding degree three del Pezzo surface are the strict
+transforms of these six lines. Finally, we get a surface of type 74 when the pi are
+rational points, of type 75 when one is rational and the two other ones defined over
+Fq2 and conjugate over Fq, and of type 76 when the three points are defined over
+Fq3 and conjugate over Fq.
+5.3. Type 36. It remains to construct a del Pezzo surface of degree three and
+arithmetic type 36. Such a surface has geometric type 3A1, and the inverse of the
+weight two part of its zeta function is Φ2
+1Φ2Φ2
+3.
+We start with a degree one del Pezzo surface S obtained by blowing up two
+degree three points p1, p2 = pσ
+1, p3 = pσ2
+1
+and q1, q2 = qσ
+1 , q3 = qσ2
+1
+in P2(Fq3) and
+a degree two point r1, r2 = rσ
+1 in P2(Fq2), such that there exists a conic passing
+through p1, p2, q1, q2, r1, r2 and defined over Fq3, but there are no other (−2)-curve
+on the resulting degree one surface than the strict transforms of this conic and its
+conjugates. We report the proof of the existence of such a configuration.
+The surface S has geometric type 3A1 by our hypothesis on its (−2)-curves
+(note that the strict transforms are separated by the blowups), and the inverse of
+the weight two part of its zeta function is (X −1)(X2 −1)(X3 −1)2 = Φ4
+1Φ2Φ2
+3. We
+can contract the strict transform of the line (r1r2), and of the conic (q1q2q3r1r2):
+these are disjoint exceptional curves that do not meet the (−2)-curves, both defined
+over Fq. In this way we get a degree three surface with geometric type 3A1 and
+the desired zeta function.
+It remains to show the existence of such a configuration of points over any finite
+field of odd characteristic.
+We start with two elements θ ∈ Fq3 \Fq, η ∈ Fq2 \Fq, whose respective minimal
+polynomials over Fq are πθ(x) := X3 + aX2 + bX + c and πη(X) = X2 + tX + n.
+We consider the points p1 = (θ : θ2 : 1), q1 = (θ : θ2 : u) and r1 = (η : 1 : 0) for
+some u ∈ Fq3 \ {0, 1}.
+From Pascal’s theorem, the points p1, p2, q2, r2, r1, q1 are conconic if and only if
+the pairs of opposite sides of the hexagon meet in three collinear points (p1p2) ∩
+(r1r2), (p2q2) ∩ (r1q1) and (p1q1) ∩ (q2r2); we get the points
+(θ − θq : θ2 − θ2q : 0), (θq : θ2q : α), (θ : θ2 : β)
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+43
+where we have set
+α := uηθ2q − θq
+ηθ2 − θ , β := uq ηqθ2 − θ
+ηqθ2q − θq
+The points are collinear if and only if we have (uβ − uqα)θ3(θq−1 − θ2(q−1)) = 0, if
+and only if uβ − uqα = 0 (note that θq−1 ̸= 1). We easily check that the preceding
+equality gives
+u ∈ (ηθ2 − θ)(ηqθ2 − θ)F×
+q = (nθ4 + tθ3 + θ2)F×
+q
+In other words, for our choice of the points, there are exactly q−1 values of u in Fq3
+such that the points p1, p2, q2, r2, r1, q1 are conconic; note that using the Frobenius
+action, we also get that the points p2, p3, q3, r2, r1, q2 are conconic, so as the points
+p1, p3, q3, r2, r1, q1.
+We remark that the conic C1 passing through p1, p2, q2, r2, r1, q1 must be irre-
+ducible. Else it would be the union of two lines d and d′. If d contains r1, it cannot
+contain r2 = rσ
+1 : it would be the line Z = 0 and the other line would contain the
+four remaining points; this is impossible since p2 does not lie on (p1q1). Thus dσ3
+contains r2, d is not defined over Fq3, and we get d′ = dσ3. In this case both d and
+dσ3 would contain p1, p2, q1, q2, this is impossible.
+It remains to verify that one can choose u such that there is no other (−2)-curve
+on the degree one surface obtained by blowing up the eight points. In other words,
+no three points can be collinear, no six can lie on a conic, and there is no cubic
+passing through the eight points which is singular at one of them.
+We first check that no three of the eight points can be collinear. If such a subset
+contains r1 and r2, this is clear. If it contains r1, then the line joining r1 to any
+degree three point is defined over Fq6, and cannot contain any other point defined
+over Fq3.
+We are reduced to the subsets of three points among the p1, p2, p3, q1, q2, q3. Any
+subset of the form {pi, pj, qk} or {qi, qj, pk} with k ∈ {i, j} cannot be collinear; else
+the corresponding line would cross one of the three irreducible conics conjugate
+to C1 above at three points. We are reduced to consider the subsets {p1, p2, p3},
+{q1, q2, q3}, and (up to Galois action) {p1, p2, q3}, {q1, q2, p3}.
+Note that a point (x : y : z) ∈ P2(Fq3) and its conjugates over Fq are collinear
+if and only if the Fq subspace generated by x, y, z is strictly contained in Fq3. As
+a consequence, it follows from the definition of p1 that the points p1, p2, p3 are not
+collinear. Now write
+u = m(nθ4 + tθ3 + θ2) = m
+�
+(1 − nb − a(t − na))θ2 − (nc + t − na)θ − (t − na)c
+�
+for some m ∈ F×
+q . From the above criterion, we get that the points q1, q2, q3 are
+collinear if and only if t = na (of course we have c ̸= 0).
+The points p1, p2, q3 are collinear if and only if
+������
+1
+θ
+θ2
+1
+θq
+θ2q
+uq2
+θq2
+θ2q2
+������
+= 0
+This equation is (semi)-linear in the variable u, and admits a unique solution.
+In the same way, the points q1, q2, p3 are collinear if and only if
+u(θ2q2+q − θq2+2q) − uq(θ2q2+1 − θq2+2) + θ2q+1 − θq+2 = 0
+
+44
+R. BLACHE AND E. HALLOUIN
+Setting U = u(θ2q2+q − θq2+2q), we can rewrite the equation U + U q + θ2q+1 −
+θq+2 = 0. Since gcd(T q + T, T q3 − T ) = T , the map U �→ U q + U is an Fq-linear
+automorphism of Fq3, and this equation admits exactly one solution.
+Now choose six of the eight points, and assume they lie on a conic. If ri is one
+of these points, then among the five remaining points, four must lie on one of the
+conjugates of C1; from Bezout theorem, the conic must be one of these, and the
+subset of six conconic points is among the three we already constructed. It remains
+to consider the six points p1, p2, p3, q1, q2, q3; any conic passing through these points
+should be defined over Fq, et thus have an equation of the form
+eX2 + hXY + iY 2 + jXZ + kY Z + lZ2
+Plugging the coordinates of p1 and q1, substracting and factoring 1 − u, we get
+jθ + kθ2 + l(1 + u) = 0
+so that the family (θ, θ2, 1 + u) does not generate the Fq-vector space Fq3. This
+happens if, and only if 1 + mc(t − na) = 0 – with u = m(nθ4 + tθ3 + θ2).
+It remains to show that there is no cubic passing through the eight points and
+singular at one of them. If such a cubic C exists, it has exactly one singular point,
+which is not defined over Fq. So it is distinct from Cσ. Now these two cubics have
+at least ten intersection points (counted with multiplicities), which is impossible.
+Summing up, we have to choose η and θ as above, such that the coefficients of
+their minimal polynomials satisfy t − na ̸= 0; then we get q − 1 possible values for
+u such that the three prescribed subsets of six points are conconic. We remove one
+possible value to verify that the six points pi, qi do not lie on a conic, then at most
+one for each of the assertions p1, p2, q3 non collinear, and q1, q2, p3 non collinear.
+We get a least q − 4 possible values for u, and the problem is settled for q ≥ 5.
+For q = 3, one can verify that if θ is a root of T 3 + T 2 + 2, and η a root of
+T 2 + T + 2, then for m = −1 the points defined above have the required position.
+Appendix A. Arithmetic types for degrees three to six
+We give below the lists of arithmetic types for degrees 3 ≤ d ≤ 6. The tables are
+organized as follows
+• in the first column, we fix a number for each type; there are also three
+types (one in degree four and two in degree three) for which we add an
+asterisk. This means that the invariant H1(Γ, Pic(X ⊗ Fq)) is non trivial
+(it is isomorphic to the Klein four-group Z/2Z × Z/2Z in each case), and
+that a surface of this type is not birational to the projective plane [21,
+Section 29].
+• in the second one, we describe the geometric types for the given degrees, in
+other words the closed and symmetric parts of a root system of type E9−d
+up to the action of the Weyl group; they come from [7], except for degree
+3 where they are given in [8, Table IV (iv)]. We give the Dynkin types of
+the singular points and (between parentheses) the number of exceptional
+divisors on the surface;
+• in the third column, we give the stabilizer attached to the geometric type;
+• in the fourth one, we give the characteristic polynomial of the action of any
+element w in the conjugacy class of the stabilizer on the geometric Picard
+group Pic(X) of a weak del Pezzo surface of the type;
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+45
+• finally, we give the characteristic polynomial of the action of any element
+w in the conjugacy class of the stabilizer on the geometric Picard group
+Pic(Xs) of a singular del Pezzo surface of a type.
+Moreover, in the table for degree 4, we add a column, in order to describe each
+conjugacy class in W(D5) by its signed cycle type.
+Table 5: Arithmetic types in degree 6
+Type T
+Geometric type
+Stab
+χw,Pic(X)
+χw,Pic(Xs)
+1
+A1 (4)
+Z/2Z
+Φ4
+1
+Φ3
+1
+2
+Φ3
+1Φ2
+Φ2
+1Φ2
+3
+A1 (3)
+S3
+Φ4
+1
+Φ3
+1
+4
+Φ3
+1Φ2
+Φ2
+1Φ2
+5
+Φ2
+1Φ3
+Φ1Φ3
+6
+2A1 (2)
+{e}
+Φ4
+1
+Φ2
+1
+7
+A2 (2)
+Z/2Z
+Φ4
+1
+Φ2
+1
+8
+Φ3
+1Φ2
+Φ1Φ2
+9
+A2A1 (1)
+{e}
+Φ4
+1
+Φ1
+Table 6: Arithmetic types in degree 5
+Type T
+Geometric type
+Stab
+χw,Pic(X)
+χw,Pic(Xs)
+1
+A1 (7)
+S3
+Φ5
+1
+Φ4
+1
+2
+Φ4
+1Φ2
+Φ3
+1Φ2
+3
+Φ3
+1Φ3
+Φ2
+1Φ3
+4
+2A1 (5)
+Z/2Z
+Φ5
+1
+Φ3
+1
+5
+Φ3
+1Φ2
+2
+Φ2
+1Φ2
+6
+A2 (4)
+Z/2Z
+Φ5
+1
+Φ3
+1
+7
+Φ4
+1Φ2
+Φ2
+1Φ2
+8
+A2A1 (3)
+{e}
+Φ5
+1
+Φ2
+1
+9
+A3 (2)
+{e}
+Φ5
+1
+Φ2
+1
+10
+A4 (1)
+{e}
+Φ5
+1
+Φ1
+Table 7: Arithmetic types in degree 4
+Type T
+Geometric type
+Stab
+W(D5)
+χw,Pic(X)
+χw,Pic(Xs)
+1
+A1 (12)
+S4 × Z/2Z
+11111
+Φ6
+1
+Φ5
+1
+2
+2111
+Φ5
+1Φ2
+Φ4
+1Φ2
+3
+2111
+Φ5
+1Φ2
+Φ4
+1Φ2
+4
+11111
+Φ4
+1Φ2
+2
+Φ3
+1Φ2
+2
+5
+221
+Φ4
+1Φ2
+2
+Φ3
+1Φ2
+2
+6
+2111
+Φ3
+1Φ3
+2
+Φ2
+1Φ3
+2
+7
+311
+Φ4
+1Φ3
+Φ3
+1Φ3
+8
+221
+Φ2
+1Φ2
+2Φ4
+Φ1Φ2
+2Φ4
+9
+1121
+Φ3
+1Φ2Φ4
+Φ2
+1Φ2Φ4
+10
+32
+Φ3
+1Φ2Φ3
+Φ2
+1Φ2Φ3
+
+46
+R. BLACHE AND E. HALLOUIN
+Table 7: Arithmetic types in degree 4
+Type T
+Geometric type
+Stab
+W(D5)
+χw,Pic(X)
+χw,Pic(Xs)
+11
+2A1 (9)
+D8
+11111
+Φ6
+1
+Φ4
+1
+12
+2111
+Φ5
+1Φ2
+Φ3
+1Φ2
+13
+221
+Φ4
+1Φ2
+2
+Φ2
+1Φ2
+2
+14
+221
+Φ4
+1Φ2
+2
+Φ3
+1Φ2
+15
+41
+Φ3
+1Φ2Φ4
+Φ2
+1Φ4
+16
+2A1 (8)
+S4 × Z/2Z
+11111
+Φ6
+1
+Φ4
+1
+17
+2111
+Φ5
+1Φ2
+Φ3
+1Φ2
+18
+11111
+Φ4
+1Φ2
+2
+Φ2
+1Φ2
+2
+19
+11111
+Φ4
+1Φ2
+2
+Φ3
+1Φ2
+20
+2111
+Φ3
+1Φ3
+2
+Φ2
+1Φ2
+2
+21∗
+11111
+Φ2
+1Φ4
+2
+Φ1Φ3
+2
+22
+311
+Φ4
+1Φ3
+Φ2
+1Φ3
+23
+2111
+Φ3
+1Φ2Φ4
+Φ2
+1Φ4
+24
+2111
+Φ3
+1Φ2Φ4
+Φ1Φ2Φ4
+25
+311
+Φ2
+1Φ2
+2Φ6
+Φ1Φ2Φ6
+26
+A2 (8)
+D8
+11111
+Φ6
+1
+Φ4
+1
+27
+2111
+Φ5
+1Φ2
+Φ3
+1Φ2
+28
+11111
+Φ4
+1Φ2
+2
+Φ2
+1Φ2
+2
+29
+2111
+Φ3
+1Φ3
+2
+Φ2
+1Φ2
+2
+30
+221
+Φ2
+1Φ2
+2Φ4
+Φ1Φ2Φ4
+31
+3A1 (6)
+(Z/2Z)2
+11111
+Φ6
+1
+Φ3
+1
+32
+2111
+Φ5
+1Φ2
+Φ2
+1Φ2
+33
+11111
+Φ4
+1Φ2
+2
+Φ2
+1Φ2
+34
+2111
+Φ3
+1Φ3
+2
+Φ1Φ2
+2
+35
+A1A2 (6)
+Z/2Z
+11111
+Φ6
+1
+Φ3
+1
+36
+2111
+Φ5
+1Φ2
+Φ2
+1Φ2
+37
+A3 (5)
+Z/2Z
+11111
+Φ6
+1
+Φ3
+1
+38
+221
+Φ4
+1Φ2
+2
+Φ2
+1Φ2
+39
+A3 (4)
+D8
+11111
+Φ6
+1
+Φ3
+1
+40
+2111
+Φ5
+1Φ2
+Φ2
+1Φ2
+41
+11111
+Φ4
+1Φ2
+2
+Φ1Φ2
+2
+42
+11111
+Φ4
+1Φ2
+2
+Φ2
+1Φ2
+43
+2111
+Φ3
+1Φ2Φ4
+Φ1Φ4
+44
+4A1 (4)
+D8
+11111
+Φ6
+1
+Φ2
+1
+45
+11111
+Φ4
+1Φ2
+2
+Φ2
+1
+46
+221
+Φ4
+1Φ2
+2
+Φ2
+1
+47
+11111
+Φ4
+1Φ2
+2
+Φ1Φ2
+48
+221
+Φ2
+1Φ2
+2Φ4
+Φ1Φ2
+49
+2A1A2 (4)
+Z/2Z
+11111
+Φ6
+1
+Φ2
+1
+50
+2111
+Φ3
+1Φ3
+2
+Φ1Φ2
+51
+A1A3 (3)
+Z/2Z
+11111
+Φ6
+1
+Φ2
+1
+52
+2111
+Φ5
+1Φ2
+Φ1Φ2
+53
+A4 (3)
+{e}
+11111
+Φ6
+1
+Φ2
+1
+54
+D4 (2)
+Z/2Z
+11111
+Φ6
+1
+Φ2
+1
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+47
+Table 7: Arithmetic types in degree 4
+Type T
+Geometric type
+Stab
+W(D5)
+χw,Pic(X)
+χw,Pic(Xs)
+55
+11111
+Φ4
+1Φ2
+2
+Φ1Φ2
+56
+2A1A3 (2)
+Z/2Z
+11111
+Φ6
+1
+Φ1
+57
+11111
+Φ4
+1Φ2
+2
+Φ1
+58
+D5 (1)
+{e}
+11111
+Φ6
+1
+Φ1
+Table 8: Arithmetic types in degree 3
+Type T
+Geometric type
+Stab
+χw,Pic(X)
+χw,Pic(Xs)
+1
+A1 (21)
+S6
+Φ7
+1
+Φ6
+1
+2
+Φ6
+1Φ2
+Φ5
+1Φ2
+3
+Φ5
+1Φ2
+2
+Φ4
+1Φ2
+2
+4
+Φ4
+1Φ3
+2
+Φ3
+1Φ3
+2
+5
+Φ3
+1Φ2
+3
+Φ2
+1Φ2
+3
+6
+Φ5
+1Φ3
+Φ4
+1Φ3
+7
+Φ4
+1Φ2Φ3
+Φ3
+1Φ2Φ3
+8
+Φ3
+1Φ2
+2Φ4
+Φ2
+1Φ2
+2Φ4
+9
+Φ4
+1Φ2Φ4
+Φ3
+1Φ2Φ4
+10
+Φ3
+1Φ5
+Φ2
+1Φ5
+11
+Φ2
+1Φ2Φ3Φ6
+Φ1Φ2Φ3Φ6
+12
+2A1 (16)
+S4 × Z/2Z
+Φ7
+1
+Φ5
+1
+13
+Φ6
+1Φ2
+Φ4
+1Φ2
+14
+Φ5
+1Φ2
+2
+Φ4
+1Φ2
+15
+Φ5
+1Φ2
+2
+Φ3
+1Φ2
+2
+16
+Φ4
+1Φ3
+2
+Φ3
+1Φ2
+2
+17∗
+Φ3
+1Φ4
+2
+Φ2
+1Φ3
+2
+18
+Φ5
+1Φ3
+Φ3
+1Φ3
+19
+Φ4
+1Φ2Φ4
+Φ3
+1Φ4
+20
+Φ4
+1Φ2Φ4
+Φ2
+1Φ2Φ4
+21
+Φ3
+1Φ2
+2Φ6
+Φ2
+1Φ2Φ6
+22
+A2 (15)
+S3 ≀ Z/2Z
+Φ7
+1
+Φ5
+1
+23
+Φ6
+1Φ2
+Φ4
+1Φ2
+24
+Φ5
+1Φ2
+2
+Φ3
+1Φ2
+2
+25
+Φ4
+1Φ3
+2
+Φ3
+1Φ2
+2
+26
+Φ5
+1Φ3
+Φ3
+1Φ3
+27
+Φ3
+1Φ2
+3
+Φ1Φ2
+3
+28
+Φ3
+1Φ2
+2Φ4
+Φ2
+1Φ2Φ4
+29
+Φ4
+1Φ2Φ3
+Φ2
+1Φ2Φ3
+30
+Φ2
+1Φ2Φ3Φ6
+Φ1Φ3Φ6
+31
+3A1 (12)
+D12
+Φ7
+1
+Φ4
+1
+32
+Φ6
+1Φ2
+Φ3
+1Φ2
+33
+Φ5
+1Φ2
+2
+Φ5
+1Φ2
+34
+Φ4
+1Φ3
+2
+Φ2
+1Φ2
+2
+35
+Φ3
+1Φ2
+3
+Φ2
+1Φ3
+36
+Φ2
+1Φ2Φ2
+3
+Φ1Φ2Φ3
+
+48
+R. BLACHE AND E. HALLOUIN
+Table 8: Arithmetic types in degree 3
+Type T
+Geometric type
+Stab
+χw,Pic(X)
+χw,Pic(Xs)
+37
+A1A2 (11)
+S3
+Φ7
+1
+Φ4
+1
+38
+Φ6
+1Φ2
+Φ3
+1Φ2
+39
+Φ5
+1Φ3
+Φ2
+1Φ3
+40
+A3 (10)
+D8
+Φ7
+1
+Φ4
+1
+41
+Φ6
+1Φ2
+Φ3
+1Φ2
+42
+Φ5
+1Φ2
+2
+Φ3
+1Φ2
+43
+Φ5
+1Φ2
+2
+Φ2
+1Φ2
+2
+44
+Φ4
+1Φ2Φ4
+Φ2
+1Φ4
+45
+4A1 (9)
+S4
+Φ7
+1
+Φ3
+1
+46
+Φ5
+1Φ2
+2
+Φ3
+1
+47
+Φ5
+1Φ2
+2
+Φ2
+1Φ2
+48
+Φ3
+1Φ2
+3
+Φ1Φ3
+49
+Φ3
+1Φ2
+2Φ4
+Φ2
+1Φ2
+50
+2A1A2 (8)
+Z/2Z
+Φ7
+1
+Φ3
+1
+51
+Φ4
+1Φ3
+2
+Φ2
+1Φ2
+52
+A1A3 (7)
+Z/2Z
+Φ7
+1
+Φ3
+1
+53
+Φ6
+1Φ2
+Φ2
+1Φ2
+54
+2A2 (7)
+D12
+Φ7
+1
+Φ3
+1
+55
+Φ6
+1Φ2
+Φ2
+1Φ2
+56
+Φ4
+1Φ3
+2
+Φ2
+1Φ2
+57∗
+Φ3
+1Φ4
+2
+Φ1Φ2
+2
+58
+Φ5
+1Φ3
+Φ1Φ3
+59
+Φ3
+1Φ2
+2Φ6
+Φ1Φ6
+60
+A4 (6)
+Z/2Z
+Φ7
+1
+Φ3
+1
+61
+Φ6
+1Φ2
+Φ2
+1Φ2
+62
+D4 (6)
+S3
+Φ7
+1
+Φ3
+1
+63
+Φ5
+1Φ2
+2
+Φ2
+1Φ2
+64
+Φ3
+1Φ2
+3
+Φ1Φ3
+65
+A12A2 (5)
+Z/2Z
+Φ7
+1
+Φ2
+1
+66
+Φ4
+1Φ3
+2
+Φ1Φ2
+67
+2A1A3 (5)
+Z/2Z
+Φ7
+1
+Φ2
+1
+68
+Φ5
+1Φ2
+2
+Φ2
+1
+69
+A1A4 (4)
+{e}
+Φ7
+1
+$Phi3
+1
+70
+A5 (3)
+Z/2Z
+Φ7
+1
+Φ2
+1
+71
+Φ6
+1Φ2
+Φ1Φ2
+72
+D5 (3)
+{e}
+Φ7
+1
+Φ2
+1
+73
+A1A5 (2)
+{e}
+Φ7
+1
+Φ1
+74
+3A2 (3)
+S3
+Φ7
+1
+Φ1
+75
+Φ4
+1Φ3
+2
+Φ1
+76
+Φ3
+1Φ2
+3
+Φ1
+77
+E6 (1)
+{e}
+Φ7
+1
+Φ1
+References
+1. M. Artin, Coverings of the rational double points in characteristic p, Complex analysis and
+algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 11–22.
+
+CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS
+49
+2. Barinder Banwait, Francesc Fit´e, and Daniel Loughran, Del Pezzo surfaces over finite fields
+and their Frobenius traces, Math. Proc. Cambridge Philos. Soc. 167 (2019), no. 1, 35–60.
+3. M. J. Bright, N. Bruin, E. V. Flynn, and A. Logan, The Brauer-Manin obstruction and X[2],
+LMS J. Comput. Math. 10 (2007), 354–377.
+4. Martin Bright, Brauer groups of singular del Pezzo surfaces, Michigan Math. J. 62 (2013),
+no. 3, 657–664.
+5. R. W. Carter, Conjugacy classes in the weyl group, Compositio Mathematica 25 (1972), no. 1,
+1–59.
+6. Arthur Cayley, A memoir on cubic surfaces, Philos. Trans. Roy. Soc. London 159 (1869),
+231–326.
+7. D. F. Coray and M. A. Tsfasman, Arithmetic on singular Del Pezzo surfaces, Proc. London
+Math. Soc. (3) 57 (1988), no. 1, 25–87.
+8. Harold S. M. Coxeter, Finite groups generated by reflections, and their subgroups generated
+by reflections, Proc Cambridge Philos. Soc. 30 (1934), 466–482.
+9. Michel Demazure, Surfaces de del pezzo in S´eminaire sur les Singularit´es des Surfaces, Lecture
+Notes in Mathematics, vol. 777, Springer, Berlin, 1980.
+10. Ulrich Derenthal, Singular del Pezzo surfaces whose universal torsors are hypersurfaces, Proc.
+Lond. Math. Soc. (3) 108 (2014), no. 3, 638–681.
+11. Igor Dolgachev and Alexander Duncan, Regular pairs of quadratic forms on odd-dimensional
+spaces in characteristic 2, Algebra Number Theory 12 (2018), no. 1, 99–130.
+12. Igor V. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge,
+2012.
+13. Patrick Du Val, On isolated singularities which do not affect the conditions of adjunction.
+III, Proc. Cambridge Philos. Soc 30 (1934), no. 3, 483–491.
+14. E. V. Flynn, Homogeneous spaces and degree 4 del Pezzo surfaces, Manuscripta Math. 129
+(2009), no. 3, 369–380.
+15. Robin Hartshorne, Algebraic geometry, Graduate Studies in Mathematics, vol. 52, Springer,
+1977.
+16. Tatsuro Kawakami and Masaru Nagaoka, Pathologies and liftability of Du Val del Pezzo
+surfaces in positive characteristic, Math. Z. 301 (2022), no. 3, 2975–3017.
+17. J´anos Koll´ar, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer
+Grenzgebiete, vol. 32, Springer-Verlag, Berlin, 1996.
+18. B. `E. Kunyavski˘ı, A. N. Skorobogatov, and M. A. Tsfasman, del Pezzo surfaces of degree four,
+M´em. Soc. Math. France (N.S.) (1989), no. 37, 113.
+19. Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique
+factorization, Inst. Hautes ´Etudes Sci. Publ. Math. (1969), no. 36, 195–279.
+20. Daniel Loughran and Andrey Trepalin, Inverse Galois problem for del Pezzo surfaces over
+finite fields, Math. Res. Lett. 27 (2020), no. 3, 845–853.
+21. Yu. I. Manin, Cubic forms, second ed., North-Holland Mathematical Library, vol. 4, North-
+Holland Publishing Co., Amsterdam, 1986.
+22. J.-Y. M´erindol, Les singularit´es simples elliptiques, leurs d´eformations, les surfaces de del
+Pezzo et les transformations quadratiques, Ann. Sci. ´Ecole Norm. Sup. (4) 15 (1982), no. 1,
+17–44.
+23. Ludwig Schl¨afli, On the distribution of surfaces of the third order into species, in reference to
+the absence or presence of singular points, and the reality of their lines, Philos. Trans. Roy.
+Soc. London 153 (1863), 193–241.
+24. Alexei Skorobogatov, del Pezzo surfaces of degree 4 and their relation to Kummer surfaces,
+Enseign. Math. (2) 56 (2010), no. 1-2, 73–85.
+25. H. P. F. Swinnerton-Dyer, The zeta function of a cubic surface over a finite field, Proc.
+Cambridge Philos. Soc. 63 (1967), 55–71.
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+32.
+27. William C. Waterhouse, Pairs of quadratic forms, Invent. Math. 37 (1976), no. 2, 157–164.
+
+50
+R. BLACHE AND E. HALLOUIN
+LAMIA, Universit´e des Antilles
+Email address: regis.blache@univ-antilles.fr
+Institut de Math´ematiques de Toulouse, UMR 5219
+Email address: hallouin@univ-tlse2.fr
+
diff --git a/w9FRT4oBgHgl3EQfhDe7/content/tmp_files/load_file.txt b/w9FRT4oBgHgl3EQfhDe7/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a0a876ba719ed6f2ef1217bfad93fba28fc9b90f
--- /dev/null
+++ b/w9FRT4oBgHgl3EQfhDe7/content/tmp_files/load_file.txt
@@ -0,0 +1,2736 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf,len=2735
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='13582v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='AG]  31 Jan 2023  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES  OVER FINITE FIELDS  R´EGIS BLACHE AND EMMANUEL HALLOUIN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In this article, we consider weak del Pezzo surfaces defined over a  finite field, and their associated, singular, anticanonical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We first define arithmetic types for such surfaces, by considering the Frobe-  nius actions on their Picard groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' this extends the classification of Swinnerton-  Dyer and Manin for ordinary del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We also show that some  invariants of the surfaces only depend on the above type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Then we study an inverse Galois problem for singular del Pezzo surfaces  having degree 3 ≤ d ≤ 6: we describe which types can occur over a given finite  field (of odd characteristic when 3 ≤ d ≤ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Contents  Introduction  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Weak del Pezzo surfaces  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A blow-up model  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Roots, exceptional curves and geometric types  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Arithmetic types  8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Singular del Pezzo surfaces  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Anticanonical models: geometric aspects  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Divisor class groups and zeta functions  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Construction of weak del Pezzo surfaces of degree at least five  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Degrees seven and eight  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Degree six  14  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Degree five  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Construction of singular del Pezzo surfaces of degree four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Pencils of quadrics, their Segre symbols, and geometric types  16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Morphisms to the projective line  18  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Galois action on the singular quadrics, and arithmetic types  23  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Quadratic module associated to a pencil of quadrics  24  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Construction of degree four del Pezzo surfaces of any arithmetic type  29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Construction of singular del Pezzo surfaces of degree three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  38  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Blowing up degree four surfaces  38  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Other constructions  39  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Type 36  42  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Arithmetic types for degrees three to six  44  References  48  Date: February 1, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Primary 11G25, 14J26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Secondary 14G10, 11E12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' del Pezzo surfaces over finite fields, zeta functions, quadratic modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  1  2  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  Introduction  In this article, we study certain del Pezzo surfaces defined over a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Recall that a smooth projective surface X is a weak del Pezzo surface when its  anticanonical divisor −KX is big and nef;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' its degree is the self-intersection number  d := K·2  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' If moreover −KX is ample, we call X an ordinary del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  When a weak del Pezzo surface X is not ordinary, it contains absolutely irreducible  curves with self intersection −2, that we shall call (−2)-curves in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The anticanonical model of a weak, non ordinary del Pezzo surface X is a singular  surface, that we denote by Xs, and call a singular del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that  Xs has Du Val singularities, and is Gorenstein;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' for these reasons such surfaces are  sometimes called Du Val del Pezzo or Gorenstein del Pezzo in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The study of complex singular cubic surfaces (degree 3 del Pezzo surfaces over  C) dates back to the nineteenth century [6, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It has been generalized to del  Pezzo surfaces along the twentieth century [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In particular we know all types of  singularities (sometimes called the Dynkin types) that can occur in characteristic  zero [12, Chapter 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Over an algebraically closed field of positive characteristic,  some new types occur, but only in characteristic 2 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The interest on del Pezzo surfaces over finite fields is more recent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' If X is an  ordinary del Pezzo surface defined over the finite field Fq, where q = pm is a power  of a prime, the Frobenius action σ∗ on Pic(X ⊗Fq) must preserve the anticanonical  class and the intersection product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The group of automorphisms of Pic(X ⊗ Fq)  with these properties is a (finite) Weyl group depending on the degree of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Thus the image of the Galois group Gal(Fq/Fq) is a cyclic subgroup generated by  the image of the Frobenius morphism σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' its conjugacy class is the arithmetic type  of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Swinnerton-Dyer [25] and Manin [21] construct tables of conjugacy classes in  these Weyl groups in order to classify ordinary del Pezzo surfaces over finite fields  (the table for degree 3 has been corrected in [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Many invariants of a del Pezzo surface only depend on its arithmetic type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' this  is the case for its zeta function [21, Theorem 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1, Corollary 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2]  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1)  Z(X, T )−1 = (1 − T )(1 − q2T ) det  �  I − qT σ∗| Pic(X ⊗ k)  �  The first aim of this paper is to extend the classification of Swinnerton-Dyer and  Manin to weak del Pezzo surfaces, by defining their arithmetic type, and to give an  expression for their zeta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We begin by classifying the possible singularities over the algebraic closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Fol-  lowing [7, 10], we define a geometric type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This is the configuration of negative  curves on the surface X ⊗ k (corresponding to lines and singularities on the anti-  canonical model), up to the action of the Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It is finer than the Dynkin  type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The types are orbits of the Weyl action on the root bases in the lattice E9−d  corresponding to the degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Coming back to the surface X, the Galois action must preserve the set Rirr of  (−2)-curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Thus the Frobenius maps to an element of its stabilizer Stab(Rirr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We define the arithmetic type of X as the conjugacy class of the Frobenius action  in this last group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Here again, many properties of a weak del Pezzo surface only depend on its  arithmetic type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' this is true for its zeta function from (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1), and we show that it  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  3  remains true for its anticanonical model Xs (note that its geometric Picard lattice  is the orthogonal of Rirr in the geometric Picard group of X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We have the equality  Z(Xs, T )−1 = (1 − T )(1 − q2T ) det  �  I − qT σ∗| Pic(Xs ⊗ k)  �  Our second aim is to construct such surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' When the finite field is small, not  all are constructible [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' For instance, over the field F2, there are no split ordinary  del Pezzo surfaces of degree d ≤ 4 since there are at most four points in general  position in P2(F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  This is the inverse Galois problem for singular del Pezzo surfaces over finite  fields: in the ordinary case, it asks for which conjugacy classes of the Weyl group  can arise as the conjugacy class of the Frobenius action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It has been solved for  ordinary degree four surfaces in [26, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='4], and for ordinary degree three  and two surfaces in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' There is also a weaker version of this problem, asking for  which integers can arise as the trace of the Frobenius action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It has been solved for  ordinary del Pezzo surfaces, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It should be easier for non ordinary del Pezzo surfaces since when we consider  them as blowups of the projective plane, we relax the condition of blowing up points  in general position to almost general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' But when the degree is at most four  and the base field is not algebraically closed, some of the surfaces are no longer  birational to the projective plane, and we have to provides other constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In this paper we solve the inverse Galois problem for singular del Pezzo surfaces  of degree at least 5 over any finite field, and for surfaces of degree 3 or 4 when the  characteristic is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We refer to Appendix A for the tables of arithmetic types for  each degree 3 ≤ d ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' There exists a weak, non ordinary del Pezzo surface of degree d and  any arithmetic type T over the finite field Fq in the following cases  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we have d ≥ 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we have d = 4 and Fq has odd characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we have d = 3, Fq has odd characteristic and (q, T) /∈ {(3, 1), (3, 12)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  There does not exist any surface of degree 3 and arithm´etic type 1 or 12 over F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The types corresponding to the degrees d ≥ 5 are not very difficult to construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It turns out that these surfaces are birational to P2, and it is sufficient to blowup  the projective plane at well chosen points, and to contract exceptional curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In the case of degree four, we no longer consider a blowup model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We exploit an  idea which is present in [3, 14, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The anticanonical model of a del Pezzo surface  a degree four is the base locus of a pencil of quadrics in P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' To such a pencil we  associate a quadratic Fq|T ]-module (equivalently, a Frobenius algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Now fixing  a geometric type, then an arithmetic type for a del Pezzo surface, gives a precise  description of the quadratic module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We construct such modules with prescribed  arithmetic properties, and their existence is sufficient to prove the second assertion  of the above theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that these quadratic modules are rather different in  even characteristic, since there a bilinear form over an odd dimensionnal vector  space must be degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This is why we do not consider that case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' However,  nice normal forms have been described in this case [11], that should help solving  this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  4  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  Note that we have used the mathematical software magma to construct explicit  models (ie a couple of quadrics in P4) for all types of degree four singular del Pezzo  surfaces over a given finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The code is freely available on the second author’s  webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Finally, we construct degree three surfaces in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Our main con-  struction is by blowing up a point – not lying on any negative curve – on a well  chosen degree four del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We count the numbers of such points for each  arithmetic type belonging to degree four;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' this allows us to show the existence of  many degree three surfaces with given arithmetic type, but also – when there is  no such point – the non existence result stated in the last sentence of the above  Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We also use other types of blow up (of the projective plane, or a degree  four surface at a point lying on one or more exceptional curve) in order to construct  the surfaces belonging to the remaining arithmetic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The paper is organised as follows: in section 1, we describe weak del Pezzo  surfaces, and recall their principal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  This allows us to introduce the  classification and to define the different types (geometric, then arithmetic) that  we use in the rest of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then we turn to the description of singular del  Pezzo surfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we briefly describe their different groups of divisors, and we show  Theorem 1 in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The next section is devoted to the proof of the first assertion  in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The fourth section is more technical: we treat the degree four del  Pezzo surfaces over finite fields of odd characteristic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' to such a variety, we associate  a quadratic Fq|T ]-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then most of the work is devoted to describing the link  between the arithmetic properties of the module and the arithmetic type of the  surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' this allows us to prove the second assertion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The last section  mainly builds on the preceding one: we blow up the degree four surfaces at well  chosen points in order to construct degree three surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We also use some direct  constructions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' this allows us to prove the last two assertions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We give the description of the different arithmetic types for degree d ≥ 3 in the  Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weak del Pezzo surfaces  We first define the smooth surfaces we shall consider in this article  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' A smooth projective surface X defined over a field k is a weak del  Pezzo surface when its anticanonical divisor −KX is  (i) big, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' K·2  X > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (ii) and nef, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' for any effective divisor D on X, (−KX) · D ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It is an ordinary del Pezzo surface when −KX is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Its degree is d := K·2  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Note that since −KX is nef, the adjunction formula ensures that for any abso-  lutely irreducible curve C on X, we have C · C ≥ C · (C + KX) = 2pa(C) − 2 ≥ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Moreover, if for such a curve this inequality is an equality, then we have C ·KX = 0,  and from the Nakai-Moishezon criterion the anticanonical divisor is not ample: the  surface X is a not an ordinary del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It is well known [9, 7] that the geometry of del Pezzo surfaces depends to a  large extent of its absolutely irreducible curves with negative self-intersection, the  so-called negative curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' These are the generalization of the celebrated 27 lines on  an smooth cubic hypersurface in P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  5  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let X denote a weak del Pezzo surface over the field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element D in the geometric Picard group Pic(X ⊗k) is an exceptional divisor  when D·2 = D · KX = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An absolutely irreducible curve C on X whose class is an exceptional divisor is  an exceptional curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element D in Pic(X ⊗k) is a root when it satisfies D·2 = −2 and D·KX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We denote by R(X) the set of roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  When C is a curve on X whose class is a root, we say that C (or its class) is an  effective root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' If moreover C is absolutely irreducible, then it is a (−2)-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We denote by A the union of the (−2)-curves on X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' this is a closed subscheme  of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We denote by U the complementary of A on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' First remark that the negative curves which are absolutely irreducible  are isomorphic to P1 from the adjunction formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The sets of exceptional divisors and roots are finite and depend up to isomor-  phism only on the degree of the surface [9, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Tables 2 et 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Note also (contrary to the case of ordinary del Pezzo surfaces) that all exceptional  divisors need not correspond to exceptional curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' see Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='11 for a numerical  criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Finally, the sets of exceptional and (−2)-curves will be crucial in this article  since they determine (up to an isomorphism) the geometric type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We first give a construction of weak del Pezzo surfaces as blow-ups of the pro-  jective plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' A blow-up model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We assume k algebraically closed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We denote by X(Σ) the surface obtained from the projective plane  by successively blowing up the points in Σ := {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , pr}  π : X(Σ)  Blpr  −→ Xr → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' → X2  Blp1  −→ X1 = P2  where each pi, 1 ≤ i ≤ r is a closed point in the surface Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  For 1 ≤ i ≤ r, we denote by Ei the total transform in X(Σ) of the exceptional  divisor of the blowing-up Blpi : Xi+1 → Xi, and we write pi ≺ pi+1 when pi+1 is  infinitely near to pi, ie when it lies on the exceptional divisor of Blpi in Xi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The Picard lattice of the surface X(Σ) is the group generated by E0 := π∗L, the  total transform of the class L of a line in P2, and the Ei, 1 ≤ i ≤ r  Pic(X(Σ)) = Zr+1 = ZE0 + ZE1 + · · · + ZEr  endowed with the intersection product given by E·2  0 = 1, E·2  i  = −1 for 1 ≤ i ≤ r  and Ei · Ej = 0 for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The canonical class is given by  KX(Σ) = −3E0 + E1 + · · · + Er  From [9, III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Theorem 1], the surface X(Σ) is a weak del Pezzo surface of degree  d = 9 − r if, and only if r ≤ 8, and the points in Σ are in almost general position:  at each stage, the point pi does not lie on a (−2)-curve on Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The converse statement is almost true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' actually we have the following description  of weak del Pezzo surfaces [7, Proposition 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='4]  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let X denote a weak del Pezzo surface of degree d over an  algebraically closed field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then we have 1 ≤ d ≤ 9, and if we set r = 9 − d, we  must have one of the following  6  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  (i) r = 1 and X ≃ P1 × P1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (ii) r = 1 and X ≃ F2 the Hirzebruch surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (iii) 0 ≤ r ≤ 8, and X ≃ X(Σ), where Σ := {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , pr} consists of points in  almost general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Roots, exceptional curves and geometric types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In this section, we con-  sider a weak del Pezzo surface X of degree d ≤ 7 over an algebraically closed field  k, and we set r := 9 − d as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that the description of del Pezzo surfaces  of degrees 8 and 9 follows immediately from the preceding result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  There exists some Σ := {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , pr} consisting of points in almost general posi-  tion such that X ≃ X(Σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' this choice allows us to identify Pic(X) ≃ Zr+1 as in the  preceding section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Recall that R(X) is the set of roots in Pic(X)  R(X) := {D ∈ Pic(X), D·2 = −2, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='KX = 0}  We denote by Reff(X) ⊂ R(X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Rirr(X) ⊂ R(X)) the subset of effective  roots (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' of (−2)-curves) in Pic(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Let R(X) denote the root module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' it is the sub-Z-module of Pic(X) generated  by Rirr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It is well known [9] that the set R(X) is a root system in the orthogonal (KX)⊥⊗  Q of the canonical divisor KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Under the identification of Pic(X) and Zr+1, it is  sent on the root system Rd with basis {E0−E1 −E2−E3, E1 −E2, · · · , Er−1 −Er}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We denote by E9−d the intersection graph of this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It is the Dynkin diagram  associated to the degree d, namely  Degree d  6  5  4  3  2  1  Dynkin diagram E9−d  A2 × A1  A4  D5  E6  E7  E8  The group of automorphisms of the Picard group Pic(X) preserving the canonical  divisor and the intersection product is isomorphic to the Weyl group associated to  E9−d, which we denote by W(E9−d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It is generated by the reflections through the  hyperplanes orthogonal to the roots, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' the sα : x �→ x + (x · α)α, α ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The following result [9, III Th´eor`eme 2] is fundamental for the geometric classi-  fication of weak del Pezzo surfaces  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let X denote a weak del Pezzo surface of degree d ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then  the set Reff(X) ∪ (−Reff(X)) is a closed and symmetric part of R(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It is a root system in the space R(X)⊗Q, of which the set Rirr(X) forms a basis  (and we call it a root basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  As a consequence, the free Z-module generated by Rirr(X) is equal to R(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An immediate consequence is that since we have R(X) ⊂ K⊥  X, and this last  module has rank r = 9−d, there are at most r (−2)-curves on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Their intersection  graph has a strong geometric significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It is sometimes called the Dynkin type  of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We are ready to define the first, geometric part of our classification  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The geometric type of X is the orbit of the image of its set of  (−2)-curves Rirr(X) under the action of W(E9−d) on the set of bases for closed and  symmetric parts of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  When X is ordinary, we say it has ordinary geometric type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  7  Note that two isomorphisms between the lattices Pic(X) and Zr+1 differ by an  element of the Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This is why we define the type as an orbit under the  action of this group: this makes it independent of the choice of such an isomorphism,  and of the blown-up points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note also that the geometric type above is equivalent  to the type from [10, Definition 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The possible orbits can be deduced recursively from a theorem by Borel and  de Siebenthal that classifies the closed symmetric parts of maximal rank in a root  system up to the action of the Weyl group [22, p 29], [12, p 404].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' They are given,  degree by degree, in [12, Chapters 8 and 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We recall them in Appendix A as a  column in the table of arithmetic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  For each orbit, one can choose a root basis in such a way that its intersection  graph is a subgraph of the Dynkin diagram E9−d when d ≥ 5, and of the extended  (or affine) Dynkin diagram �E9−d when 3 ≤ d ≤ 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' this is no longer true when  d ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The geometric type we have just defined is a finer invariant than  the Dynkin type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' For instance, if d = 6 there are two orbits of Dynkin type A1,  depending on whether the (−2)-curve lies on the A1 or the A2 component of the  root system: surfaces in the corresponding geometric types differ by the number of  exceptional curves they contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' There are also two orbits for each of the Dynkin  types 2A1 and A3 when d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Note that for d ≥ 3, the possible geometric types of del Pezzo surfaces are in  bijection with the orbits given by Borel-de Siebenthal theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This is no longer  true when d ≤ 2: for instance, the orbits corresponding to certain types for d = 1  or d = 2 only occur as geometric types of del Pezzo surfaces in characteristic 2 [16,  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A convenient way to represent the geometric type is to consider a new graph,  containing the intersection graph of the (−2)-curves as a subgraph [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We define the graph of negative curves associated to the surface  X as the graph whose vertices are circles corresponding to (−2)-curves, and points  corresponding to exceptional curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Two vertices corresponding to curves C et C′  are joined by n edges if we have C · C′ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Since the Weyl group preserves the intersection product, two surfaces sharing  the same geometric type have the same graph of negative curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The converse  is true [10, Remark 4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' actually the geometric type of a weak del Pezzo surface  is completely determined by its degree, its Dynkin type and the number of its  exceptional curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' As a consequence, this is the way we will encode it in the tables  of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Here is a criterion for an exceptional divisor to be irreducible [9, Corollaire au  Th´eor`eme III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2]  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let D denote an exceptional divisor of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then D is the class of  an exceptional curve if, and only if we have D · R ≥ 0 for any effective root (or  (−2)-curve) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We end with [9, Lemme IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2], that will be useful we we decribe the fibers of the  desingularization of a songular del Pezzo surface  8  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Each connected component B of A is the support of a unique fun-  damental cycle, which is the least effective root C such that for any irreducible  component D of B we have C · D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The fundamental cycle depends only on the corresponding connected component  of the Dynkin type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Actually it is the highest root of the root system associated to  this component [12, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Arithmetic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We assume here that k = Fq is a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We denote  by σ a generator of the absolute Galois group Γ := Gal(k/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Let X denote a weak del Pezzo surface defined over k, having degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We  denote by Rirr(X) the set of the (−2)-curves of X ⊗ k, and by Rirr a representative  of the geometric type of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We fix an isomorphism between Pic(X ⊗ k) and  ZE0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' + ZE9−d that send Rirr(X) to Rirr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The automorphism σ induces the automorphism Id ×σ of the surface X ⊗ k,  and an automorphism (Id ×σ)∗ of the group Pic(X ⊗ k), that we denote by σ∗ in  the following;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' it preserves the intersection pairing, and the canonical class since X  is defined over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Under the action of the isomorphism between Pic(X ⊗ k) and  ZE0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='+ ZE9−d, the image of σ∗ is an element of the Weyl group W(E9−d), and  we get a morphism from Γ to W(E9−d), whose image is a finite cyclic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Finally, σ∗ preserves the set of (−2)-curves, and its image in W(E9−d) must lie  in Stab(Rirr), the stabilizer of Rirr in W(E9−d) which is the subgroup consisting of  the θ such that θRirr = Rirr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  This motivates the following  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let X denote a weak del Pezzo surface defined over k, and Rirr  the image of the set of its (−2)-curves described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The arithmetic type T of X is the conjugacy class of the image of σ∗ in Stab(Rirr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Two isomorphisms between Pic(X ⊗ k) and ZE0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' + ZE9−d sending the  (−2)-curves of X to Rirr differ by an element of the above stabilizer, which is a  subgroup of W(E9−d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Defining the type as a conjugacy class in this group makes  it independent of the choice of such an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We will see below that two singular del Pezzo surfaces such that the  Frobenius actions on the Picard groups of their associated weak del Pezzo surfaces  lie in the same conjugacy class in W(E9−d) (not in the stabilizer) can have different  arithmetic properties, in particular different zeta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This is easily seen on  the tables given in Appendix A that describe the different arithmetic types for  degrees 3 ≤ d ≤ 6, in particular in degree 4 where we precise the corresponding  conjugacy classes in W(D5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  To end this section, we remark that two weak del Pezzo surfaces sharing the  same arithmetic type have the same zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Actually, since a weak del Pezzo  surface is rational, we have the following [21, Theorem 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1, Corollary 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2]  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' For any n ≥ 1, the number of rational points of the weak del  Pezzo surface X over the finite field Fqn is  #X(Fqn) = q2n + qn Tr(σ∗n) + 1  As a consequence, the zeta function of the weak del Pezzo surface X is  Z(X, T )−1 = (1 − T )(1 − q2T ) det  �  I − qT σ∗| Pic(X ⊗ k)  �  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Singular del Pezzo surfaces  In this section, we consider a weak del Pezzo surface X of degree d defined over  the finite field k = Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  If X is not ordinary, its anticanonical divisor is no longer ample, and the mor-  phism it (or one of its multiples) defines is no longer an embedding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' its image Xs  is singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In this section, we first describe the geometry of this image, then we determine  the divisor class groups of X and the associated singular variety in order to prove  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Anticanonical models: geometric aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The anticanonical model of the surface X is the variety  Xs := Proj  ∞  �  n=0  H0(X, −nKX)  and we denote by ϕ : X → Xs the associated morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The variety Xs just defined is the singular del Pezzo surface associated to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' One can also consider for i ≥ 1, the plurianticanonical linear system  | − iKX| and the image X(i) of the morphism it defines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This variety is isomorphic  to Xs as long as di ≥ 3 [9, V Th´eor`eme 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' As a consequence, the variety Xs can  be identified with the anticanonical image of X when d ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The anticanonical model of a degree d del Pezzo surface is [17, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5]  (i) if d = 4, the complete intersection of two quadrics in P4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (ii) if d = 3, a cubic in P3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (iii) if d = 2, a degree four hypersurface in weighted projective space P(1, 1, 1, 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (iv) if d = 1, a degree six hypersurface in P(1, 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We list below some properties of the surface Xs, and of the morphism ϕ [9, IV  Th´eor`eme 1, V Proposition 1, V Th´eor`emes 1 et 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Recall that A is the union of  the (−2)-curves on X, and U is its complementary  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1) The schematic fibers of the morphism ϕ are the points of U and  the fundamental cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' As a consequence, ϕ is birational, and we have ϕ∗OX =  OXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  2) for all n ∈ Z, i > 0, we have Riϕ∗O(nKX) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  3) The surface Xs is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The singular points of Xs are the images of the  fundamental cycles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' they are rational double points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  4) If we set O(KXs) := ϕ∗O(KX), then O(KXs) is locally free of rank 1, and  for every integer n we have canonical isomorphisms  O(nKXs) = ϕ∗O(nKX), O(nKX) = ϕ∗O(nKXs)  In other words, ϕ is an isomorphism from U on its image Us, and each connected  component of A is sent to a point which is a rational double point on Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We obtain  all singular points of Xs in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The map ϕ is a minimal resolution of the  singularities of Xs, and the Dynkin type of X is the dual graph to this resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Thus the singularity type of Xs is exactly the Dynkin type of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The type of a singularity x of Xs is the type of the connected component corre-  sponding to x in the intersection graph of (−2)-curves of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Since all singularities  are rational double points, their types fall in the ADE classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  10  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  Note also that since O(KXs) is locally free of rank 1, this invertible sheaf corre-  sponds to a Cartier divisor KXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We end by recalling the following result [9, V Corollaire 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let F denote a locally free sheaf on Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then, for any i ≥ 0, we  have  Hi(Xs, F) = Hi(X, ϕ∗F)  Moreover, we have Serre duality  Hi(Xs, F) = H2−i(Xs, O(KXs) ⊗ F∨)∨  From this result, we can describe the global sections of the Cartier divisors on  Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Moreover we see that the sheaf O(KXs) is dualizing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' since it is locally free, the  surface Xs is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Divisor class groups and zeta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let X denote a weak non ordi-  nary del Pezzo surface defined over k = Fq, and Xs its anticanonical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then  ϕ and Xs are defined over k since the anticanonical divisor is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In the same way,  the varieties Sing(Xs) of dimension zero, and A of dimension 1 are defined over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Recall that the geometric Picard group (the group of classes of Cartier divisors)  Pic(X ⊗ k) identifies to the free Z-module generated by E0 and E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , Er, with  r = 9 − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It is equal to the group of classes of Weil divisors Cl(X ⊗ k) since X is  smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Using this identification, we will no longer mention the dependance on X  of some objects such as the root modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We first describe the groups Cl(Xs ⊗ k) and Pic(Xs ⊗ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that since Xs is  normal, the groups of Cartier divisors and of invertible sheaves coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The restriction to U⊗k of Weil divisors X⊗k is surjective [15, Proposition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  its kernel consists of divisors whose support is contained in the complementary of  U ⊗ k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' in A ⊗ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This last group is R, since it is generated by the irreducible  components of A ⊗ k, and these are exactly the (−2)-curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  No principal divisor on X ⊗ k has support contained in A ⊗ k [19, p 225].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Thus  R remains the kernel of the restriction of classes of Weil divisors from Cl(X ⊗ k)  to Cl(U ⊗ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The morphism ϕ induces an isomorphism from U to Us, and we have Cl(Us⊗k) =  Cl(U ⊗ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Now Xs ⊗ k \\ Us ⊗ k has codimension 2 in Xs ⊗ k, and we deduce [15,  Proposition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5] that Cl(Us ⊗ k) = Cl(Xs ⊗ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1)  0  � R  � Cl(X ⊗ k) = Pic(X ⊗ k)  � Cl(Xs ⊗ k)  � 0  We come to the Picard group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The pull-back ϕ∗ : Pic(Xs ⊗ k) → Pic(X ⊗ k)  gives rise to the exact sequence [4, Proposition 1]  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2)  0  � Pic(Xs ⊗ k)  ϕ∗  � Pic(X ⊗ k)  θ  � R∨  � Br(Xs ⊗ k)  ϕ∗  � Br(X ⊗ k)  where we have set R∨ := Hom(R, Z), and θ comes from the intersection product:  for any D ∈ Pic(X ⊗ k), θ(D) : R → Z is defined by θ(D)(R) = D · R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  11  In other words, the group Pic(Xs ⊗ k) identifies to the following subgroup of  Pic(X ⊗ k)  Pic(Xs ⊗ k)  =  {D ∈ Pic(X ⊗ k), ∀R ∈ R, D · R = 0}  =  {D ∈ Pic(X ⊗ k), ∀R ∈ Rirr, D · R = 0}  Since X ⊗ k is a rational surface, its Brauer group is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We deduce the  equality Coker θ = Br(Xs ⊗ k), and we have the following explicit description of  this last group [4, Proposition 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The root module R is a subgroup of K⊥  X, and  Br(Xs ⊗ k) is the torsion subgroup of the quotient  Coker θ = Br(Xs ⊗ k) =  �  K⊥  X/R  �  tors  It depends on the geometric type defined above (not only on the Dynkin type in  general: for instance there are two orbits for the Dynkin type 4A1 when d = 2,  one gives a trivial cokernel, the other an order 2 cokernel), and the different cases  are described in [4, Theorem 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that θ is often surjective (this is always true  when d ≥ 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We turn to the study of the zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We first determine the zeta function  of the union A of the (−2)-curves in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Since the set Rirr is stable under the action of Γ, the action of σ∗ on Pic(X ⊗ k)  restricts to an action on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Moreover this action preserves the intersection product,  and it induces an automorphism on the singular graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' as a consequence it permutes  the (−2)-curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The matrix of the action of σ∗ over R with respect to the basis  Rirr is a permutation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We can express the zeta function of A in terms of the characteristic polynomial  of this action  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We have the equality  Z(A, T ) = Z(Sing(Xs), T ) det(I − qT σ∗|R)−1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Write A as the disjoint union of its connected components Ax, where for  any x ∈ Sing(Xs)(Fq), Ax is the fiber (seen as a set) ϕ−1({x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Let n ≥ 1 an integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' since ϕ is defined over k, we have Axσn = Aσn  x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' If we have  x /∈ Sing(Xs)(Fqn), then Aσn  x ∩ Ax = ∅, and we deduce that Ax(Fqn) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Now assume x ∈ Sing(Xs)(Fqn), and let us denote by Cx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , Cxk the absolutely  irreducible components of Ax, whuch form the Coxeter-Dynkin graph associated to  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  For any two curves Cxi and Cxj defined over Fqn, there is a unique (since the  graph has no cycle) chain with minimal length in the graph between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' As the  extremities of this chain are fixed by σn, the whole chain is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We deduce from  this fact that the subgraph of the Cxi, 1 ≤ i ≤ k defined over Fqn is connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' let  us denote Nnx the number of its vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' As the (−2)-curves have normal crossings,  we deduce  12  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  #Ax(Fqn)  =  Nnx  �  i=1  #Cxi(Fqn) −  �  1≤i<j≤Nn  #(Cxi ∩ Cxj)(Fqn)  =  Nnx(1 + qn) −  �  1≤i<j≤Nnx  Cxi · Cxj  =  1 + Nnxqn  where the last equality comes from the fact that the sum of the intersection  products is the number of edges in the subgraph whose vertices are the Cxi defined  over Fqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Since this graph is connected, has Nnx vertices and type A, D, E, it  contains exactly Nnx − 1 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Finally, if none of the absolutely irreducible components of Ax is defined over  Fqn, then σn acts without fixed points over the Coxeter-Dynkin graph associated  to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The only graphs admitting such an action are the A2k, and the action is the  symmetry around the center of the graph, of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In this case, the unique point  defined over Fqn is the intersection of the effective roots Cxk and Cxk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Summing up, we have  #Ax(Fqn) =  � 1 + Nnxqn  if x ∈ Sing(Xs)(Fqn)  0  else  Thus, for any n ≥ 1 we have  #A(Fqn) = # Sing(Xs)(Fqn) + Nnqn  where Nn is the number of (−2)-curves on X defined over Fqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Replacing this expression in the usual expansion of zeta functions, we get  Z(A, T ) = exp  \uf8eb  \uf8ed�  n≥1  #A(Fqn)T n  n  \uf8f6  \uf8f8 = Z(Sing(Xs), T ) exp  \uf8eb  \uf8ed�  n≥1  Nn  (qT )n  n  \uf8f6  \uf8f8  Since the matrix of σ∗n with respect to the basis Rirr(X) is a permutation matrix,  its trace is exactly the number of elements in the basis which remain invariant under  the action of σ∗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We deduce that Nn = Tr(σ∗n|R), and the result comes from  the expansion of the characteristic polynomial of an endomorphism in terms of the  traces of its iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  □  We are ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' From the above results, the following equalities hold for any n ≥ 1  #Xs(Fqn)  =  #Us(Fqn) + # Sing(Xs)(Fqn)  excision for Xs  =  #U(Fqn) + # Sing(Xs)(Fqn)  isomorphim U ≃ Us  =  #U(Fqn) + #A(Fqn) − qn Tr(σ∗n|R)  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5  =  #X(Fqn) − qn Tr(σ∗n|R)  excision for X  Now from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='15, we have #X(Fqn) = q2n + qn Tr(σ∗n) + 1, and  #Xs(Fqn) = q2n + qn (Tr(σ∗n) − Tr(σ∗n|R)) + 1  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  13  If we pass to zeta functions as in the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='15, we get  Z(Xs, T )−1 = (1 − T )(1 − q2T ) det  �  I − qT σ∗| Pic(X ⊗ k)  �  det(I − qT σ∗|R)−1  Tensoring the exact sequence of Z-modules (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2) with C, the torsion disappears  and we get the exact sequence of vector spaces  0  � Pic(Xs ⊗ k) ⊗ C  � Pic(X ⊗ k) ⊗ C  � R∨ ⊗ C  � 0  Finally, the action of σ∗ on R∨ ⊗C is adjoint to that of σ∗−1 over R ⊗C, and they  share the same characteristic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Finally the matrix of σ∗ in the basis Rirr  of the the Z-module R is a permutation matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' thus the actions of σ∗−1 and σ∗  also share the same characteristic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get the equality  det  �  I − qT σ∗| Pic(X ⊗ k)  �  = det(I − qT σ∗|R) det  �  I − qT σ∗| Pic(Xs ⊗ k)  �  This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Construction of weak del Pezzo surfaces of degree at least five  The aim of this section is to show that for any finite field Fq and arithmetic  type belonging to a degree d ≥ 5, there exists a weak del Pezzo surface of this type  defined over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In order to do this, we give explicit constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  These surfaces are birational to the projective plane P2 [7, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3], and our  constructions are sequences of blowups and contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The fundamental remark  is that the configurations of points to be blown-up exist over any finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' For  instance, we will often need three collinear points, but this is possible over any  finite field since a line defined over Fq contains q + 1 rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We will often use the graphs of negative curve from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='10 in order to  represent the geometric type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Degrees seven and eight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' First note that in this case, there is only one  geometric type (of Dynkin type A1), and one arithmetic type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  From Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5, the only degree 8 weak, non ordinary del Pezzo surface is  the Hirzebruch surface F2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' the only degree 8 singular del Pezzo surface is its image  by the contraction of its negative section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We construct these surfaces from degree  7 surfaces below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Again from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5, all the degree 7 del Pezzo surfaces over an alge-  braically closed field are obtained from the projective plane P2 by two successive  blowups at points p1, p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Such a surface is not ordinary if and only if the center of  the second blowup is a point p2 ≻ p1 of the exceptional divisor of the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In  this case, the effective root is the strict transform of this exceptional divisor, whose  class in the Picard group is E1 − E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get the following graph of negative curves  E1 − E2  E2  E0 − E1 − E2  In order to get a non ordinary del Pezzo surface of degree 7 defined over the finite  field Fq, we have to choose rational points p1, p2 for the centers of the blowups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The Hirzebruch surface can be obtained from the degree 7 non ordinary del Pezzo  surface by contracting the strict transform of the line (p1p2) (the unique line in P2  passing trough p1 and whose strict transform after the first blowup passes trough  p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  14  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Degree six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Recall that the Weyl group is generated by the reflections sE1−E2,  sE2−E3 and sE0−E1−E2−E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It is isomorphic to the dihedral group of order 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The possible geometric types are given in [12, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2] or [7, Proposition  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Except in one case, we follow the choices made in this last article for the set of  roots representing the corresponding orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' As a consequence, most of the graphs  of negative curve are already described in the above mentioned article, and we do  not recall them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We refer to the corresponding table in the appendix for the different arithmetic  types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A1(4) to obtain the split surface (arithmetic type 1) we blowup three non collinear  points rational points p1 ≺ p2 and p3 (in other words, the strict transform  of the line (p1p3) after the first blowup does not pass through p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The stabilizer of the root E1−E2 is the subgroup of order 2 generated by  the reflection sE0−E1−E2−E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Its elements fall into two conjugacy classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In order to construct a surface of arithmetic type 2, we will contract  a line on a weak degree 5 del Pezzo surface X(Σ) of geometric type A1,  obtained by blowing-up four points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , p4 in P2, the first three being  collinear (see the degree 5 surface of geometric type A1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  If we contract the exceptional curve E0 − E3 − E4 on such a surface, the  vertices E3 and E4 disappear, and we obtain a degree 6 del Pezzo surface  with the graph of negative curves  E0 − E1 − E2 − E3  E1  E0 − E1 − E4  E0 − E2 − E4  E2  Assume that the first two points p1, p2 ∈ P2(Fq2) are conjugate over  Fq, and p3, p4 are defined over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The exceptional curve E0 − E3 − E4  is defined over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Contracting it, we get a surface X defined over Fq of  degree 6, of the geometric type under consideration, which is not split: this  is a surface of arithmetic type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A1(3) we blow-up three collinear points in P2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we get the three exceptional curves  E1, E2, E3, and the effective root E0 − E1 − E2 − E3 which corresponds to  the strict transform of the line through the pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The stabilizer of the root in the Weyl group W(E6) is the subgroup  generated by the reflections sE2−E1 and sE2−E3, which is isomorphic to the  symmetric group S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  There are three conjugacy classes in this group, corresponding to the  three arithmetic types 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We construct a surface of each type in the  following way  3 : the points p1, p2, p3 are defined over Fq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  4 : the points p1, p2 = pσ  1 are defined over Fq2, and p3 over Fq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  5 : the points p1, p2 = pσ  1, p3 = pσ2  1  are defined over Fq3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  2A1 we blowup three collinear points p1 ≺ p2 and p3 (the point p2 is the in-  tersection of the strict transform of the line (p1p3) with the exceptional  curve of the first blowup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The exceptional curves on the resulting surface  are E2, E3, and the (−2)-curves correspond to the classes E1 − E2 and  E0 − E1 − E2 − E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The stabilizer of the set {E1 −E2, E0 −E1 −E2 −E3} in the Weyl group  is trivial, and there is exactly one arithmetic type, with number 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  15  A2 we blowup three non collinear points p1 ≺ p2 ≺ p3 (the point p3 is not the  intersection of the strict transform of (p1p2) with the exceptional divisor  of the second blowup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The exceptional curves correspond to the classes  E0 − E1 − E2, E3, and the (−2)-curves to E1 − E2, E2 − E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The stabilizer of the set {E1−E2, E2−E3} in the Weyl group is generated  by the reflection sE0−E1−E2−E3, it has order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' As a consequence, there are  two arithmetic types, one split that can be constructed by choosing rational  points above, the other non-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In order to construct the non-split one, we start with a degree 5 del  Pezzo surface of geometric type A2 and arithmetic type 7 (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' If  we contract the exceptional curve with class E0 −E1 −E2 (which is defined  over Fq), we get a degree 6 del Pezzo surface of arithmetic type 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A1A2 we blowup three collinear points p1 ≺ p2 ≺ p3, where the point p3 is the  intersection of the strict transform of (p1p2) with the exceptional divisor  of the second blowup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get an exceptional curve of class E3, and (−2)-  curves corresponding to the classes E1 − E2, E2 − E3, E0 − E1 − E2 − E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Once again, the stabilizer is trivial and the only arithmetic type (number  9) is split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Degree five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Here the Weyl group is generated by the reflections sE1−E2,  sE2−E3, sE3−E4 and sE0−E1−E2−E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It is isomorphic to the symmetric group S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In this section, we shall not systematically follow the choices made in [7, Propo-  sition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5] for the classes of (−2)-curves representing each geometric type (see also  [12, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1] for a list of these types).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The reason is that there are many ways  to construct a del Pezzo surface of a given geometric type by playing on the con-  figuration of the points we blow up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Here we adopt the configuration that seems  best adapted to the description of the arithmetic types (in some sense the most  “symmetric ” one), and it need not coincide with the one described in the above  article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We rewrite the graphs when they differ from [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We refer to the corresponding table in the Appendix for the different arithmetic  types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A1 We blowup four points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , p4 in P2 with the first three collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The  only (−2)-curve is the strict transform of the line through these points, its  class is E0 − E1 − E2 − E3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' moreover there exist seven exceptional curves,  and the graph of negative curves is  E0 − E1 − E2 − E3  E1  E2  E3  E0 − E1 − E4  E0 − E2 − E4  E0 − E3 − E4  E4  The stabilizer of the root E0−E1−E2−E3 is generated by the reflections  sE1−E2, sE2−E3, and is isomorphic to the symmetric group S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' There are  three conjugacy classes, giving rise to three arithmetic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The corresponding surfaces are easily constructed, playing on the defi-  nition fields of the points p1, p2, p3 (we always choose p4 ∈ P2(Fq))  1 : the three points are in P2(Fq);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  2 : there are two conjugate points in P2(Fq2), and one in P2(Fq);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  16  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  3 : the three points are conjugate in P2(Fq3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  2A1 Here we follow [7, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we blowup p1 ≺ p2, p3 ≺ p4 such  that the strict transform of the line (p1p3) does not meet the exceptional  divisor of the first blowup at p2 nor of the third at p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get two disjoint  (−2)-curves with classes E1 − E2 and E3 − E4, and five exceptional curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The stabilizer of the set of negative curves is the order two subgroup of  W(A4) generated by sE1−E3 ◦ sE2−E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We get the two arithmetic types in the following way  4 : if p1 ≺ p2, p3 ≺ p4 are defined over Fq, the surface is split;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  5 : we choose couples of conjugate points p1 ≺ p2, p3 = pσ  1 ≺ p4 = pσ  2  defined over Fq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A2 We blow up p1, p3, p4 collinear, then p2 ≻ p1 which does not lie on the strict  transform of the line (p1p3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we obtain the following graph of negative curves  E1 − E2  E0 − E1 − E3 − E4  E0 − E1 − E2  E2  E3  E4  The stabilizer of the set of (−2)-curves is the subgroup of the Weyl group  generated by sE3−E4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' it has order 2 and we get two arithmetic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' These  are easily described  6 : if the pi are defined over Fq, the surface is split;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  7 : if we choose p4 = pσ  3 defined over Fq2, it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A1A2 We blowup p1 ≺ p2 ≺ p3 collinear and p4 ∈ P2 outside the line (p1p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The stabilizer is trivial, and the only arithmetic type is split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A3 We blowup p1 ≺ p2 ≺ p3 ≺ p4 where p1, p2, p3 are not collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The  stabilizer is trivial, and the only arithmetic type is split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A4 We blowup p1 ≺ p2 ≺ p3 ≺ p4 where p1, p2, p3 are collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The stabilizer  is trivial, and the only arithmetic type is split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Construction of singular del Pezzo surfaces of degree four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We adopt a different point of view in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Actually not all weak degree  four del Pezzo surfaces are birational to the projective plane over a finite field, and  the preceding constructions, which use sequences of blowups and contractions, no  longer suffice to describe all types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We shall mostly use a well known property of these surfaces: their anticanonical  model is the base locus of a pencil of quadrics in projective space P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The classi-  fication of these objects is different in characteristic two, and we assume that the  characteristic of our base field is odd in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We shall show that singular del Pezzo surfaces of every arithmetic type exist  over any finite field with odd cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Pencils of quadrics, their Segre symbols, and geometric types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In  this section, we work over the field k, an algebraic closure of a finite field of odd  characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We recall [12, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The results there are described over the field of com-  plex numbers, and our aim is to show that they remain valid over any algebraically  closed field of odd characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  17  The anticanonical model of a weak del Pezzo surface of degree 4 is the base locus  of a pencil of quadrics in projective space P4 [17, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We denote by Q0  and Q∞ two distinct quadrics of this pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This gives us two quadratic forms over  the vector space k5, and two symmetric 5×5 matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we denote the characteristic  polynomial of the pencil corresponding to this basis by F(λ, µ) := det(λQ0+µQ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A theorem of Kronecker [27, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1] tells us that the vector space k5 can  be written as an orthogonal direct sum of subspaces of two types:  a non singular space, say of dimension d, over which the determinant has  degree d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  some basic singular spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' here the equations of the quadrics take a very  special form with respect to a well chosen basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A direct calculation from such equations shows that the base locus of a pencil of  quadrics has one-dimensional singular locus as long as the space contains a basic  singular space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Thus the space k5 must be non-singular for the pair of quadratic  forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that this is no longer true in characteristic two, even for ordinary  degree 4 del Pezzo surfaces [11], since a quadratic form over an odd dimensional  vector space must be degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We assume in the following that the quadric Q∞ is non-singular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' thus the poly-  nomial P(t) := F(1, t) has degree 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This is a harmless assumption since we assume  k algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We follow Waterhouse [27, Section 1] and associate to our pair of quadratic forms  a quadratic form Φ on a k[T ]-module of finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' If ϕ0, ϕ∞ denote the bilinear  forms on k5 associated to Q0 and Q∞, then ϕ∞ is non degenerate, and there is  an unique endomorphism u of k5 such that ϕ∞(u(x), y) = ϕ0(x, y) for any vectors  x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Moreover u is symmetric with respect to ϕ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The vector space k5, endowed with the action of u, becomes a k[T ]-module of  finite length that we denote by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The non degenerate form ϕ∞ defines an isomor-  phism Φ of k[T ]-modules between M and its dual M ∗ := Homk[T ](M, k(T )/k[T ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Consider the factorisation of P over k, P(T ) = �t  i=1(T − θi)mi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we get an  isomorphism of k[T ]-modules  M = ⊕t  i=1 ⊕ei  j=1 (k[T ]/((T − θi)mij))nij  As a consequence, we can associate to M the following Segre symbol  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1)  [(m11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' m11  �  ��  �  n11 ×  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' m1e1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' m1e1  �  ��  �  n1e1 ×  ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' (mt1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' mt1  �  ��  �  nt1×  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' mtet .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' mtet  �  ��  �  ntet ×  )]  Note that we remove the parentheses around the ith term when it contains exactly  one integer, ie when we have ei = ni1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  As in [27, Section 2], one can write the restrictions of the quadratic forms to a  submodule of the form k[T ]/(T − θ)m as (here res denotes the residue in the sense  of Laurent series)  Q0(x) = res(T (T − θ)−mx2), Q∞(x) = res((T − θ)−mx2)  since every element in (k[T ]/(T − θ)m)× is a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In the basis ((T −θ)m−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , T −θ, 1), the expressions of these forms are exactly  those given in [12, Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='21)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Now the analytic expressions for the rational  double points of type An or Dn (the only ones occuring for degree 4 singular del  Pezzo surfaces) are exactly the same in characteristic 0 or odd [1], and we deduce  18  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  that the following one to one correspondance between the geometric types and  (some of the) Segre symbols [12, Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='6] remains true in our setting  Type  Number  Segre  of singularity  of lines  symbol  Ordinary  16  [11111]  A1  12  [2111]  2A1  9  [221]  2A1  8  [(11)111]  A2  8  [311]  3A1  6  [(11)21]  A1A2  6  [32]  A3  5  [41]  Type  Number  Segre  of singularity  of lines  symbol  A3  4  [(21)11]  4A1  4  [(11)(11)1]  2A1A2  4  [3(11)]  A1A3  3  [(21)2]  A4  3  [5]  D4  2  [(31)1]  2A1A3  2  [(21)(11)]  D5  1  [(41)]  Some Segre symbols do not give rise to singular degree 4 del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' For  instance the symbol [(11111)] gives rise to two collinear quadratic forms over k5,  and the base locus of the pencil is a quadric in P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We shall often use the following remark along this section  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' These symbols carry the following informations: the number of differ-  ent terms (ie the number t in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1), each couple of parentheses counts for one term)  is equal to the number of singular quadrics in the pencil, and for such a quadric,  its co-rank is the number of terms in the corresponding parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Actually we  remark that the possible ranks for the singular quadrics in the pencil are 3 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In the following, we shall often use the Segre symbol instead of the geometric  type for our contructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Moreover, we always choose the same set of (−2)-curves  as in [7, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1] to represent the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Morphisms to the projective line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We continue to work over an alge-  braically closed field of odd characteristic in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We keep on with the  notations of the preceding section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  A convenient way to classify the arithmetic types for an ordinary del Pezzo  surface is to visualize the Galois action on a graph consisting of its conic bundles  [18, pages 8 and 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' A non ordinary del Pezzo surface no longer has conic bundles,  due to the presence of (−2)-curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' the aim of this section is to define a new graph  for each geometric type, which is very close to the original one, whose vertices  represent  the fibers of rational maps or morphisms from X to the projective line,  the classes of (−2)-curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We will use these graphs in the next sections in order to control the arithmetic  types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Let ϕ : X → P1 denote a surjective morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Since X is a rational surface, the  generic fiber of ϕ is isomorphic to P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Moreover all its fibers are linearly equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We deduce from the adjunction formula that their class C satisfies C·2 = 0 and  C · KX = −2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' moreover we have ϕ = ϕ|C|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Recall [3, Theorem 2]: there exist exactly ten classes in Pic(X) satisfying the  above equations (note that they form complementary couples in the sense that we  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  19  have Ci + C′  i = −KX for each i)  Ci = E0 − Ei, C′  i = 2E0 −  5  �  j=1  Ej + Ei, 1 ≤ i ≤ 5  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We set C := ∪5  i=1{Ci, C′  i} in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We represent these  classes in the following graph, in which the couples are materialized by vertical  dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  In the case of singular del Pezzo surfaces, the elements of C no longer correspond  to conic bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' But they still correspond to rational maps from X to the projec-  tive line, as we shall see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We will adapt the graph according to the geometric  type, by adding vertices corresponding to (−2)-curves, and an edge between a class  in C and such a curve when they intersect positively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  From the description of the elements in C on one hand, and of the (−2)-curves  (recall that their classes have the form Ei − Ej or E0 − Ei − Ej − Ek) on the other,  we see that we must have C · R ∈ {−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Let C ∈ C be such that we have C · R = −1 for some (−2)-curve R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we deduce  that C − R ∈ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Thus R is a fixed component of the linear system |C|, and ϕ|C| is  a rational map, but not a morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We deduce that there exists at most  NX := #{C ∈ C , ∀R ∈ Rirr(X), C · R ≥ 0}  morphisms from X to P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Our strategy is to construct exactly NX such morphisms by considering the  singular quadrics in the pencil defining the anticanonical model Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Recall that  P denotes the characteristic polynomial of the pencil, and that θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , θt are its  (pairwise distinct) roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Let θi denote a simple root of P, if any;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' the ith part of the Segre symbol (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1) is  a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' From remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1, the quadric Qi := Q0 − θiQ∞ is a singular quadric in P4 of  rank 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Thus it is a cone whose vertex is a point vi and base a non singular quadric  qi := Qi ∩ Hi for some hyperplane Hi not containing vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Moreover the vertex vi is  not contained in Xs since θi is a simple root of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The projection P4 → Hi from vi restricts to a double covering Xs → qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Now qi  is non singular, thus it is isomorphic to P1 × P1, and the projections on the factors  give rise to two morphisms from Xs to P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Composing them with the anticanonical  morphism gives two morphisms ϕi1, ϕi2 : X → P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Note that we have morphisms from Xs to P1, thus their fibers are Cartier divisors  on Xs, and their classes in Pic(X) satisfy C · R = 0 for all (−2)-curves R on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Finally, the fibers of the above morphisms from Xs to P1 are of the form Π∩Q∞,  where Π is some plane containing one of the lines of qi and passing through vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  These are plane conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' For any point p ∈ Xs, the hyperplane H of P4 tangent  to Qi at p intersects Qi along the union of two planes Π1 and Π2, and we have  H ∩ Xs = H ∩ Qi ∩ Q∞ = (Π1 ∩ Q∞) ∪ (Π2 ∩ Q∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We deduce that the sum of the  20  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  classes of the fibers of these two morphisms is exactly the class of the anticanonical  divisor: they are complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Now assume that θi is a multiple root of P, and that Qi has rank 4 (this means  that the ith part of the Segre symbol (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1) satisfies ei = ni1 = 1 and mi1 = mi > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The same construction as in the preceding case yields two morphisms from Xs\\{vi}  to P1 (note that the vertex vi of Qi is a singular point of Xs here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Consider the blowup Blvi(P4) → P4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' then Blvi(Xs) is the strict transform of  Xs [15, Corollary II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='15], and the two morphisms above become morphisms from  Blvi(Xs) to P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The fiber of vi in the blowup Blvi(Xs) → Xs contains only (−2)-  curves, and we must have a morphism X → Blvi(Xs) since X is the minimal  desingularization of Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Composing, we get two morphisms ϕi1, ϕi2 : X → P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The last case to be considered is when θi is a multiple root of P, and Qi has rank  3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' here the ith part of the Segre symbol (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1) contains exactly two terms between  parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In this case Qi is a cone with vertex ℓi ≃ P1 and base a smooth  plane conic ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The projection with vertex ℓi from P4 \\ ℓi to this plane induces a  morphism from Xs \\ ℓi ∩ Xs to ci ≃ P1 (a rational map Xs ��� P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The blowup Blℓi(P4) → P4 restricts to Blℓi∩Xs(Xs) → Xs as above, and we  get a morphism Blℓi∩Xs(Xs) → P1 from the above rational map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Once again,  the fibers above the points in ℓi ∩ Xs in the morphism Blℓi∩Xs(Xs) → Xs only  contain (−2)-curves, and we get a morphism X → Blℓi∩Xs(Xs) from the minimal  desingularization, from which we get a morphism ϕi1 : X → P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Summing up, we have constructed 2a + 2b + c morphisms from X to P1, where  a is the number of simple roots of P, b the number of multiple roots corresponding  to a rank 4 quadric in the pencil, and c the number of rank 3 quadrics in the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The following table contains the graphs we mentioned at the beginning of the  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It is sufficient to prove the next proposition for each geometric type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Note that for each type we use an empty square to denote the classes in C  that intersect negatively a (−2)-curve, and a full square for the other ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we  also denote by ri the curve with class Ei − Ei+1, and by rijk the curve with class  E0 − Ei − Ej − Ek  Table 1: Conics-(−2)-curves graphs  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Type  Conics-(−2) Curves Graph  N  (a, b, c)  (−2)-curves  A1 [2111]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R  8  (3, 1, 0)  R = r45  2A1(9) [221]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R1  R2  6  (1, 2, 0)  R1 = r23  R2 = r45  2A1(8) [111(11)]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R1  R2  7  (3, 0, 1)  R1 = r123  R2 = r45  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  21  Table 1: Conics-(−2)-curves graphs  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Type  Conics-(−2) Curves Graph  N  (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' c)  (−2)-curves  A2 [311]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R1  R2  6  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  R1 = r34  R2 = r45  3A1 [(11)21]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R1  R2  R3  5  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1)  R1 = r23  R2 = r45  R3 = r123  A1A2 [32]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R1  R2  R3  4  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  R1 = r12  R2 = r34  R3 = r45  A3(5) [41]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R1  R2  R3  4  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  R1 = r23  R2 = r34  R3 = r45  A3(4) [(21)11]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R3  R2  R1  5  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1)  R1 = r45  R2 = r34  R3 = r123  4A1 [(11)(11)1]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R4  R1  R2  R3  4  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 2)  R1 = r12  R2 = r345  R3 = r45  R4 = r123  2A1A2 [3(11)]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R1  R2  R4  R3  3  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1)  R1 = r12  R2 = r23  R3 = r45  R4 = r123  A1A3 [(21)2]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R1  R2  R4  R3  3  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1)  R1 = r12  R2 = r34  R3 = r45  R4 = r123  A4 [5]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R1  R2  R3  R4  2  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  R1 = r12  R2 = r23  R3 = r34  R4 = r45  22  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  Table 1: Conics-(−2)-curves graphs  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Type  Conics-(−2) Curves Graph  N  (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' c)  (−2)-curves  D4 [(31)1]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R4  R1  R2  R3  3  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1)  R1 = r23  R2 = r34  R3 = r45  R4 = r123  2A1A3 [(21)(11)]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R5  R1  R3  R4  R2  2  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 2)  R1 = r12  R2 = r345  R3 = r34  R4 = r45  R5 = r123  D5 [(41)]  C1  C2  C3  C4  C5  C′  1  C′  2  C′  3  C′  4  C′  5  R1  R2  R3  R4  R5  1  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1)  R1 = r12  R2 = r23  R3 = r34  R4 = r45  R5 = r123  We summarize the results of this section in the following  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let X denote a degree four weak del Pezzo surface, and P a  characteristic polynomial for the pencil of quadrics defining its anticanonical model  in P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Denote by aX the number of simple roots of P, bX ( resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' cX) the number  of multiple roots corresponding to a rank 4 ( resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' rank 3) quadric in the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  If NX is the number of classes in C not intersecting any (−2)-curve negatively,  we have NX = 2aX + 2bX + cX, and this is the number of surjective morphisms  from X to the projective line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  For each of these classes C, the morphism ϕ|C| : X → P1 has fibers linearly  equivalent to C, and the type of map ϕ|C| defines on Xs depends on the following  numerical criterion  (a) 2aX of them satisfy C · R = 0 for all (−2)-curves R ∈ Rirr(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' for those  ones ϕ|C| factors through the anticanonical morphism and yields a mor-  phism from Xs to P1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' moreover these classes lie in Pic(Xs), and they form  aX couples of complementary classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (b) 2bX + cX satisfy C · R ≥ 0 for all (−2)-curves R and C · R > 0 for at least  one, and for those ϕ|C| does not define a morphism from Xs to P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that in the ordinary case, we have aX = 5 and bX = cX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Then the fibers of the 10 morphisms form the conic bundles, and we recover the  original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Another consequence of the discussion above is that, for a fixed class  C ∈ C which intersects all the (−2)-curves nonnegatively, we have two possibilities  for the relative position of a (−2)-curve with class R with the fibers of the morphism  ϕ|C|  (i) when we have C · R = 1, R is transverse to the fibration, and ϕ|C| restricts  to an isomorphism from R to P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The images in Xs of its fibers all pass  through the singular point which is the image of R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  23  (ii) when we have C · R = 0, R is contained in a fiber of the morphism ϕ|C|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Galois action on the singular quadrics, and arithmetic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In this  section we work over the finite field k = Fq, q a power of an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It remains to provide information on the arithmetic type of a degree four del  Pezzo surface X from arithmetic information on the rank four singular quadrics  in the pencil defining Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We begin by using the results of the preceding section  to describe the Galois action on the morphisms to the projective line from this  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Later, we will transpose this action to an action on the graph of  conics and (−2)-curves, which will naturally lead to a (“part” of a) conjugacy class  in the Weyl group W(D5), that we describe below, and that is given in the fourth  column of the table for degree four surfaces in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We begin by a definition (note that it is independant of the choice of the sup-  plementary)  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let Q denote a rank four quadric in P4, defined over Fqd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' its  restricted discriminant is the discriminant of its base in F×  qd/F×2  qd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In other word,  it is the discriminant of the restriction of quadratic form corresponding to Q to a  supplementary of its kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The Weyl group W(D5) can be identified with the group H5 ⋊ S5, where H5 is  the subgroup of (Z/2Z)5 which is the kernel of the map ε = (ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , ε5) �→ �5  i=1 εi,  and S5 acts on H5 by σ · ε = (εσ−1(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , εσ−1(5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Recall from [5, Proposition 25] that the conjugacy classes in W(D5) are even  signed cycle-types;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' they form a partition of 5 where each integer in the partition is  overlined or not, and there is an even number of overlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The conjugacy class of  an element (ε, σ) has partition corresponding to the conjugacy class of σ in S5, and  a number in this partition is overlined when the sum of the εi where i describes the  support of the corresponding cycle is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that if σ is a d-cycle, the conjugacy  class of an element of the form (ε, σ) is d when its d-th power is trivial, and d else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Note that W(D5) is a subgroup of W(B5) = (Z/2Z)5 ⋊ S5, whose conjugacy  classes can be represented by (even or odd) signed cycle-types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Note also that an arithmetic type is a conjugacy class in some subgroup of  W(D5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' As a consequence, different arithmetic types can be contained in the same  conjugacy class in W(D5), and we will have to be more precise in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  With these notations (and the ones used in the preceding sections) at hand, we  begin by describing the Galois action on the part of the graph coming from the  morphisms associated to the simple roots of P  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let X be a weak degree 4 del Pezzo surface over Fq whose anti-  canonical model is the intersection of two quadrics Q0 and Q∞ in P4, both defined  over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let G ∈ Fq[T ] be a simple irreducible divisor of degree d of the character-  istic polynomial P(T ) and let θ ∈ Fqd be a root of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We denote by ∆(θ) ∈ F×  qd the restricted discriminant of Q = Q0 − θQ∞, ie the  discriminant of its base q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  To G, one can associate d couples of complementary classes in C , and the action  of the Frobenius on those classes induces a permutation of signed sub-type d or d  depending on whether ∆(θ) is or is not a square in F×  qd (note that this does not  depend on the choice of the root θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  24  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let θ1 = θ, θ2 = θq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , θd = θqd−1 ∈ Fqd denote the roots of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' From the  results in the preceding section, we obtain d rank 4 quadrics Qi := Q0 − θiQ∞,  and 2d morphisms ϕij from Xs to P1, 1 ≤ i ≤ d, 1 ≤ j ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It follows from  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3 that the classes fij of the fibers of these morphisms form d couples  of complementary classes in C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The action of Frobenius sends Qi on Qi+1 (the indices are read modulo d), and  the couple {fi1, fi2} to the couple {fi+1,1, fi+1,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Thus the permutation associated  to the Frobenius (seen as an element in W(D5)) is the cycle (1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' d) of length d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The base q1 of the quadric Q1 is defined over Fqd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' it is well known that it is either  hyperbolic (isomorphic to P1 × P1 over Fqd) or elliptic (it becomes isomorphic to  P1 × P1 only after a quadratic extension of the base field) depending on whether  its discriminant is or is not a square in Fqd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In the hyperbolic case (then any of the qi is hyperbolic), the d-th power of the  Frobenius stabilizes each one of the two P1, and the fibers f11 and f12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' the Galois  action has order d, and type d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Else the d-th power of the Frobenius exchanges the  two P1 and the fibers f11, f12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' the Galois action has order 2d, and type d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  □  We now describe the Galois action on the morphisms associated to the rank 4  quadrics in the pencil coming from multiple roots of its characteristic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let θ ∈ Fq be a multiple root of P, say of multiplicity m, such that  the associated quadric Q := Q0 − θQ∞ has rank 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Denote by G the minimal  polynomial of θ over Fq, by d its degree, and by ∆(θ) the restricted discriminant of  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The d vertices of Q and its conjugates Qσ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , Qσd−1 are singular points on Xs,  and these singularities have Dynkin type dAm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' To these correspond 2d morphisms  from X to P1, whose fibers f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , f σd−1  1  , f σd−1  2  form 2d classes in C , and the  following cases occur  (a) d = 1, 2 ≤ m ≤ 4: the Galois action fixes f1 and f2 if ∆(θ) is a square in  F×  q , and exchanges them otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (b) d = 2, m = 2 and the Galois action acts on the fibers as the bitransposition  (f1f σ  1 )(f2f σ  2 ) if ∆(θ) is a square in F×  q2, and as the four cycle (f1f σ  1 f2f σ  2 )  else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Since P has degree 5 and Gm is a divisor, we must have md ≤ 5, which  leaves us with the listed possibilities, and d = 1, m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' But we see from the  Segre symbols that this case corresponds to a singularity of Dynkin type A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' For  this geometric type, there is only one arithmetic type, and the Galois action is not  relevant: it will be sufficient to construct any del Pezzo surface over Fq with this  singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The assertion on the Dynkin type of the singularities comes from the Segre sym-  bols: the number m appearing as a part of it corresponds to an Am−1 singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The assertion on the fibers comes from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Finally the arithmetic assertions on the Galois action are easily deduced from  the proof of the preceding lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Quadratic module associated to a pencil of quadrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We continue to  work over the finite field k = Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The anticanonical model of a degree four del Pezzo surface is the base locus of a  pencil of quadrics in P4, thus it is defined by the vanishing of a pair of quadratic  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  25  forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We follow [27] in this section, and define a quadratic module (here a k[T ]-  module of finite length endowed with a non degenerate bilinear form) from the  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We give a normal form for such a module and we deduce arithmetic  information on the singular quadrics of the pencil from this normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  As usual we denote by {Q0, Q∞} a basis for the pencil of quadratic forms defining  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We assume Q∞ is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that this requires the existence of a  non-degenerate quadric of the pencil which is defined over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We will have to drop  this assumption farther in some very particular cases when q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  If ϕ0, ϕ∞ denote the bilinear forms on k5 associated respectively to Q0 and Q∞,  then ϕ∞ is non degenerate, and there is an unique endomorphism u of k5 such that  ϕ∞(u(x), y) = ϕ0(x, y) for any vectors x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Moreover u is symmetric with respect  to ϕ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The vector space V = k5, endowed with the action of u becomes a k[T ]-module  of finite length that we denote by Vu, and ϕ∞ is a nondegenerate symmetric k[T ]-  bilinear form on Vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We call the pair (Vu, ϕ∞) a quadratic k[T ]-module in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Since k[T ] is a principal ideal domain, the module Vu decomposes as a direct  sum of primary cyclic components:  Vu =  t  �  i=1  \uf8eb  \uf8ed  ni  �  j=1  k[T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='xij  \uf8f6  \uf8f8 ,  ann (xij) = P mij  i  k[T ],  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2)  where the xij are elements of Vu, the polynomials P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , Pt ∈ k[T ] are irreducible  and pairwise distinct, and the exponents mi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' miti are (not necessarily distinct)  positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that when k is algebraically closed we recover the Segre  symbol (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Waterhouse has proved that the cyclic primary components of the decomposition  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2) can be chosen in such a way that they are two-by-two ϕ∞-orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Moreover, it turns out that each cyclic component can be explicitly described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  To this end, we need to introduce some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let F ∈ k[T ] be a non constant,  unitary polynomial and let us consider the quotient algebra k[T ]/(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We put  t = T mod F, we choose some δ ∈ k[T ]/(F) and we define λF , ΦF , δ ·λF and δ ·ΦF  as  the linear form λF : k[T ]/(F) → k defined as the last vector of the dual  basis of (1, t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , tdeg(F )−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  the symmetric bilinear form ΦF : k[T ]/(F) × k[T ]/(F) → k defined by  ΦF (x, y) = λF (xy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  the linear form defined by δ · λF (x) = λF (δx) and  the bilinear form defined by δ · ΦF (x, y) = λF (δxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Then the linear form λF is a dualizing form and the algebra k[T ]/(F) with its du-  alizing form is called a Frobenius algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' by definition, this means that the bilinear  form ΦF is non degenerate or equivalently that for every λ ∈ Homk (k[T ]/(F), k),  there exists α ∈ k[T ]/(F) such that λ(x) = α · λF (x) = λF (αx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The quadratic module [[F, δ]] is the k[T ]-module k[T ]/(F) en-  dowed with the bilinear form ϕ = δ · ΦF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  26  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  This is an example of quadratic module with cyclic underlying k[T ]-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Note that the pair of quadratic forms on the corresponding k-vector space is given  by Q∞(x) = λF (δx2) and Q0(x) = λF (tδx2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In fact this is a generic example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let (Vu, ϕ) be k[T ]-quadratic module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Suppose that the k[T ]-  module Vu is cyclic, let x ∈ Vu be such that Vu = k[T ] · x, and let F ∈ k[T ] be a  generator of its annihilator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then there exists δ ∈ k[T ]/(F) such that for all p, q in  k[T ]/(F) we have ϕ(p · x, q · x) = δ · ΦF (p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' By definition of F, the map ι defined by p �→ p · x is an isomorphism of  k[T ]-modules from k[T ]/(F) to Vu = k[T ]·x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The map ψ defined by ψ(p, q) = ϕ(p·  x, q · x) for every p, q ∈ k[T ]/(F) is bilinear symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Since u is ϕ-symmetric, the  multiplication by t endomorphism is ψ-symmetric, and we have ψ(tp, q) = ψ(p, tq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In particular, we get ψ(p, q) = ψ(1, pq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' But we know that there exists δ ∈ k[T ]/(F)  such that ψ(1, q) = δ · λF (q) for every q ∈ k[T ]/(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We deduce that  ϕ(p · x, q · x) = ψ(p, q) = ψ(1, pq) = δ · λF (pq) = δ · ΦF (p, q)  and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  □  We can use the quadratic modules we have just defined to give a normal form  for all quadratic modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' From [27, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1 and Section 2], we have  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='12 (Waterhouse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let (V, ϕ) be a non-degenerate k[T ]-quadratic mod-  ule of finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Let  �  P eij  i  �  i,j, 1 ≤ i ≤ r, 1 ≤ j ≤ nij, be the elementary divisors of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then there  exists δij ∈ (k[T ]/(P eij  i  ))× such that the pair (V, ϕ) is isometric to the orthogonal  sum �  i,j[[P eij  i  , δij]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Thus we can associate to the anticanonical model of any degree 4 del Pezzo  surface (with at least one non-singular quadric defined over Fq in the pencil) a  k[T ]-quadratic module of the form �  i[[Fi, δi]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Conversely, to such a module we associate the del Pezzo surface defined by the  vanishing of the following two quadratic forms defined for x = (xi) by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3)  Q∞(x) =  �  i  λFi(δix2  i ), Q0(x) =  �  i  λFi(T δix2  i )  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that this association is in no way an application: we have arbi-  trarily chosen the basis {Q0, Q∞}, and one could replace some δi by x2δi without  changing the resulting del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Also replacing all δi by aδi for some  a ∈ F×  q gives the quadratic forms aQ∞, aQ0 and does not change the del Pezzo  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  For this reason, we will frequently be able to put additional restrictions on the δi  in the last section, in order to ease the computation of the restricted discriminants  of the singular quadrics of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Moreover we have chosen to use the elementary divisors of the k[T ]-module in  the above presentation, but we could have chosen the invariant factors instead, or  any other possible decomposition in cyclic submodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We have seen that one of the key tools to compute the arithmetic type of a Del  Pezzo surface of degree 4 is to control the discriminants of the bases of the rank 4  singular quadrics in the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' From the Frobenius algebras point of view, we have  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  27  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let F ∈ k[T ] be a unitary polynomial and let δ ∈ k[T ]/(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We denote by NF : k[T ]/(F) → k the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (1) The discriminant of the bilinear form δ · ΦF equals  (−1)  deg(F )(deg(F )−1)  2  NF (δ)  (2) Let θ ∈ k be a root of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then the bilinear form (θ − T )δ ·ΦF is degenerate  of rank deg F − 1 and its restricted discriminant equals  (−1)  deg(F )(deg(F )−1)  2  NF (δ)δ(θ)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' (1) Put n = deg(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let C = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , tn−1) be the canonical basis of k[T ]/(F),  and let D = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , fn) be its dual basis with respect to λF , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  such that  λF (ti−1fj) = δij, 1 ≤ i, j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The coordinates of the elements of D in the  basis C are given by the columns of the matrix:  P =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  a1  · ·  an−1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  1  an−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  where the ai’s are the coefficients of F = T n+an−1T n−1+· · ·+a1T +a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Moreover,  for any x ∈ k[T ]/(F), one has x = �n  i=1 λF (xfi)ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let us compare the matrices  of the bilinear form δ · φF and of the linear map mδ (multiplication by δ) in the  basis C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' by definition they are equal to  Mat (δ · ΦF , C) = (λF (δeiej))1≤i,j≤n  and  Mat (mδ, C) = (λF (δfiej))1≤i,j≤n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Thus they are related by the formula Mat (mδ, C) = tP Mat (δ · ΦF , C) and the  result follows from the equality of the determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (2) Let e denote the multiplicity of the root θ for F, and set F = (T − θ)eG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  there exist U, V ∈ k[T ] satisfying U(T − θ)e + V G = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then for any x ∈ k[T ]/(F),  we have  λF (x)  =  λF (xU(T − θ)e) + λF (xV G)  =  λF ((xU mod G)(T − θ)e) + λF ((xV mod (T − θ)e)G)  =  λG(xU) + λ(T −θ)e(xV )  where the last equality holds since G and (T − θ)e are unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We deduce the  following orthogonal decomposition  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='4)  (θ − T )δ · ΦF = (θ − T )V δ · Φ(T −θ)e ⊕ (θ − T )Uδ · ΦG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In order to compute the restricted discriminant of the first component, we note  that (θ − T )V δ · λ(T −θ)e((T − θ)a) = −λ(T −θ)e((T − θ)a+1δV ) is equal to 0 as long  as a ≥ e − 1, and to −δ(θ)V (θ) when a = e − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get  Mat  �  (θ − T )V δ · Φ(T −θ)e,  �  (T − θ)e−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , 1  ��  =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  −V (θ)δ(θ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  0  −V (θ)δ(θ)  · ·  ⋆  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  28  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  This form has rank e − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' its kernel is generated by (T − θ)e−1, and its restricted  discriminant equals:  (−1)  (e−1)(e−2)  2  × (−1)e−1 × V (θ)e−1δ(θ)e−1 = (−1)  e(e−1)  2  δ(θ)e−1  G(θ)e−1  because V (θ)G(θ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  For the second component of the decomposition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='4), putting d = deg(G), and  using (1), its discriminant equals  (−1)d(d−1)NG ((θ − T )Uδ) = (−1)  d(d−1)  2  +ed  NG(δ)  NG(θ − T )e−1  since U(θ − T ) × (−1)e(θ − T )e−1 = 1 mod G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We have NG(θ − T ) = G(θ), and the product of these two discriminants gives  the result as NF (δ) = NG(δ)N(T −θ)e(δ) = NG(δ)NT −θ(δ)e = NG(δ)δ(θ)e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' At this point, we make the following observation: when the underly-  ing k[T ]-module is cyclic, all singular quadrics in the correspondong pencil (defined  by equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3)) have corank one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The converse is true: when the underlying k[T ]-module is not cyclic, it admits  at least two invariant factors, and for all roots of the one of smallest degree the  above result shows that the corresponding singular quadric has corank equal to the  number of invariant factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We end this section with a criterion that determines the field of definition of  certain singular points of the del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We will use it in the next section in  order to determine the arithmetic types when the pencil contains a rank 3 quadric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Let X be a degree four del Pezzo surface over Fq such that for some  θ ∈ Fq, the non-degenerate quadratic k[T ]-module associated to X can be written  as an orthogonal sum [[T − θ, δ1]] ⊕ [[T − θ, δ2]] ⊕ M, where the annihilator P of the  k[T ]-module M satisfies P(θ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Then the quadric of the pencil with equation Q0 − θQ∞ = 0 has rank 3, and  if ℓ ≃ P1 is its vertex, the singular del Pezzo variety Xs meets ℓ at two distinct  points, which are defined over Fq if −δ1δ2 is a square in F×  q , and conjugate over  Fq else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We choose a basis (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , e5) of k5 which is adapted to the orthogonal  decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In this basis, from the expressions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3) of the forms Q0, Q∞, their  matrices can be written in block diagonal form as respectively  Mat(Q∞) =  \uf8eb  \uf8ed  δ1  0  0  0  δ2  0  0  0  A∞  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=', Mat(Q0) =  \uf8eb  \uf8ed  θδ1  0  0  0  θδ2  0  0  0  A0  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Since the annihilator P of the k[T ]-module M satisfies P(θ) ̸= 0, the matrix A0 −  θA∞ is invertible, and the quadric with equation Q0 − θQ∞ has rank 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Its vertex is the line (x1 : x2 : 0 : 0 : 0) in P4, the intersection of which with the  quadric Q∞ = 0 is defined by the equation δ1x2  1 + δ2x2  2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Now we have δ1δ2 ̸= 0  since the form Q∞ is assumed non degenerate, and we get two solutions since the  characteristic is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The last assertion is classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  □  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  29  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Construction of degree four del Pezzo surfaces of any arithmetic  type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We are ready to show Theorem 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Our strategy is the following : in order  to construct a surface of a given arithmetic type, we construct a quadratic module  as a sum of cyclic modules in normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We proceed geometric type by geometric  type, and for each one we do the following  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' the Segre symbol gives the factorisation of the characteristic polynomial  over Fq, and we deduce from it a decomposition of a quadratic k[T ]-module  associated to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We choose it cyclic when this is possible, but in general  we choose the decomposition that have seemed us best adapted to the com-  putation of the restricted discriminants from lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='14, and of the infor-  mation needed in lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We sometimes put additional restrictions on  the δi in order to ease this computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we describe all possible Galois actions on the graph of conics and (−2)-  curves given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' they give the possible arithmetic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' For each  one, we use lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='7, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='8 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16 to describe the factorization type of  the characteristic polynomial over Fq, the restricted discriminants of rank  4 singular quadrics and the squareness of certain invariants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we choose some δi in the cyclic quadratic submodules such that the invari-  ants associated to the singular quadrics in the pencil satisfy the conditions  listed at the second point above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The existence of a polynomial with this factorization, and of some δi with the  claimed properties, is sufficient to ensure that the del Pezzo surface associated to  the quadratic k[T ]-module we have constructed has the arithmetic type we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' As already mentioned in the introduction, we have used the math-  ematical software magma to construct a couple of quadrics in P4 for each type of  degree four singular del Pezzo surfaces over a given finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' There is a slight  difference between the procedure used in this program and the description of the  quadratic modules presented below: here we decompose the modules using a biggest  possible cyclic module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In the program we use another decomposition, with more  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' There is a Chinese remainder theorem for quadratic modules –not presented  here– that gives an equivalence between the two descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Ordinary geometric type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) = �5  i=1(T −  θi) with distinct θi, 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  All singular quadrics in the pencil have rank four, and we can associate to the  del Pezzo surface X a cyclic quadratic module [[P, δ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We will construct δ such  that NP (δ) is a square in F×  q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' then from lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='14, the restricted discriminants  of the four singular quadrics are the ∆(θi) = NP (δ)δ(θi) ∈ Fq(θi)× and we have  ∆(θi) ≡ δ(θi) mod Fq(θi)×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Applying lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='7, we see that the signed types representing the conjugacy  classes in W(D5) (which are the arithmetic types of ordinary degree four del Pezzo  surfaces) give  the degrees of the irreducible factors of P in Fq[T ] just by removing the  bars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  the reduced discriminants ∆(θi) mod Fq(θi)×2, ie the classes δ(θi) mod  Fq(θi)×2 from our convention on NP (δ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  30  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  For each geometric type, we present the correspondance between signed types  and properties of P and δ in a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In the last column, we write {□, · · · , □  �  ��  �  d times  } when  the corresponding d roots θi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , θi+d−1 are conjugate over Fq and δ(θi) is a square  in F×  qd, and {⊠, · · · , ⊠  �  ��  �  d times  } when δ(θi) is not a square in F×  qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Signed type  (δ(θ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' δ(θ5))  11111  (□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □)  11111  (□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠)  11111  (□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content=' ⊠)  32  ({⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' {⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠})  Note that a polynomial P and an element δ with the desired properties exist as  long as we have q ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' But when we have q = 3, one cannot construct a polynomial  with five roots over Fq, and some types do not exist [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) = �3  i=1(T −  θi)(T − θ4)2 with distinct θi, 1 ≤ i ≤ 4, and θ4 ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  All singular quadrics in the pencil have rank four, and we can associate to the  del Pezzo surface X a cyclic quadratic module [[P, δ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' For each arithmetic type, we  will choose some δ such that NP (δ) is a square in F×  q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' then from lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='14, the  restricted discriminants of the four singular quadrics are the ∆(θi) = NP (δ)δ(θi) ∈  Fq(θi)× and we have ∆(θi) ≡ δ(θi) mod Fq(θi)×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element of Stab(Rirr) must act on the last two couples of complementary  conics as the identity (of signed type 11) or the bitransposition (C4C′  5)(C5C′  4) (of  signed type 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' note that both have even signed types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Since it is an element of  W(D5), it must act on the first three couples as an element of even signed type, ie  as an element of H3 ⋊ S3, where H3 is the subgroup of Z/2Z3 which is the kernel  of the map ε = (ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' , ε3) �→ �3  i=1 εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get the following ten conjugacy classes  in Stab(Rirr)  111 · 11  111 · 11  21 11  21 · 11  3 · 11  111 · 2  111 · 2  21 · 2  21 · 2  3 · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  31  Write P(T ) = F(T )(T − θ4)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' applying lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='7, we see that the first part of  the above signed types give  the degrees of the irreducible factors of F in Fq[T ] just by removing the  bars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  for 1 ≤ i ≤ 3, the reduced discriminants ∆(θi) mod Fq(θi)×2, ie the classes  δ(θi) mod Fq(θi)×2 from our convention on NP (δ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Then, from lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='8, the classes C4 and C′  5, which correspond to the fibers of the  morphisms defined by the last singular quadric Q0 − θ4Q∞, are fixed by Frobenius  when δ(θ4) is a square in F×  q , and exchanged else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This gives us the second part of  the signed type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We obtain the following table  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  (δ(θ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' δ(θ4))  1  111 · 11  11111  (□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □)  2  111 · 2  2111  (□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠)  3  21 · 11  2111  ({□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □)  4  111 · 11  11111  (⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □)  5  21 · 2  221  ({□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠)  6  111 · 2  2111  (⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠)  7  3 · 11  311  ({□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □)  8  21 · 2  221  ({⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠)  9  21 · 11  2111  ({⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □)  10  3 · 2  32  ({□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠)  Note that we can always construct a polynomial with the claimed factorization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  except when q = 3 in cases 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 4 and 6 since then P needs to have four distinct  roots in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Elements δ ∈ k[T ]/(P) with the required properties always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Finally note that we have NP (δ) ∈ F×2  q  in all cases, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It remains to construct surfaces of types 1,2,4 and 6 over F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We mimic the above  construction: we consider the quadratic forms with the following block diagonal  matrices  Mat(Q0) =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  δ∞  1  1  0  0  0  δ0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  , Mat(Q∞) =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  1  −1  0  δ0  δ0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  The four quadrics of the pencil which are defined over F3 (namely Qt = Q0 +  tQ∞, t ∈ F3 and Q∞) are singular of rank four, and the vertex of Q0 –here (0 : 0 :  0 : 1 : 0)– is the unique singular point of the base of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We deduce that  Xs has geometric type A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The restricted discriminants are, modulo squares  ∆(Q0) ≡ δ0δ∞, ∆(Q1) ≡ ∆(Q2) ≡ δ∞, ∆(Q∞) ≡ 1  and we get the types 1, 2, 4 and 6 respectively choosing (δ0, δ∞) of the form (□, □),  (⊠, □), (⊠, ⊠) and (□, ⊠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type 2A1 with 9 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) =  (T − θ1)(T − θ2)2(T − θ3)2 with distinct θi, 1 ≤ i ≤ 3, and θ1 ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  32  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  All singular quadrics in the pencil have rank four, and we can associate to the  del Pezzo surface X a cyclic quadratic module [[P, δ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We will always choose some  δ such that NP (δ) (or equivalently δ(θ1)) is a square in F×  q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' then we have the  congruence ∆(θi) ≡ δ(θi) mod Fq(θi)×2 for the restricted discriminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element of Stab(Rirr) can  fix the (−2)-curves, and the sets {C2, C′  2, C3, C′  3} and {C4, C′  4, C5, C′  5};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' in  this case it acts on the last four couples of complementary conics as  – the identity, of signed type 11 · 11 or  – a bitransposition such as (C2C′  3)(C2C′  3), of signed type 2 · 11 or  – four transpositions such as (C2C′  3)(C2C′  3)(C4C′  5)(C4C′  5), of signed type  2 · 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  exchange the (−2)-curves, and the sets {C2, C′  2, C3, C′  3} and {C4, C′  4, C5, C′  5};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  in this case it acts on the last four couples of complementary conics as  – four transpositions such as (C2C4)(C′  2C′  4)(C3C5)(C′  3C′  5), of signed type  we note here {2 · 2} or  – a permutation such as (C3C5C′  2C′  4)(C′  3C′  5C2C4), of signed type 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Note that all have even signed types, and the action must be trivial on the remaining  couple of conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get the following five conjugacy classes in Stab(Rirr)  1 · 11 · 11,  1 · 2 · 11,  1 · 2 · 2,  1 · {2 · 2},  1 · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Applying lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='8, we get the following table  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  (δ(θ1), δ(θ2), δ(θ3))  11  1 · 11 · 11  11111  (□, □, □)  12  1 · 2 · 11  2111  (□, ⊠, □)  13  1 · 2 · 2  221  (□, ⊠, ⊠)  14  1 · {2 · 2}  221  (□, {□, □})  15  1 · 4  41  (□, {⊠, ⊠})  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type 2A1 with 8 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) =  �3  i=1(T − θi)(T − θ4)2 with distinct θi, 1 ≤ i ≤ 4, and θ4 ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  There is a rank three singular quadric in the pencil, corresponding to the root  θ4, and we can associate to the del Pezzo surface X a quadratic module of the form  [[F, δ]] ⊕ [[T − θ4, δ1]] ⊕ [[T − θ4, δ2]], where F(T ) = �3  i=1(T − θi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  With the help of lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='14, and using a block diagonal form as in the proof of  lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16, we get that the rank four singular quadrics in the pencil have restricted  discriminants  ∆(θi) = (θi − θ4)2δ1δ2(−1)3NF (δ)δ(θi) ≡ −δ1δ2NF (δ)δ(θi) mod Fq(θi)×2  for 1 ≤ i ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We shall construct quadratic modules satisfying δ2 = −1 and  δ1NF (δ) ∈ F×2  q , so that ∆(θi) ≡ δ(θi) mod Fq(θi)×2 for 1 ≤ i ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element of Stab(Rirr) must act on the last two couples of complementary  conics as the identity or as (C5C′  5) of signed type 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In the first case it acts on  the first three couples as an even signed type permutation, ie as an element of  H3 ⋊ S3, and in the second as an odd signed type permutation, ie as an element  of Z/2Z3 ⋊ S3 \\ H3 ⋊ S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get the following conjugacy classes in Stab(Rirr)  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  33  111 · 11, 111 · 11, 21 · 11, 21 · 11, 3 · 11, 111 · 11, 111 · 11, 21 · 11, 21 · 11, 3 · 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The second part of the signed type is 11 exactly when the singular points are  defined over Fq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16, applied with our convention δ2 = −1, tells us that  this happens exactly when δ1 is a square in F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It remains to use lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='7 to determine the factorization pattern of F and the  δ(θi) mod Fq(θi)×2 from the signed type of the action on the first three couples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We get the following table  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  (δ(θ1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' δ(θ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' δ(θ3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' δ1)  16  111 · 11  11111  (□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □)  17  21 · 11  2111  ({□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □)  18  111 · 11  11111  (⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □)  19  111 · 11  11111  (⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠)  20  21 · 11  2111  ({□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠)  21  111 · 11  11111  (⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠)  22  3 · 11  311  ({□,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □)  23  21 · 11  2111  ({⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠)  24  21 · 11  2111  ({⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' □)  25  3 · 11  311  ({⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ⊠)  Note that we can always construct a polynomial over Fq with the claimed factor-  ization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' except when q = 3 in cases 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 19 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Elements δ ∈ k[T ]/(F) with  the required properties always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Finally note that we have δ1NF (δ) ∈ F×2  q  in  all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It remains to construct surfaces of types 16, 18, 19 and 21 over F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We mimic  the above construction: we consider the quadratic forms with the following block  diagonal matrices  Mat(Q0) =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  δ∞  δ1  1  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  , Mat(Q∞) =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  δ1  −1  δ0  −1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  The quadrics with equations Q∞, Q0 + Q∞ and Q0 − Q∞ have rank 4, and the  quadric Q0 has rank 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The singular points of Xs are the intersections of the vertex  of Q0 with Q∞, that is the points (0 : 0 : 0 : a : b) with δ0a2 − b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get  a surface of geometric type 2A1 with 8 lines as above, since the line joining the  singularities is not contained in Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Computing the restricted discriminants, and applying lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16, we see that  we get the types 16, 18, 19 and 21 respectively choosing (δ0, δ1, δ∞) of the form  (□, □, □), (□, ⊠, ⊠), (⊠, ⊠, ⊠) and (⊠, □, □).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) = (T − θ1)(T −  θ2)(T − θ3)3 with distinct θi, 1 ≤ i ≤ 3, and θ3 ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  All singular quadrics in the pencil have rank four, and we can associate to the  del Pezzo surface X a cyclic quadratic module [[P, δ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We will construct δ such that  NP (δ) is a square in F×  q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' then we have the congruence ∆(θi) ≡ δ(θi) mod Fq(θi)×2  for the restricted discriminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  34  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  An element of Stab(Rirr) must act on the last three couples of complementary  conics as the identity (of signed type 111) or the permutation (C5C′  3)(C3C′  5)(C4C′  4)  (of signed type 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It will act on the first two couples as en even signed permutation  in the first case, as an odd signed one in the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get the following five  conjugacy classes in Stab(Rirr)  11 · 111,  11 · 111,  2 · 111,  11 · 21,  2 · 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Note that the signed type is completely determined by its first part, and we just  have to describe the action on it via lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get the following table, where  we have chosen δ(θ3) so that the convention NP (δ) = δ(θ1)δ(θ2)δ(θ3)3 ∈ F×2  q  holds  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  (δ(θ1), δ(θ2), δ(θ3))  26  11 · 111  11111  (□, □, □)  27  2 · 111  2111  ({□, □}, □)  28  11 · 111  11111  (⊠, ⊠, □)  29  11 · 21  2111  (⊠, □, ⊠)  30  2 · 21  221  ({⊠, ⊠}, ⊠)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type 3A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) = (T − θ1)(T −  θ2)2(T − θ3)2 with distinct θi, 1 ≤ i ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We assume that the rank three conic in  the pencil corresponds to θ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We can write the quadratic k[T ]-module [[F, δ]]⊕[[T −  θ3, δ1]] ⊕ [[T − θ3, δ2]] where F(T ) := (T − θ1)(T − θ2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  With the help of lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='14, and using a block diagonal form as in the proof of  lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16, we get that the rank four singular quadrics in the pencil have restricted  discriminants  ∆(θi) = (θi − θ3)2δ1δ2(−1)3NF (δ)δ(θi) ≡ −δ1δ2δ(θ1)δ(θi) mod F×2  q , 1 ≤ i ≤ 2  We shall construct quadratic modules satisfying δ2 = −1 and δ1δ(θ1) ∈ F×2  q , so  that ∆(θi) ≡ δ(θi) mod F×2  q  for 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element of Stab(Rirr) must act on the last two couples of complementary  conics as the identity (of signed type 11) or the permutation (C5C′  5) (of signed type  11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' On the second and third couples, it acts as the identity (of signed type 11) or  as the bitransposition (C2C′  3)(C3C′  2) of signed type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We then have to “complete”  the action by fixing C1 and C′  1 if C5 and C′  5 are fixed, and by transposing them  else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get the following four conjugacy classes in Stab(Rirr)  1 · 11 · 11,  1 · 11 · 11,  1 · 2 · 11,  1 · 2 · 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  As above, we use lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='8 (a) to read the restricted discriminant ∆(θ2) from  the action on the second and third couples, and lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16 to get δ1 from the  action on the last couple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  (δ(θ1), δ(θ2), δ1)  31  1 · 11 · 11  11111  (□, □, □)  32  1 · 2 · 11  2111  (□, ⊠, □)  33  1 · 11 · 11  11111  (⊠, □, ⊠)  34  1 · 2 · 11  2111  (⊠, ⊠, ⊠)  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  35  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type A1A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) = (T −  θ1)2(T − θ2)3 with distinct θi, 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' There is no rank three conic in the  pencil, and we can write the quadratic k[T ]-module in cyclic form [[P, δ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='14 tells us that the restricted discriminants are ∆(θi) = NP (δ)δ(θi) =  δ(θ1)2δ(θ2)3δ(θi) ≡ δ(θ2)δ(θi) mod F×2  q  for 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We shall construct quadratic  modules satisfying δ(θ2) ∈ F×2  q , so that ∆(θi) ≡ δ(θi) mod F×2  q  for 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element of Stab(Rirr) must act on the first two couples of complementary  conics as the identity (of signed type 11) or a bitransposition (of signed type 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  On the last three couples, it acts as the identity in order to have en even signed  type (recall that the non trivial action on these three couples has signed type 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We get two conjugacy classes in Stab(Rirr), namely 11 · 111 and 2 · 111, and using  lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='8 (a), we get the table  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  (δ(θ1), δ(θ2))  35  11 · 111  11111  (□, □)  36  2 · 111  2111  (⊠, □)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type A3 with five lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) =  (T − θ1)(T − θ2)4 with distinct θi, 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' There is no rank three conic in the  pencil, and we can write the quadratic k[T ]-module in cyclic form [[P, δ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='14 tells us that the restricted discriminants are ∆(θi) = NP (δ)δ(θi) ≡  δ(θ1)δ(θi) mod F×2  q  for 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We shall construct quadratic modules satisfying  δ(θ1) ∈ F×2  q , so that ∆(θi) ≡ δ(θi) mod F×2  q  for 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element of Stab(Rirr) must act on the last four couples of complementary  conics as the identity (of signed type 1111) or as (C2C′  5)(C′  2C5)(C3C′  4)(C′  3C4) (of  signed type 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Thus it must act as the identity on the first couple in order to have  en even signed type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get two conjugacy classes in Stab(Rirr), namely 1 · 1111  and 1 · 22, and using lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='8 (a), we get the table  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  (δ(θ1), δ(θ2))  37  1 · 1111  11111  (□, □)  38  1 · 22  221  (□, ⊠)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type A3 with four lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) =  (T − θ1)(T − θ2)(T − θ3)3 with distinct θi, 1 ≤ i ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We can write the quadratic  k[T ]-module [[F, δ]] ⊕ [[T − θ3, δ1]] ⊕ [[(T − θ3)2, δ2]] where F(T ) := (T − θ1)(T − θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  With the help of lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='14, and using a block diagonal form as in the proof of  lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16, we get that the rank four singular quadrics in the pencil have restricted  discriminants  ∆(θi) = (θi − θ3)3δ1δ2  2NF (δ)δ(θi) ≡ (θi − θ3)δ1δ(θ1)δ(θ2)δ(θi) mod F×2  q , 1 ≤ i ≤ 2  We shall construct quadratic modules satisfying δ1 = (θ1 − θ3)(θ2 − θ3), so that  ∆(θi) ≡ (θj − θ3)δ(θj) mod F×2  q  for 1 ≤ i ̸= j ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element of Stab(Rirr) must act on the last three couples of complementary  conics as the identity (of signed type 111) or the transposition (C5C′  5) (of signed  type 111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We deduce the possible actions on the first two couples, and the following  five conjugacy classes in Stab(Rirr)  11 · 111,  11 · 111,  2 · 111,  11 · 111,  2 · 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  36  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  The type is completely determined by the action on the first two couples: it is  sufficient to use lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='7 to obtain the table  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  ((θ1 − θ3)δ(θ1), (θ2 − θ3)δ(θ2))  39  11 · 111  11111  (□, □)  40  2 · 111  2111  ({□, □})  41  11 · 111  11111  (⊠, ⊠)  42  11 · 111  11111  (□, ⊠)  43  2 · 111  2111  ({⊠, ⊠})  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type 4A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) = (T −θ1)(T −  θ2)2(T − θ3)2 with distinct θi, 1 ≤ i ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We can write the quadratic k[T ]-module  [[T − θ1, δ]] ⊕ [[F, δ1]] ⊕ [[F, δ2]] where F(T ) := (T − θ2)(T − θ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We will construct  such a module satisfying the additional assumption δ2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element in Stab(Rirr) acts on the four (−2)-curves, and we deduce from this  its action on the couples {Ci, C′  i}, i ∈ {1, 2, 4, 5}  (a) it fixes the four (−2)-curves, and the eight conics classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' then its signed  type is 1111;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (b) it fixes two (−2)-curves (necessarily corresponding to the same rank 3  quadric of the pencil, thus in the same “component” of the graph), and  permutes the other ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' for instance the transposition (R3R4) corresponds  to (C5C′  5), of signed type 1111;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (c) it acts as the bitransposition (R1R2)(R3R4) on (−2)-curves, and on conic  classes as (C2C′  2)(C5C′  5), its signed type is 1111  (d) it acts as the bitransposition (R1R3)(R2R4) on (−2)-curves, and on conic  classes as (C1C4)(C2C5)(C′  1C′  4)(C′  2C′  5) of signed type 22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  (e) it acts as the four-cycle (R1R3R2R4) on (−2)-curves and on conic classes  as (C1C4)(C′  1C′  4)(C2C5C′  2C′  5) of signed type 22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Then we complete with the correct action on {C3, C′  3} in order to get an even signed  permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get five conjugacy classes  1 · 1111,  1 · 1111,  1 · 1111,  1 · 22,  1 · 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The two rank three conics can be defined over Fq (we have θ2, θ3 ∈ Fq), or over  Fq2, and conjugate over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Applying lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16 to X in the first case, and to  X ⊗ Fq2 in the second, we get the following table under the assumption δ2 = −1  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  (δ1(θ2), δ1(θ3))  44  1 · 1111  11111  (□, □)  45  1 · 1111  11111  (⊠, ⊠)  46  1 · 22  221  ({□, □})  47  1 · 1111  11111  (⊠, □)  48  1 · 22  221  ({⊠, ⊠})  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type 2A1A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) = (T −  θ1)3(T − θ2)2 with distinct θi, 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We can write the quadratic k[T ]-module  [[(T −θ1)3, δ]]⊕[[T −θ2, δ1]]⊕[[T −θ2, δ2]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We will construct such a module satisfying  the additional assumptions δ = 1, δ2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We have already seen that an element in Stab(Rirr) can act on the first three  couples with a signed type 111 or 21, and on the last two with a signed type 11 or  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This leaves us with two conjugacy classes, namely 11 · 111 and 21 · 11, which  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  37  are determined by the action on the last couple, ie on the two A1-singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Applying lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16, and from our assumption δ2 = −1, we get the table  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  δ1  49  111 · 11  11111  □  50  21 · 11  2111  ⊠  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type A1A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) = (T −  θ1)2(T − θ2)3 with distinct θi, 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We can write the quadratic k[T ]-module  [[(T − θ1)2, δ]] ⊕ [[(T − θ2)2, δ1]] ⊕ [[T − θ2, δ2]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We will construct such a module  satisfying the additional assumptions δ1 = δ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The restricted discriminant of  the rank four quadric in the pencil (corresponding to root θ1) is  ∆(θ1) = δ(θ1)(θ1 − θ2)3δ2  1δ2 ≡ δ(θ1)(θ1 − θ2) mod F×2  q  An element in Stab(Rirr) acts on the first two couples with a signed type 11 or  2, and on the last three with a signed type 111 or 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This leaves us with two  conjugacy classes, namely 111·11 and 2·111, and they are determined by the action  on the first two couples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Applying lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='8 (a), and from our assumptions, we  get the table  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  (θ1 − θ2)δ(θ1)  51  111 · 11  11111  □  52  2 · 111  2111  ⊠  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) = (T − θ1)5,  and we can consider the cyclic quadratic k[T ]-module [[(T − θ1)5, δ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element in Stab(Rirr) can only act on the graph trivially (the other automor-  phism of this graph is (C1C′  5)(C′  1C5)(C2C′  4)(C′  2C4)(C3C′  3) which has odd signed  type 221).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Thus for any choice of δ we get the arithmetic type 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type D4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) = (T − θ1)(T −  θ2)4 with distinct θi, 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We can write the quadratic k[T ]-module [[T −  θ1, δ]] ⊕ [[(T − θ2)3, δ1]] ⊕ [[T − θ2, δ2]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We will construct such a module satisfying  the additional assumptions δ = δ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In this case the restricted discriminant of  the rank four quadric in the pencil (corresponding to root θ1) is  ∆(θ1) = −(θ2 − θ1)4δ3  1δ2 ≡ δ1 mod F×2  q  An element in Stab(Rirr) acts on the last four couples with a signed type 1111  or 1111, and on the first one with a signed type 1 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This leaves us with two  conjugacy classes, namely 1 · 1111 and 1 · 1111, and they are determined by the  action on the first couple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Applying lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='7 to the value of ∆(θ1) above, we get  the table  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  δ1  54  1 · 1111  11111  □  55  1 · 1111  11111  ⊠  38  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type 2A1A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) = (T −  θ1)2(T − θ2)3 with distinct θi, 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We can write the quadratic k[T ]-module  [[T − θ1, δ1]] ⊕ [[T − θ1, δ2]] ⊕ [[(T − θ2)2, η1]] ⊕ [[T − θ2, η2]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We will construct such a  module satisfying the additional assumptions δ2 = −1 and η1 = η2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element in Stab(Rirr) acts on the first two couples with a signed type 11  or 11, and on the last three with a signed type 111 or 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This leaves us with  two conjugacy classes, namely 11 · 111 and 11 · 111, and they are determined by  the action on the first two couples, ie by the action on the two A1-singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Applying lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16 with δ2 = −1, we get the table  N ◦  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Weyl  δ1  56  11 · 111  11111  □  57  11 · 111  11111  ⊠  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Geometric type D5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The factorization of P over Fq is P(T ) = (T −θ)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We  can write the quadratic k[T ]-module [[(T − θ)4, δ1]] ⊕ [[T − θ, δ2]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  An element in Stab(Rirr) can only act on the graph trivially (the other auto-  morphism of this graph is (C5C′  5) which has odd signed type 11111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Thus for any  choice of δi we get the arithmetic type 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Construction of singular del Pezzo surfaces of degree three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The aim of this section is to prove the last two assertions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We blow up the degree four surfaces from the preceding section at well chosen  rational points in order to construct degree three surfaces, but we also use some  direct constructions by blowing up some well chosen configurations of points in the  projective plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Blowing up degree four surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Our first construction of degree three  del Pezzo surfaces is by blowing up a rational point not lying on any of the negative  curves of a degree four del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In this way we get a surface of the same  Dynkin type (note that for degree three, the geometric type coincides with the  Dynkin type), whose zeta function is the one of the degree four surface, divided  by 1 − qT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This (and the Galois action on the (−2)-curves) makes it very easy to  control the arithmetic type of the new surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In order to do this, we count the number of rational points not lying on any  negative curve on a degree four weak del Pezzo surface of a given arithmetic type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  If a rational point lies on a negative curve, then the curve must be defined over Fq,  or the point is the intersection of two Galois conjugate curves (thus defined over  Fq2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We deduce that the number we are looking for is (note that there is no 3-cycle  in any graph of negative curves,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' and such three curves cannot be concurrent)  N  =  #X(Fq) − (#N(Fq)(q + 1) − I1) − I2  =  q2 − tq + 1 − (#N(Fq)(q + 1) − I1) − I2  where t is the trace of the action of the Frobenius operator on Pic(X⊗Fq),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' N(Fq) is  the number of negative curves defined over Fq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' I1 is the number of their intersection  points (which is readily computed as an intersection number from the N(Fq) curves  above),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' and I2 is the number of couples of conjugate negative curves intersecting  themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  39  We compute these data and present them in table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1, which is constructed as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' For each geometric type of degree four del Pezzo surfaces, we consider the  graph of negative curves given in [7, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We denote the curves in the  same way, except for the (−2)-curves : recall that we denote by ri the curve with  class Ei − Ei+1, and by rijk the curve with class E0 − Ei − Ej − Ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  For each arithmetic type in degree four, we give in the second column an element  of the Weyl group in the conjugacy class corresponding to the Frobenius action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We present it as a composite of reflections, where we denote by sij (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' sijk) the  reflection around the root Ei − Ej (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' E0 − Ei − Ej − Ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In the third column  we present the action of this element on the negative curves, as a product of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  As explained above, this is sufficient to determine the numbers t, N(Fq), I1, I2 and  N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' they are given in the following columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The last column gives the type of the degree three surface obtained by blowing  up a rational point not lying on any negative curve, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Note that this is sufficient to prove the existence of a degree three del Pezzo  surface of the type given in the last column over the finite field Fq as long as the  number given in the last but one column is positive for the given q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Assume that there exists a weak del Pezzo surface of degree three and arithmetic  type 1, defined over Fq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' the Galois action on the Picard group is trivial, and all its  negative curves are defined over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It contains an exceptional curve that does not  meet the (−2)-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Contracting this curve gives a degree three del Pezzo surface  of arithmetic type 1, and the image of the curve is a point that does not lie on any  negative curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' But such a point does not exist when q = 3 from the first line of  the above table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  One shows in the same way that a weak del Pezzo surface of degree three and  arithmetic type 12 does not exist over F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The only difference is that contracting  an exceptional curve that does not meet the (−2)-curve gives a point outside the  negative curves on a weak del Pezzo surface of degree four and arithmetic type 11  or 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' such a point does not exist when q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Other constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' There remains 77 − 48 = 29 types to be constructed  over any finite field with odd characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  For those we give an alternative contruction: we often present them as blow ups  of the projective plane at well-chosen rational points, and sometimes as blow ups of  a degree four del Pezzo surface at some point lying on one or two negative curve(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  First remark that we get all arithmetic types for the geometric type A1 over Fq  by blowing up the points of a degree 6 zero-dimensional subscheme of a smooth  conic, everything being defined over Fq, when q is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that the only  (−2)-curve is the strict transform of the conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Playing on the fields of definition  of the points in the subscheme, we get all partitions of the integer 6, and all types  (note the stabilizer here is S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In this way we construct a surface for each of  the remaining types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get type 5 by blowing up two points defined over Fq3,  type 10 by blowing up one point defined over Fq and one over Fq5, and type 5 by  blowing up one point defined over Fq6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Since a smooth conic defined over Fq has  qk + 1 points over Fqk, such configurations exist over any finite field (also of even  characteristic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  40  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  Type  Frobenius in W(D5)  Galois action on neg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' curves  (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' #N(Fq),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' I1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' I2)  N  Type  1  A1(12)  Id  (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 − 7q + 12  1  2  s12  (ℓ1ℓ2)(ℓ13ℓ23)(ℓ14ℓ24)  (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 − 3q  2  3  s123  (ℓ45q)(ℓ1ℓ23)(ℓ2ℓ13)(ℓ3ℓ12)  (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 − q − 4  2  4  s12 ◦ s345  (ℓ12q)(ℓ1ℓ2)(ℓ3ℓ45)(ℓ5ℓ34)(ℓ14ℓ24)(ℓ13ℓ23)  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 + q + 4  3  5  s12 ◦ s123  (ℓ3q)(ℓ2ℓ13)(ℓ1ℓ23)(ℓ5ℓ34)(ℓ12ℓ45)  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 2)  q2 − q − 2  3  6  s12 ◦ s345 ◦ s123  (ℓ1ℓ13)(ℓ3q)(ℓ2ℓ23)(ℓ5ℓ34)(ℓ12ℓ45)(ℓ14ℓ24)  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 4)  q2 − q − 4  4  7  s12 ◦ s23  (ℓ1ℓ2ℓ3)(ℓ13ℓ12ℓ23)(ℓ14ℓ24ℓ34)  (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 − q  6  8  s13 ◦ s23 ◦ s145 ◦ s123  (ℓ1qℓ3ℓ13)(ℓ2ℓ23ℓ45ℓ12)(ℓ14ℓ5ℓ34ℓ24)  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 − q  8  9  s345 ◦ s23 ◦ s12  (ℓ23qℓ12ℓ13)(ℓ2ℓ1ℓ45ℓ3)(ℓ14ℓ5ℓ34ℓ24)  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 + q  9  10  s12 ◦ s23 ◦ s123  (ℓ14ℓ24ℓ34)(ℓ45q)(ℓ1ℓ13ℓ3ℓ23ℓ2ℓ12)  (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 − q  7  11  2A1(9)  Id  (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 − 5q + 6  12  12  s123  (qℓ45)(ℓ1ℓ23)(ℓ3ℓ12)  (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 − q − 2  13  13  s123 ◦ s145  (qℓ1)(ℓ3ℓ12)(ℓ5ℓ14)(ℓ45ℓ23)  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 2)  q2 − q − 2  15  14  s24 ◦ s35  (ℓ3ℓ5)(ℓ45ℓ23)(ℓ14ℓ12)(r2r4)  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1)  q2 − q − 2  14  15  s145 ◦ s24 ◦ s35  (qℓ23ℓ1ℓ45)(ℓ3ℓ14ℓ12ℓ5)(r2r4)  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 + q  19  16  2A1(8)  Id  (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content=' 2)  q2 − 1  51  51  A1A3(3)  Id  (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 − q  52  52  s345  (ℓ34ℓ5)  (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 − q  53  53  A4(3)  Id  (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 − q  60  54  D4(2)  Id  (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2  62  55  s234 ◦ s15  (ℓ5ℓ2)(r4r123)  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2  63  56  2A1A3(2)  Id  (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 − q  67  57  s134 ◦ s25  (ℓ5ℓ2)(r1r345)(r123r4)  (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2 + q  68  58  D5(1)  Id  (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 0)  q2  72  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Galois action on negative curves in degree 4  We get all arithmetic types for the geometric type A2 over Fq by blowing up  the points of two degree 3 zero-dimensional subschemes lying on two different lines,  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  41  but not containing their intersection point, everything being defined over Fq for q  large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that the (−2)-curves are the strict transforms of the two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  As above, the fields of definitions of the points, and the lines, give the arithmetic  type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We get types  26 when we choose the two lines to be defined over Fq, with three points  defined over Fq on one, and three conjugate points defined over Fq3 on the  other line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  27 when we choose the two lines to be defined over Fq, with three conjugate  points defined over Fq3 on the both lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  29 when we choose the two lines to be defined over Fq, with one point defined  over Fq and two defined over Fq2 on one, and three conjugate points defined  over Fq3 on the other line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  30 when we choose the two lines to be conjugate, defined over Fq2, with one  point defined over Fq6 on one, and all its conjugates points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We get type 35 by blowing up three conjugate points defined over Fq3 in the  projective plane, then three infinitely near conjugate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that the (−2)-  curves are the strict tranforms of the exceptional divisors of the first blow up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In order to construct a surface of type 39, we start from a degree four del Pezzo  surface of arithmetic type 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Such a surface contains two exceptional curves defined  over Fq, that meet together and each one meets exactly one (−2)-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We blow  up a point defined over Fq on one of these exceptional curve, different from their  intersection point and not lying on a (−2)-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This is possible for any q, and we  get the desired surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In order to get a degree three surface of geometric type 4A1, we start with four  lines in general position in the projective plane, their union being defined over  Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then we blow up their six intersection points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' the (−2)-curves are the strict  transforms of the lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then we get the arithmetic types by playing on the fields  of definition of the lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' For instance we get type 48 by choosing three conjugates  lines defined over Fq3 and the last one defined over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  If we start from a degree four surface of geometric type 2A1 defined over Fq,  and we blow up a rational point which lies at the intersection of two exceptional  curve (ie corresponding to a “middle” edge of the graph of negative curves), we  get a degree three surface of geometric type 2A2 over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In this way we get  types 54, 55, 56, 57, 58 59 in degree three respectively from types 16, 17, 20, 21, 22  et 25 in degree four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Note that in any case these degree four surfaces contain two  intersecting exceptional curves which are either defined over Fq, or defined over  Fq2 and conjugate over Fq: their intersection point is always a rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We get type 61 by blowing up p1 ≺ p2 ≺ p3 ≺ p4 four infinitely near points with  the first three not collinear, and a point of degree 2 on a line passing through p1  but whose strict transform by the first blow up does not contain p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We get a degree three surface of type 64 by blowing up three collinear points  p1, p2, p3 defined over Fq3 and conjugate over Fq then three conjugate points  p4, p5, p6 with pi ≺ pi+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We get a degree three del Pezzo surface of geometric type A12A2 when we blow  up the point ℓ1 ∩ ℓ14 on a degree four del Pezzo surface of type 3A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This point is  defined over Fq when the degree four surface has one of the types 31 or 34, and we  get a surface of type 65 or 66 in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  42  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  Most of the remaining constructions are completely geometric and can be found  in [12, pp 493-494].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Namely we get type  69 by blowing up p1 ≺ p2 ≺ p3 ≺ p4 ≺ p5 with p1, p2, p3 not collinear and p6  on the (smooth) conic defined by the first five ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  70 by blowing up p1 ≺ p2 ≺ p3 ≺ p4 ≺ p5 ≺ p6 with p1, p2, p3 not collinear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  73 by blowing up p1 ≺ p2 ≺ p3 ≺ p4 ≺ p5 ≺ p6 with p1, p2, p3 not collinear  and p6 on the conic defined by the first five ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  77 by blowing up p1 ≺ p2 ≺ p3 ≺ p4 ≺ p5 ≺ p6 with p1, p2, p3 collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Type 71 in degree three is obtained by blowing up a rational point lying on the  exceptional curve ℓ2 (but not on any (−2)-curve) in a degree four del Pezzo surface  of arithmetic type 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We end with the geometric type 3A2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we blow up three points p1, p2, p3 in  general position in the projective plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Then, on the resulting degree 6 ordinary  del Pezzo surface, we blow up the intersection points ℓ1 ∩ ℓ12, ℓ2 ∩ ℓ23 and ℓ3 ∩ ℓ13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The (−2)-curves on the corresponding degree three del Pezzo surface are the strict  transforms of these six lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Finally, we get a surface of type 74 when the pi are  rational points, of type 75 when one is rational and the two other ones defined over  Fq2 and conjugate over Fq, and of type 76 when the three points are defined over  Fq3 and conjugate over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Type 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It remains to construct a del Pezzo surface of degree three and  arithmetic type 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Such a surface has geometric type 3A1, and the inverse of the  weight two part of its zeta function is Φ2  1Φ2Φ2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We start with a degree one del Pezzo surface S obtained by blowing up two  degree three points p1, p2 = pσ  1, p3 = pσ2  1  and q1, q2 = qσ  1 , q3 = qσ2  1  in P2(Fq3) and  a degree two point r1, r2 = rσ  1 in P2(Fq2), such that there exists a conic passing  through p1, p2, q1, q2, r1, r2 and defined over Fq3, but there are no other (−2)-curve  on the resulting degree one surface than the strict transforms of this conic and its  conjugates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We report the proof of the existence of such a configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The surface S has geometric type 3A1 by our hypothesis on its (−2)-curves  (note that the strict transforms are separated by the blowups), and the inverse of  the weight two part of its zeta function is (X −1)(X2 −1)(X3 −1)2 = Φ4  1Φ2Φ2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We  can contract the strict transform of the line (r1r2), and of the conic (q1q2q3r1r2):  these are disjoint exceptional curves that do not meet the (−2)-curves, both defined  over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In this way we get a degree three surface with geometric type 3A1 and  the desired zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It remains to show the existence of such a configuration of points over any finite  field of odd characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We start with two elements θ ∈ Fq3 \\Fq, η ∈ Fq2 \\Fq, whose respective minimal  polynomials over Fq are πθ(x) := X3 + aX2 + bX + c and πη(X) = X2 + tX + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We consider the points p1 = (θ : θ2 : 1), q1 = (θ : θ2 : u) and r1 = (η : 1 : 0) for  some u ∈ Fq3 \\ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  From Pascal’s theorem, the points p1, p2, q2, r2, r1, q1 are conconic if and only if  the pairs of opposite sides of the hexagon meet in three collinear points (p1p2) ∩  (r1r2), (p2q2) ∩ (r1q1) and (p1q1) ∩ (q2r2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' we get the points  (θ − θq : θ2 − θ2q : 0), (θq : θ2q : α), (θ : θ2 : β)  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  43  where we have set  α := uηθ2q − θq  ηθ2 − θ , β := uq ηqθ2 − θ  ηqθ2q − θq  The points are collinear if and only if we have (uβ − uqα)θ3(θq−1 − θ2(q−1)) = 0, if  and only if uβ − uqα = 0 (note that θq−1 ̸= 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We easily check that the preceding  equality gives  u ∈ (ηθ2 − θ)(ηqθ2 − θ)F×  q = (nθ4 + tθ3 + θ2)F×  q  In other words, for our choice of the points, there are exactly q−1 values of u in Fq3  such that the points p1, p2, q2, r2, r1, q1 are conconic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' note that using the Frobenius  action, we also get that the points p2, p3, q3, r2, r1, q2 are conconic, so as the points  p1, p3, q3, r2, r1, q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We remark that the conic C1 passing through p1, p2, q2, r2, r1, q1 must be irre-  ducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Else it would be the union of two lines d and d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' If d contains r1, it cannot  contain r2 = rσ  1 : it would be the line Z = 0 and the other line would contain the  four remaining points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' this is impossible since p2 does not lie on (p1q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Thus dσ3  contains r2, d is not defined over Fq3, and we get d′ = dσ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In this case both d and  dσ3 would contain p1, p2, q1, q2, this is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It remains to verify that one can choose u such that there is no other (−2)-curve  on the degree one surface obtained by blowing up the eight points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' In other words,  no three points can be collinear, no six can lie on a conic, and there is no cubic  passing through the eight points which is singular at one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We first check that no three of the eight points can be collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' If such a subset  contains r1 and r2, this is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' If it contains r1, then the line joining r1 to any  degree three point is defined over Fq6, and cannot contain any other point defined  over Fq3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We are reduced to the subsets of three points among the p1, p2, p3, q1, q2, q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Any  subset of the form {pi, pj, qk} or {qi, qj, pk} with k ∈ {i, j} cannot be collinear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' else  the corresponding line would cross one of the three irreducible conics conjugate  to C1 above at three points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We are reduced to consider the subsets {p1, p2, p3},  {q1, q2, q3}, and (up to Galois action) {p1, p2, q3}, {q1, q2, p3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Note that a point (x : y : z) ∈ P2(Fq3) and its conjugates over Fq are collinear  if and only if the Fq subspace generated by x, y, z is strictly contained in Fq3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' As  a consequence, it follows from the definition of p1 that the points p1, p2, p3 are not  collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Now write  u = m(nθ4 + tθ3 + θ2) = m  �  (1 − nb − a(t − na))θ2 − (nc + t − na)θ − (t − na)c  �  for some m ∈ F×  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' From the above criterion, we get that the points q1, q2, q3 are  collinear if and only if t = na (of course we have c ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  The points p1, p2, q3 are collinear if and only if  ������  1  θ  θ2  1  θq  θ2q  uq2  θq2  θ2q2  ������  = 0  This equation is (semi)-linear in the variable u, and admits a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  In the same way, the points q1, q2, p3 are collinear if and only if  u(θ2q2+q − θq2+2q) − uq(θ2q2+1 − θq2+2) + θ2q+1 − θq+2 = 0  44  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  Setting U = u(θ2q2+q − θq2+2q), we can rewrite the equation U + U q + θ2q+1 −  θq+2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Since gcd(T q + T, T q3 − T ) = T , the map U �→ U q + U is an Fq-linear  automorphism of Fq3, and this equation admits exactly one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Now choose six of the eight points, and assume they lie on a conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' If ri is one  of these points, then among the five remaining points, four must lie on one of the  conjugates of C1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' from Bezout theorem, the conic must be one of these, and the  subset of six conconic points is among the three we already constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' It remains  to consider the six points p1, p2, p3, q1, q2, q3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' any conic passing through these points  should be defined over Fq, et thus have an equation of the form  eX2 + hXY + iY 2 + jXZ + kY Z + lZ2  Plugging the coordinates of p1 and q1, substracting and factoring 1 − u, we get  jθ + kθ2 + l(1 + u) = 0  so that the family (θ, θ2, 1 + u) does not generate the Fq-vector space Fq3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This  happens if, and only if 1 + mc(t − na) = 0 – with u = m(nθ4 + tθ3 + θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  It remains to show that there is no cubic passing through the eight points and  singular at one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' If such a cubic C exists, it has exactly one singular point,  which is not defined over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' So it is distinct from Cσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Now these two cubics have  at least ten intersection points (counted with multiplicities), which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Summing up, we have to choose η and θ as above, such that the coefficients of  their minimal polynomials satisfy t − na ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' then we get q − 1 possible values for  u such that the three prescribed subsets of six points are conconic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We remove one  possible value to verify that the six points pi, qi do not lie on a conic, then at most  one for each of the assertions p1, p2, q3 non collinear, and q1, q2, p3 non collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  We get a least q − 4 possible values for u, and the problem is settled for q ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  For q = 3, one can verify that if θ is a root of T 3 + T 2 + 2, and η a root of  T 2 + T + 2, then for m = −1 the points defined above have the required position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Arithmetic types for degrees three to six  We give below the lists of arithmetic types for degrees 3 ≤ d ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' The tables are  organized as follows  in the first column, we fix a number for each type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' there are also three  types (one in degree four and two in degree three) for which we add an  asterisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' This means that the invariant H1(Γ, Pic(X ⊗ Fq)) is non trivial  (it is isomorphic to the Klein four-group Z/2Z × Z/2Z in each case), and  that a surface of this type is not birational to the projective plane [21,  Section 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  in the second one, we describe the geometric types for the given degrees, in  other words the closed and symmetric parts of a root system of type E9−d  up to the action of the Weyl group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' they come from [7], except for degree  3 where they are given in [8, Table IV (iv)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' We give the Dynkin types of  the singular points and (between parentheses) the number of exceptional  divisors on the surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  in the third column, we give the stabilizer attached to the geometric type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  in the fourth one, we give the characteristic polynomial of the action of any  element w in the conjugacy class of the stabilizer on the geometric Picard  group Pic(X) of a weak del Pezzo surface of the type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  45  finally, we give the characteristic polynomial of the action of any element  w in the conjugacy class of the stabilizer on the geometric Picard group  Pic(Xs) of a singular del Pezzo surface of a type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Moreover, in the table for degree 4, we add a column, in order to describe each  conjugacy class in W(D5) by its signed cycle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Table 5: Arithmetic types in degree 6  Type T  Geometric type  Stab  χw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Pic(X)  χw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Pic(Xs)  1  A1 (4)  Z/2Z  Φ4  1  Φ3  1  2  Φ3  1Φ2  Φ2  1Φ2  3  A1 (3)  S3  Φ4  1  Φ3  1  4  Φ3  1Φ2  Φ2  1Φ2  5  Φ2  1Φ3  Φ1Φ3  6  2A1 (2)  {e}  Φ4  1  Φ2  1  7  A2 (2)  Z/2Z  Φ4  1  Φ2  1  8  Φ3  1Φ2  Φ1Φ2  9  A2A1 (1)  {e}  Φ4  1  Φ1  Table 6: Arithmetic types in degree 5  Type T  Geometric type  Stab  χw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Pic(X)  χw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Pic(Xs)  1  A1 (7)  S3  Φ5  1  Φ4  1  2  Φ4  1Φ2  Φ3  1Φ2  3  Φ3  1Φ3  Φ2  1Φ3  4  2A1 (5)  Z/2Z  Φ5  1  Φ3  1  5  Φ3  1Φ2  2  Φ2  1Φ2  6  A2 (4)  Z/2Z  Φ5  1  Φ3  1  7  Φ4  1Φ2  Φ2  1Φ2  8  A2A1 (3)  {e}  Φ5  1  Φ2  1  9  A3 (2)  {e}  Φ5  1  Φ2  1  10  A4 (1)  {e}  Φ5  1  Φ1  Table 7: Arithmetic types in degree 4  Type T  Geometric type  Stab  W(D5)  χw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Pic(X)  χw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Pic(Xs)  1  A1 (12)  S4 × Z/2Z  11111  Φ6  1  Φ5  1  2  2111  Φ5  1Φ2  Φ4  1Φ2  3  2111  Φ5  1Φ2  Φ4  1Φ2  4  11111  Φ4  1Φ2  2  Φ3  1Φ2  2  5  221  Φ4  1Φ2  2  Φ3  1Φ2  2  6  2111  Φ3  1Φ3  2  Φ2  1Φ3  2  7  311  Φ4  1Φ3  Φ3  1Φ3  8  221  Φ2  1Φ2  2Φ4  Φ1Φ2  2Φ4  9  1121  Φ3  1Φ2Φ4  Φ2  1Φ2Φ4  10  32  Φ3  1Φ2Φ3  Φ2  1Φ2Φ3  46  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  Table 7: Arithmetic types in degree 4  Type T  Geometric type  Stab  W(D5)  χw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Pic(X)  χw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Pic(Xs)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2A1 (9)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='D8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='11111  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Φ6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Φ4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2111  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Φ5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1Φ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Φ3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1Φ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='221  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Φ4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1Φ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Φ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1Φ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='221  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Φ4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1Φ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Φ3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1Φ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='41  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Φ3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1Φ2Φ4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='Φ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='1Φ4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Artin, Coverings of the rational double points in characteristic p, Complex analysis and  algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 11–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  CLASSIFICATION OF SINGULAR DEL PEZZO SURFACES OVER FINITE FIELDS  49  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Barinder Banwait, Francesc Fit´e, and Daniel Loughran, Del Pezzo surfaces over finite fields  and their Frobenius traces, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Cambridge Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content=' Martin Bright, Brauer groups of singular del Pezzo surfaces, Michigan Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content=' Michel Demazure, Surfaces de del pezzo in S´eminaire sur les Singularit´es des Surfaces, Lecture  Notes in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content=' J´anos Koll´ar, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer  Grenzgebiete, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Skorobogatov, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Tsfasman, del Pezzo surfaces of degree four,  M´em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content=' France (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content=' Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique  factorization, Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' M´erindol, Les singularit´es simples elliptiques, leurs d´eformations, les surfaces de del  Pezzo et les transformations quadratiques, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' ´Ecole Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' (4) 15 (1982), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1,  17–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Ludwig Schl¨afli, On the distribution of surfaces of the third order into species, in reference to  the absence or presence of singular points, and the reality of their lines, Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' London 153 (1863), 193–241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Alexei Skorobogatov, del Pezzo surfaces of degree 4 and their relation to Kummer surfaces,  Enseign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' (2) 56 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 1-2, 73–85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Swinnerton-Dyer, The zeta function of a cubic surface over a finite field, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  Cambridge Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 63 (1967), 55–71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Andrey Trepalin, Del Pezzo surfaces over finite fields, Finite Fields Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 68 (2020), 101741,  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' William C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Waterhouse, Pairs of quadratic forms, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 37 (1976), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' 2, 157–164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='  50  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' BLACHE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content=' HALLOUIN  LAMIA, Universit´e des Antilles  Email address: regis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='blache@univ-antilles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
+page_content='fr  Institut de Math´ematiques de Toulouse, UMR 5219  Email address: hallouin@univ-tlse2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FRT4oBgHgl3EQfhDe7/content/2301.13582v1.pdf'}
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